[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.50,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.50,0:00:03.21,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Sorry.
Dialogue: 0,0:00:03.21,0:00:05.82,Default,,0000,0000,0000,,I really don't mind if you\Nwalk in a little bit late.
Dialogue: 0,0:00:05.82,0:00:08.13,Default,,0000,0000,0000,,I know that you guys come\Nfrom other buildings,
Dialogue: 0,0:00:08.13,0:00:12.36,Default,,0000,0000,0000,,and some professors\Nkeep you overtime.
Dialogue: 0,0:00:12.36,0:00:15.64,Default,,0000,0000,0000,,So as long as you\Nquietly enter the room,
Dialogue: 0,0:00:15.64,0:00:19.78,Default,,0000,0000,0000,,I have no problem with\Nwalking in a little bit late.
Dialogue: 0,0:00:19.78,0:00:24.45,Default,,0000,0000,0000,,Would anybody want to\Nstart an attendance sheet?
Dialogue: 0,0:00:24.45,0:00:25.71,Default,,0000,0000,0000,,Who wants to be the one?
Dialogue: 0,0:00:25.71,0:00:26.39,Default,,0000,0000,0000,,Roberto, please.
Dialogue: 0,0:00:26.39,0:00:28.28,Default,,0000,0000,0000,,Thank you so much.
Dialogue: 0,0:00:28.28,0:00:28.91,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:00:28.91,0:00:33.53,Default,,0000,0000,0000,,We went through chapter\N12 on Monday fast.
Dialogue: 0,0:00:33.53,0:00:39.90,Default,,0000,0000,0000,,And I would like to start with\Na review of 12.1, 12.2, 12.3.
Dialogue: 0,0:00:39.90,0:00:42.75,Default,,0000,0000,0000,,So two thing we will do today.
Dialogue: 0,0:00:42.75,0:00:55.13,Default,,0000,0000,0000,,Part one will be review of\Nchapter 12, sections to 12.1,
Dialogue: 0,0:00:55.13,0:01:07.09,Default,,0000,0000,0000,,12.3 from the book and\Nstarting chapter 12,
Dialogue: 0,0:01:07.09,0:01:13.63,Default,,0000,0000,0000,,section 12.4 today later.
Dialogue: 0,0:01:13.63,0:01:14.77,Default,,0000,0000,0000,,What is that about?
Dialogue: 0,0:01:14.77,0:01:19.78,Default,,0000,0000,0000,,This is about the surface\Nintegrals, surface area,
Dialogue: 0,0:01:19.78,0:01:21.20,Default,,0000,0000,0000,,and [INAUDIBLE].
Dialogue: 0,0:01:21.20,0:01:25.01,Default,,0000,0000,0000,,
Dialogue: 0,0:01:25.01,0:01:25.60,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:01:25.60,0:01:28.29,Default,,0000,0000,0000,,What have you seen\Nin 12.1, 12.3?
Dialogue: 0,0:01:28.29,0:01:31.06,Default,,0000,0000,0000,,Let's review quickly\Nwhat you've learned.
Dialogue: 0,0:01:31.06,0:01:35.05,Default,,0000,0000,0000,,You've learned about\Nhow to interpret
Dialogue: 0,0:01:35.05,0:01:38.64,Default,,0000,0000,0000,,an integral with a positive\Nfunction that is smooth.
Dialogue: 0,0:01:38.64,0:01:41.20,Default,,0000,0000,0000,,Well, we said\Ncontinuous-- that would
Dialogue: 0,0:01:41.20,0:01:44.60,Default,,0000,0000,0000,,be enough-- over a\Nrectangular region.
Dialogue: 0,0:01:44.60,0:01:48.94,Default,,0000,0000,0000,,And the geometric meaning\Nof such a problem,
Dialogue: 0,0:01:48.94,0:01:54.04,Default,,0000,0000,0000,,integrate f of x, y positive\Nover a domain was what?
Dialogue: 0,0:01:54.04,0:02:00.36,Default,,0000,0000,0000,,The volume of a body under the\Ngraph and above that domain,
Dialogue: 0,0:02:00.36,0:02:03.86,Default,,0000,0000,0000,,so projected down, protecting\Ndown on the domain.
Dialogue: 0,0:02:03.86,0:02:04.85,Default,,0000,0000,0000,,Evaluate that body.
Dialogue: 0,0:02:04.85,0:02:06.74,Default,,0000,0000,0000,,How did we do it?
Dialogue: 0,0:02:06.74,0:02:11.00,Default,,0000,0000,0000,,Double integral of f\Nof x, y, dxdy or dA.
Dialogue: 0,0:02:11.00,0:02:15.28,Default,,0000,0000,0000,,But then we said, OK, if you\Nhave a rectangular region
Dialogue: 0,0:02:15.28,0:02:18.10,Default,,0000,0000,0000,,on the ground, then it's easy.
Dialogue: 0,0:02:18.10,0:02:19.79,Default,,0000,0000,0000,,You apply the Fubini theorem.
Dialogue: 0,0:02:19.79,0:02:22.82,Default,,0000,0000,0000,,And then you'll have\Nintegral from A to B,
Dialogue: 0,0:02:22.82,0:02:26.00,Default,,0000,0000,0000,,integral from C to\ND, fixed end points.
Dialogue: 0,0:02:26.00,0:02:28.92,Default,,0000,0000,0000,,When you didn't have\Na rectangular region
Dialogue: 0,0:02:28.92,0:02:33.66,Default,,0000,0000,0000,,to integrate over, you\Nwould have such a type one,
Dialogue: 0,0:02:33.66,0:02:36.58,Default,,0000,0000,0000,,type two regions, who\Nare easy to deal with,
Dialogue: 0,0:02:36.58,0:02:43.34,Default,,0000,0000,0000,,which were the case of regions\Nlike the ones between two
Dialogue: 0,0:02:43.34,0:02:48.50,Default,,0000,0000,0000,,straight lines\Nand two functions.
Dialogue: 0,0:02:48.50,0:02:53.46,Default,,0000,0000,0000,,And then you had the type\Ntwo, two straight lines
Dialogue: 0,0:02:53.46,0:02:55.59,Default,,0000,0000,0000,,and two functions,\Nwhere the functions
Dialogue: 0,0:02:55.59,0:03:04.81,Default,,0000,0000,0000,,were assumed differentiable\Nactually in our examples.
Dialogue: 0,0:03:04.81,0:03:05.84,Default,,0000,0000,0000,,Type one, type two.
Dialogue: 0,0:03:05.84,0:03:08.27,Default,,0000,0000,0000,,What did we do after that?
Dialogue: 0,0:03:08.27,0:03:13.14,Default,,0000,0000,0000,,After that, we said, well,\Nwhat if you're not so lucky
Dialogue: 0,0:03:13.14,0:03:16.77,Default,,0000,0000,0000,,and have such nice domains?
Dialogue: 0,0:03:16.77,0:03:21.95,Default,,0000,0000,0000,,Or maybe you have\Nsomething with a corner.
Dialogue: 0,0:03:21.95,0:03:24.28,Default,,0000,0000,0000,,What do you do if\Nyou have a corner?
Dialogue: 0,0:03:24.28,0:03:29.76,Default,,0000,0000,0000,,Well, you'd still be able to\Ndivide the surface into two,
Dialogue: 0,0:03:29.76,0:03:32.62,Default,,0000,0000,0000,,where you have two\Nseparate areas.
Dialogue: 0,0:03:32.62,0:03:38.12,Default,,0000,0000,0000,,And then you integrate on them\Nseparately at the same time.
Dialogue: 0,0:03:38.12,0:03:39.68,Default,,0000,0000,0000,,And you have an\Nadditive integral.
Dialogue: 0,0:03:39.68,0:03:41.47,Default,,0000,0000,0000,,The integral would be additive.
Dialogue: 0,0:03:41.47,0:03:43.34,Default,,0000,0000,0000,,Those are easy to deal with.
Dialogue: 0,0:03:43.34,0:03:47.54,Default,,0000,0000,0000,,Well, what if you had something\Nthat is more sophisticated,
Dialogue: 0,0:03:47.54,0:03:52.52,Default,,0000,0000,0000,,like a disk or an annulus?
Dialogue: 0,0:03:52.52,0:03:56.19,Default,,0000,0000,0000,,And in that case, it's\Nreally a big headache,
Dialogue: 0,0:03:56.19,0:04:02.81,Default,,0000,0000,0000,,considering how to do this\Nusing one of the previous steps.
Dialogue: 0,0:04:02.81,0:04:05.04,Default,,0000,0000,0000,,So we had to introduce\Npolar coordinates.
Dialogue: 0,0:04:05.04,0:04:08.32,Default,,0000,0000,0000,,And we have to\Nthink, what change
Dialogue: 0,0:04:08.32,0:04:13.23,Default,,0000,0000,0000,,do I have from x, y to r,\Ntheta, polar coordinates
Dialogue: 0,0:04:13.23,0:04:14.82,Default,,0000,0000,0000,,back and forth?
Dialogue: 0,0:04:14.82,0:04:18.61,Default,,0000,0000,0000,,And when we did the double\Nintegral over a domain f of x,
Dialogue: 0,0:04:18.61,0:04:23.56,Default,,0000,0000,0000,,y function positive dA in\Nthe Cartesian coordinates.
Dialogue: 0,0:04:23.56,0:04:25.24,Default,,0000,0000,0000,,When we switched to\Npolar coordinates,
Dialogue: 0,0:04:25.24,0:04:30.40,Default,,0000,0000,0000,,we had a magic thing\Nhappen, which was what?
Dialogue: 0,0:04:30.40,0:04:35.14,Default,,0000,0000,0000,,Some f of x of r,\Ntheta, y of r, theta.
Dialogue: 0,0:04:35.14,0:04:36.31,Default,,0000,0000,0000,,I say theta.
Dialogue: 0,0:04:36.31,0:04:37.39,Default,,0000,0000,0000,,I put phi.
Dialogue: 0,0:04:37.39,0:04:38.68,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:04:38.68,0:04:42.27,Default,,0000,0000,0000,,Let me put theta if\Nyou prefer theta.
Dialogue: 0,0:04:42.27,0:04:44.88,Default,,0000,0000,0000,,A change of\Ncoordinates, a Jacobian.
Dialogue: 0,0:04:44.88,0:04:45.66,Default,,0000,0000,0000,,That was what?
Dialogue: 0,0:04:45.66,0:04:47.90,Default,,0000,0000,0000,,Do you guys remember that?
Dialogue: 0,0:04:47.90,0:04:48.57,Default,,0000,0000,0000,,r.
Dialogue: 0,0:04:48.57,0:04:49.07,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:04:49.07,0:04:50.21,Default,,0000,0000,0000,,I'm proud of you, r.
Dialogue: 0,0:04:50.21,0:04:52.09,Default,,0000,0000,0000,,And then drd theta.
Dialogue: 0,0:04:52.09,0:04:55.64,Default,,0000,0000,0000,,So you're ready do that\Nkind of homework, integrals,
Dialogue: 0,0:04:55.64,0:04:58.81,Default,,0000,0000,0000,,double integrals in\Npolar coordinates.
Dialogue: 0,0:04:58.81,0:05:03.25,Default,,0000,0000,0000,,dr will be between\Ncertain values,
Dialogue: 0,0:05:03.25,0:05:05.31,Default,,0000,0000,0000,,hopefully fixed\Nvalues because that
Dialogue: 0,0:05:05.31,0:05:08.75,Default,,0000,0000,0000,,will make the Fubini-Tonelli\Na piece of cake.
Dialogue: 0,0:05:08.75,0:05:10.55,Default,,0000,0000,0000,,Theta, also fixed values.
Dialogue: 0,0:05:10.55,0:05:13.61,Default,,0000,0000,0000,,But not always will you have\Nfixed values, especially
Dialogue: 0,0:05:13.61,0:05:14.55,Default,,0000,0000,0000,,in the first part.
Dialogue: 0,0:05:14.55,0:05:18.74,Default,,0000,0000,0000,,You may have some function\Nof r, function of r.
Dialogue: 0,0:05:18.74,0:05:20.88,Default,,0000,0000,0000,,And here, theta 1 and theta 2.
Dialogue: 0,0:05:20.88,0:05:23.41,Default,,0000,0000,0000,,So I want to see a\Nfew more examples
Dialogue: 0,0:05:23.41,0:05:30.90,Default,,0000,0000,0000,,before I move on to section 12.4\Nbecause, as the Romans said,
Dialogue: 0,0:05:30.90,0:05:34.31,Default,,0000,0000,0000,,review is the\Nmother of studying,
Dialogue: 0,0:05:34.31,0:05:40.55,Default,,0000,0000,0000,,which is [LATIN], which means\Ngo ahead and do a lot of review
Dialogue: 0,0:05:40.55,0:05:42.62,Default,,0000,0000,0000,,if you really want to\Nmaster the concepts.
Dialogue: 0,0:05:42.62,0:05:45.20,Default,,0000,0000,0000,,
Dialogue: 0,0:05:45.20,0:05:45.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:05:45.91,0:05:50.25,Default,,0000,0000,0000,,I'm going to take the plunge\Nand go ahead and help you
Dialogue: 0,0:05:50.25,0:05:51.32,Default,,0000,0000,0000,,with your homework.
Dialogue: 0,0:05:51.32,0:05:53.94,Default,,0000,0000,0000,,I've been pondering\Nabout this a lot.
Dialogue: 0,0:05:53.94,0:05:58.75,Default,,0000,0000,0000,,We've done problems that I made\Nup, like the ones in the book.
Dialogue: 0,0:05:58.75,0:06:01.76,Default,,0000,0000,0000,,And I also took problems\Nstraight out of the book.
Dialogue: 0,0:06:01.76,0:06:06.68,Default,,0000,0000,0000,,But I would like to go over\Nsome homework type problems
Dialogue: 0,0:06:06.68,0:06:11.29,Default,,0000,0000,0000,,in order to assist you in more\Neasily doing your homework.
Dialogue: 0,0:06:11.29,0:06:14.18,Default,,0000,0000,0000,,
Dialogue: 0,0:06:14.18,0:06:18.05,Default,,0000,0000,0000,,In chapter 12, homework\Nfour-- am I right,
Dialogue: 0,0:06:18.05,0:06:19.58,Default,,0000,0000,0000,,homework number four?
Dialogue: 0,0:06:19.58,0:06:24.75,Default,,0000,0000,0000,,You have a big array of\Nproblems, all sorts of problems
Dialogue: 0,0:06:24.75,0:06:29.42,Default,,0000,0000,0000,,because mathematicians\Nhave all sorts of problems.
Dialogue: 0,0:06:29.42,0:06:31.66,Default,,0000,0000,0000,,For example, an easy\None that you're not
Dialogue: 0,0:06:31.66,0:06:34.46,Default,,0000,0000,0000,,going to have a problem with--\Nand I'm using my own end
Dialogue: 0,0:06:34.46,0:06:35.79,Default,,0000,0000,0000,,points.
Dialogue: 0,0:06:35.79,0:06:39.01,Default,,0000,0000,0000,,Your end points may be\Ndifferent in the homework.
Dialogue: 0,0:06:39.01,0:06:44.70,Default,,0000,0000,0000,,It would be homework four,\Nchapter 12, number four.
Dialogue: 0,0:06:44.70,0:06:47.63,Default,,0000,0000,0000,,And you say-- most of\Nyou should say, oh,
Dialogue: 0,0:06:47.63,0:06:48.59,Default,,0000,0000,0000,,that's a piece of cake.
Dialogue: 0,0:06:48.59,0:06:54.43,Default,,0000,0000,0000,,I don't know why she even talks\Nabout such a trivial problem,
Dialogue: 0,0:06:54.43,0:06:55.25,Default,,0000,0000,0000,,right?
Dialogue: 0,0:06:55.25,0:06:57.50,Default,,0000,0000,0000,,Many of you have said that.
Dialogue: 0,0:06:57.50,0:07:04.60,Default,,0000,0000,0000,,Well, I am willing\Nto review everything
Dialogue: 0,0:07:04.60,0:07:09.04,Default,,0000,0000,0000,,so that you have a better\Ngrasp of the material.
Dialogue: 0,0:07:09.04,0:07:14.21,Default,,0000,0000,0000,,On this one, since it's so\Neasy, I want you to help me.
Dialogue: 0,0:07:14.21,0:07:17.46,Default,,0000,0000,0000,,What kind of problem is that?
Dialogue: 0,0:07:17.46,0:07:20.51,Default,,0000,0000,0000,,As I said, mathematicians have\Nall sorts of problems, right?
Dialogue: 0,0:07:20.51,0:07:26.67,Default,,0000,0000,0000,,So a problem where you\Nhave a product inside
Dialogue: 0,0:07:26.67,0:07:30.26,Default,,0000,0000,0000,,as an integrand, where the\Nvariables are completely
Dialogue: 0,0:07:30.26,0:07:33.21,Default,,0000,0000,0000,,separated-- what does it mean?
Dialogue: 0,0:07:33.21,0:07:36.51,Default,,0000,0000,0000,,The function\Nunderneath is a product
Dialogue: 0,0:07:36.51,0:07:39.86,Default,,0000,0000,0000,,of two functions, one\Nfunction of x only,
Dialogue: 0,0:07:39.86,0:07:42.45,Default,,0000,0000,0000,,the other function\Nof y only, which
Dialogue: 0,0:07:42.45,0:07:45.44,Default,,0000,0000,0000,,is a blessing in disguise.
Dialogue: 0,0:07:45.44,0:07:46.86,Default,,0000,0000,0000,,Why is that a blessing?
Dialogue: 0,0:07:46.86,0:07:51.20,Default,,0000,0000,0000,,I told you last time that you\Ncan go ahead and write this
Dialogue: 0,0:07:51.20,0:07:54.98,Default,,0000,0000,0000,,as product of integrals.
Dialogue: 0,0:07:54.98,0:07:58.66,Default,,0000,0000,0000,,Is there anybody seeing already\Nwhat those integrals will be?
Dialogue: 0,0:07:58.66,0:08:02.75,Default,,0000,0000,0000,,Let's see how much you\Nmastered the material.
Dialogue: 0,0:08:02.75,0:08:04.58,Default,,0000,0000,0000,,STUDENT: x over 2y times--
Dialogue: 0,0:08:04.58,0:08:07.16,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NFrom 1 to 2, you said?
Dialogue: 0,0:08:07.16,0:08:08.28,Default,,0000,0000,0000,,STUDENT: Yeah, from 1 to 2.
Dialogue: 0,0:08:08.28,0:08:11.20,Default,,0000,0000,0000,,I'm sorry.
Dialogue: 0,0:08:11.20,0:08:12.60,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Of what?
Dialogue: 0,0:08:12.60,0:08:20.57,Default,,0000,0000,0000,,X, dx times the integral\Nfrom 0 to pi of what?
Dialogue: 0,0:08:20.57,0:08:21.32,Default,,0000,0000,0000,,STUDENT: Cosine y.
Dialogue: 0,0:08:21.32,0:08:23.44,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Cosine y.
Dialogue: 0,0:08:23.44,0:08:25.30,Default,,0000,0000,0000,,Do we need to\Nre-prove this result?
Dialogue: 0,0:08:25.30,0:08:26.42,Default,,0000,0000,0000,,No, we proved it last time.
Dialogue: 0,0:08:26.42,0:08:29.06,Default,,0000,0000,0000,,But practically, if\Nyou forget, the idea
Dialogue: 0,0:08:29.06,0:08:30.72,Default,,0000,0000,0000,,is a very simple thing.
Dialogue: 0,0:08:30.72,0:08:32.94,Default,,0000,0000,0000,,When you integrate\Nwith respect to y,
Dialogue: 0,0:08:32.94,0:08:35.44,Default,,0000,0000,0000,,Mr. X said, I'm\Nnot married to y.
Dialogue: 0,0:08:35.44,0:08:36.30,Default,,0000,0000,0000,,I'm out of here.
Dialogue: 0,0:08:36.30,0:08:37.26,Default,,0000,0000,0000,,I'm out of the picture.
Dialogue: 0,0:08:37.26,0:08:38.70,Default,,0000,0000,0000,,I'm going for a walk.
Dialogue: 0,0:08:38.70,0:08:44.76,Default,,0000,0000,0000,,So the integral of cosine is\Nin itself to be treated first,
Dialogue: 0,0:08:44.76,0:08:45.69,Default,,0000,0000,0000,,independently.
Dialogue: 0,0:08:45.69,0:08:47.51,Default,,0000,0000,0000,,And it's inside,\Nand it's a constant.
Dialogue: 0,0:08:47.51,0:08:50.22,Default,,0000,0000,0000,,And it pulls out in the end.
Dialogue: 0,0:08:50.22,0:08:51.96,Default,,0000,0000,0000,,And since it pulls\Nout, what you're
Dialogue: 0,0:08:51.96,0:08:56.54,Default,,0000,0000,0000,,going to be left with afterwards\Nwill be that integral of 1
Dialogue: 0,0:08:56.54,0:08:59.36,Default,,0000,0000,0000,,to 2x dx.
Dialogue: 0,0:08:59.36,0:09:01.78,Default,,0000,0000,0000,,So we've done that\Nlast time as well.
Dialogue: 0,0:09:01.78,0:09:02.55,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:09:02.55,0:09:03.92,Default,,0000,0000,0000,,STUDENT: So you\Nwould-- would you
Dialogue: 0,0:09:03.92,0:09:06.19,Default,,0000,0000,0000,,not be able to do that\Nif it was cosine x, y?
Dialogue: 0,0:09:06.19,0:09:07.48,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Absolutely.
Dialogue: 0,0:09:07.48,0:09:09.71,Default,,0000,0000,0000,,If you had cosine\Nx, y, it's bye, bye.
Dialogue: 0,0:09:09.71,0:09:12.04,Default,,0000,0000,0000,,STUDENT: So it's only when\Nthey're completely separate--
Dialogue: 0,0:09:12.04,0:09:13.91,Default,,0000,0000,0000,,DR. MAGDALENA TODA: When\Nyou are lucky enough
Dialogue: 0,0:09:13.91,0:09:16.86,Default,,0000,0000,0000,,to have a functional of only\Nthat's a function of y only.
Dialogue: 0,0:09:16.86,0:09:21.06,Default,,0000,0000,0000,,And if you had another\Nexample, sine of x plus y,
Dialogue: 0,0:09:21.06,0:09:25.84,Default,,0000,0000,0000,,anything that mixes them up--\Nthat would be a bad thing.
Dialogue: 0,0:09:25.84,0:09:27.36,Default,,0000,0000,0000,,Do I have to compute this?
Dialogue: 0,0:09:27.36,0:09:29.89,Default,,0000,0000,0000,,Not if I'm smart.
Dialogue: 0,0:09:29.89,0:09:32.95,Default,,0000,0000,0000,,At the blink of an eye,\NI can sense that maybe I
Dialogue: 0,0:09:32.95,0:09:34.35,Default,,0000,0000,0000,,should do this one first.
Dialogue: 0,0:09:34.35,0:09:35.15,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:09:35.15,0:09:36.77,Default,,0000,0000,0000,,Integral of cosine is sine.
Dialogue: 0,0:09:36.77,0:09:40.35,Default,,0000,0000,0000,,And sine is 0 at both 0 and pi.
Dialogue: 0,0:09:40.35,0:09:41.78,Default,,0000,0000,0000,,So it's a piece of pie.
Dialogue: 0,0:09:41.78,0:09:46.85,Default,,0000,0000,0000,,So if I have 0, and\Nthe answer is 0.
Dialogue: 0,0:09:46.85,0:09:48.50,Default,,0000,0000,0000,,So you say, OK,\Ngive us something
Dialogue: 0,0:09:48.50,0:09:50.63,Default,,0000,0000,0000,,like that on the midterm\Nbecause this problem is
Dialogue: 0,0:09:50.63,0:09:52.59,Default,,0000,0000,0000,,a piece of cake.
Dialogue: 0,0:09:52.59,0:09:53.35,Default,,0000,0000,0000,,Uh, yeah.
Dialogue: 0,0:09:53.35,0:09:54.51,Default,,0000,0000,0000,,I can do that.
Dialogue: 0,0:09:54.51,0:09:58.84,Default,,0000,0000,0000,,Probably you will have something\Nlike that on the midterm,
Dialogue: 0,0:09:58.84,0:10:03.21,Default,,0000,0000,0000,,on the April 2 midterm.
Dialogue: 0,0:10:03.21,0:10:08.64,Default,,0000,0000,0000,,So since Alex just\Nentered, I'm not
Dialogue: 0,0:10:08.64,0:10:12.94,Default,,0000,0000,0000,,going to erase this for a while\Nuntil you are able to copy it.
Dialogue: 0,0:10:12.94,0:10:17.75,Default,,0000,0000,0000,,I announced starting the\Nsurface area integral today.
Dialogue: 0,0:10:17.75,0:10:21.20,Default,,0000,0000,0000,,Section 12.4, we'll\Ndo that later on.
Dialogue: 0,0:10:21.20,0:10:24.88,Default,,0000,0000,0000,,And I will move on to\Nanother example right now.
Dialogue: 0,0:10:24.88,0:10:26.45,Default,,0000,0000,0000,,Oh, now they learn.
Dialogue: 0,0:10:26.45,0:10:28.56,Default,,0000,0000,0000,,Look, they learned about me.
Dialogue: 0,0:10:28.56,0:10:32.06,Default,,0000,0000,0000,,They learned about me,\Nthat I have lots of needs.
Dialogue: 0,0:10:32.06,0:10:35.17,Default,,0000,0000,0000,,And I don't complain.
Dialogue: 0,0:10:35.17,0:10:39.29,Default,,0000,0000,0000,,But they noticed that these\Nwere disappearing really fast.
Dialogue: 0,0:10:39.29,0:10:44.51,Default,,0000,0000,0000,,Everybody else told me that\NI write a lot on the board,
Dialogue: 0,0:10:44.51,0:10:45.72,Default,,0000,0000,0000,,compared to other professors.
Dialogue: 0,0:10:45.72,0:10:47.62,Default,,0000,0000,0000,,So I don't know if that is true.
Dialogue: 0,0:10:47.62,0:10:54.17,Default,,0000,0000,0000,,But I really need\Nthis big bottle.
Dialogue: 0,0:10:54.17,0:10:54.67,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:10:54.67,0:10:57.26,Default,,0000,0000,0000,,
Dialogue: 0,0:10:57.26,0:10:59.48,Default,,0000,0000,0000,,So you can actually\Nsolve this by yourself.
Dialogue: 0,0:10:59.48,0:11:00.68,Default,,0000,0000,0000,,You just don't realize it.
Dialogue: 0,0:11:00.68,0:11:03.87,Default,,0000,0000,0000,,I'm not going to take\Nany credit for that.
Dialogue: 0,0:11:03.87,0:11:06.86,Default,,0000,0000,0000,,And I'm going to go ahead\Nand give you something
Dialogue: 0,0:11:06.86,0:11:11.20,Default,,0000,0000,0000,,more challenging, see if\Nyou are ready for the review
Dialogue: 0,0:11:11.20,0:11:13.66,Default,,0000,0000,0000,,and for the midterm.
Dialogue: 0,0:11:13.66,0:11:14.16,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:11:14.16,0:11:17.02,Default,,0000,0000,0000,,That's number nine\Non your homework
Dialogue: 0,0:11:17.02,0:11:20.41,Default,,0000,0000,0000,,that may have again\Nthe data changed.
Dialogue: 0,0:11:20.41,0:11:23.50,Default,,0000,0000,0000,,But it's the same\Ntype of problem.
Dialogue: 0,0:11:23.50,0:11:27.20,Default,,0000,0000,0000,,Now you cannot ask me about\Nnumber nine anymore directly
Dialogue: 0,0:11:27.20,0:11:30.64,Default,,0000,0000,0000,,from WeBWork, because I'll\Nsay, I did that in class.
Dialogue: 0,0:11:30.64,0:11:35.76,Default,,0000,0000,0000,,And if you have\Ndifficulty with it,
Dialogue: 0,0:11:35.76,0:11:37.100,Default,,0000,0000,0000,,that means you did\Nnot cover the notes.
Dialogue: 0,0:11:37.100,0:11:40.98,Default,,0000,0000,0000,,
Dialogue: 0,0:11:40.98,0:11:42.04,Default,,0000,0000,0000,,This is pretty.
Dialogue: 0,0:11:42.04,0:11:44.09,Default,,0000,0000,0000,,You've seen that one before.
Dialogue: 0,0:11:44.09,0:11:46.42,Default,,0000,0000,0000,,And I would suspect\Nthat you're not
Dialogue: 0,0:11:46.42,0:11:50.85,Default,,0000,0000,0000,,going to even let me\Ntalk, because look at it.
Dialogue: 0,0:11:50.85,0:11:52.59,Default,,0000,0000,0000,,Evaluate the following integral.
Dialogue: 0,0:11:52.59,0:12:00.63,Default,,0000,0000,0000,,
Dialogue: 0,0:12:00.63,0:12:02.57,Default,,0000,0000,0000,,And it doesn't\Nmatter what numbers
Dialogue: 0,0:12:02.57,0:12:07.65,Default,,0000,0000,0000,,we are going to put on that\Nand what funny polynomial I'm
Dialogue: 0,0:12:07.65,0:12:08.40,Default,,0000,0000,0000,,going to put here.
Dialogue: 0,0:12:08.40,0:12:13.85,Default,,0000,0000,0000,,
Dialogue: 0,0:12:13.85,0:12:16.26,Default,,0000,0000,0000,,You are going to have\Nall sorts of numbers.
Dialogue: 0,0:12:16.26,0:12:18.63,Default,,0000,0000,0000,,Maybe these are not\Nthe most inspired ones,
Dialogue: 0,0:12:18.63,0:12:20.70,Default,,0000,0000,0000,,but this is WeBWork.
Dialogue: 0,0:12:20.70,0:12:24.44,Default,,0000,0000,0000,,It creates problems at\Nrandom, and every student
Dialogue: 0,0:12:24.44,0:12:27.07,Default,,0000,0000,0000,,may have a different\Nproblem, that is,
Dialogue: 0,0:12:27.07,0:12:29.98,Default,,0000,0000,0000,,in order to minimize cheating.
Dialogue: 0,0:12:29.98,0:12:31.05,Default,,0000,0000,0000,,And that's OK.
Dialogue: 0,0:12:31.05,0:12:33.90,Default,,0000,0000,0000,,The type of the problem\Nis what matters.
Dialogue: 0,0:12:33.90,0:12:36.70,Default,,0000,0000,0000,,So if we were in\NCalc 1 right now,
Dialogue: 0,0:12:36.70,0:12:41.27,Default,,0000,0000,0000,,and somebody would say, go\Nahead and take an integral of e
Dialogue: 0,0:12:41.27,0:12:45.21,Default,,0000,0000,0000,,to the x squared dx and compute\Nit by hand, see what you get,
Dialogue: 0,0:12:45.21,0:12:46.30,Default,,0000,0000,0000,,you already know.
Dialogue: 0,0:12:46.30,0:12:48.59,Default,,0000,0000,0000,,They don't know, poor people.
Dialogue: 0,0:12:48.59,0:12:49.36,Default,,0000,0000,0000,,They don't know.
Dialogue: 0,0:12:49.36,0:12:54.12,Default,,0000,0000,0000,,But you know because I told\Nyou that this is a headache.
Dialogue: 0,0:12:54.12,0:12:56.92,Default,,0000,0000,0000,,
Dialogue: 0,0:12:56.92,0:13:00.72,Default,,0000,0000,0000,,You need another way out.
Dialogue: 0,0:13:00.72,0:13:04.08,Default,,0000,0000,0000,,You cannot do that in Calc 2.
Dialogue: 0,0:13:04.08,0:13:08.17,Default,,0000,0000,0000,,And you cannot do that in\Nan elementary way by hand.
Dialogue: 0,0:13:08.17,0:13:11.99,Default,,0000,0000,0000,,This is something that MATLAB\Nwould solve numerically for you
Dialogue: 0,0:13:11.99,0:13:15.66,Default,,0000,0000,0000,,in no time if you gave\Ncertain values and so on.
Dialogue: 0,0:13:15.66,0:13:21.91,Default,,0000,0000,0000,,But to find an explicit\Nform of that anti-derivative
Dialogue: 0,0:13:21.91,0:13:23.51,Default,,0000,0000,0000,,would be a hassle.
Dialogue: 0,0:13:23.51,0:13:26.51,Default,,0000,0000,0000,,The same thing would happen\Nif I had the minus here.
Dialogue: 0,0:13:26.51,0:13:32.46,Default,,0000,0000,0000,,In that case, I wouldn't be able\Nto express the anti-derivative
Dialogue: 0,0:13:32.46,0:13:35.64,Default,,0000,0000,0000,,as an elementary\Nfunction at all.
Dialogue: 0,0:13:35.64,0:13:36.77,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:13:36.77,0:13:39.82,Default,,0000,0000,0000,,So this is giving\Nme a big headache.
Dialogue: 0,0:13:39.82,0:13:42.09,Default,,0000,0000,0000,,I'm going to make a face.
Dialogue: 0,0:13:42.09,0:13:43.58,Default,,0000,0000,0000,,And I'll say, oh, my god.
Dialogue: 0,0:13:43.58,0:13:45.66,Default,,0000,0000,0000,,I get a headache.
Dialogue: 0,0:13:45.66,0:13:50.66,Default,,0000,0000,0000,,Unless you help me get out of\Ntrouble, I cannot solve that.
Dialogue: 0,0:13:50.66,0:13:52.78,Default,,0000,0000,0000,,MATLAB can do that for me.
Dialogue: 0,0:13:52.78,0:13:56.26,Default,,0000,0000,0000,,On Maple, I can go in and\Nplug in the endpoints and hope
Dialogue: 0,0:13:56.26,0:14:00.85,Default,,0000,0000,0000,,and pray that I'm\Ngoing to get the best
Dialogue: 0,0:14:00.85,0:14:03.06,Default,,0000,0000,0000,,numerical approximation\Nfor the answer.
Dialogue: 0,0:14:03.06,0:14:07.85,Default,,0000,0000,0000,,But what if I want a precise\Nanswer, not a numerical answer?
Dialogue: 0,0:14:07.85,0:14:10.35,Default,,0000,0000,0000,,Then I better put my\Nmind, my own mind,
Dialogue: 0,0:14:10.35,0:14:16.46,Default,,0000,0000,0000,,my own processor to work and\Nnot rely on MATLAB or Maple.
Dialogue: 0,0:14:16.46,0:14:17.45,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:14:17.45,0:14:18.41,Default,,0000,0000,0000,,Hmm.
Dialogue: 0,0:14:18.41,0:14:21.48,Default,,0000,0000,0000,,Understandable, precise answer.
Dialogue: 0,0:14:21.48,0:14:24.47,Default,,0000,0000,0000,,And I leave it unsimplified\Nhopefully, yes.
Dialogue: 0,0:14:24.47,0:14:27.34,Default,,0000,0000,0000,,We need to think of what\Ntechnique in this case?
Dialogue: 0,0:14:27.34,0:14:28.51,Default,,0000,0000,0000,,STUDENT: Changing the order.
Dialogue: 0,0:14:28.51,0:14:30.68,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Change\Nthe order of integration.
Dialogue: 0,0:14:30.68,0:14:31.47,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:14:31.47,0:14:32.83,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:14:32.83,0:14:35.52,Default,,0000,0000,0000,,And in that case, the\Nintegrand stays the same.
Dialogue: 0,0:14:35.52,0:14:39.14,Default,,0000,0000,0000,,
Dialogue: 0,0:14:39.14,0:14:43.72,Default,,0000,0000,0000,,These two guys are\Nswapped, and the end points
Dialogue: 0,0:14:43.72,0:14:47.18,Default,,0000,0000,0000,,are changing\Ncompletely because I
Dialogue: 0,0:14:47.18,0:14:54.34,Default,,0000,0000,0000,,will have to switch from one\Ndomain to the other domain.
Dialogue: 0,0:14:54.34,0:14:58.16,Default,,0000,0000,0000,,The domain that's given here by\Nthis problem is the following.
Dialogue: 0,0:14:58.16,0:15:06.89,Default,,0000,0000,0000,,x is between 7y 7, and\Ny is between 0 and 1.
Dialogue: 0,0:15:06.89,0:15:11.42,Default,,0000,0000,0000,,So do they give you horizontal\Nstrip or vertical strip domain?
Dialogue: 0,0:15:11.42,0:15:15.36,Default,,0000,0000,0000,,
Dialogue: 0,0:15:15.36,0:15:16.23,Default,,0000,0000,0000,,Horizontal.
Dialogue: 0,0:15:16.23,0:15:17.16,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:15:17.16,0:15:19.46,Default,,0000,0000,0000,,I wasn't sure if\NI heard it right.
Dialogue: 0,0:15:19.46,0:15:22.62,Default,,0000,0000,0000,,But anyway, what is this\Nfunction and that function?
Dialogue: 0,0:15:22.62,0:15:25.55,Default,,0000,0000,0000,,So x equals 7 would be what?
Dialogue: 0,0:15:25.55,0:15:27.39,Default,,0000,0000,0000,,x equals 7 will be far away.
Dialogue: 0,0:15:27.39,0:15:34.31,Default,,0000,0000,0000,,I have to do one, two, three,\Nfour-- well, five, six, seven.
Dialogue: 0,0:15:34.31,0:15:39.90,Default,,0000,0000,0000,,Then is a vertical\Nline. x equals 7.
Dialogue: 0,0:15:39.90,0:15:41.62,Default,,0000,0000,0000,,That's the x-axis.
Dialogue: 0,0:15:41.62,0:15:42.48,Default,,0000,0000,0000,,That's the y-axis.
Dialogue: 0,0:15:42.48,0:15:44.87,Default,,0000,0000,0000,,I'm trying to draw the domain.
Dialogue: 0,0:15:44.87,0:15:48.18,Default,,0000,0000,0000,,And what is x equals 7y?
Dialogue: 0,0:15:48.18,0:15:53.92,Default,,0000,0000,0000,,X equals 7y is the\Nsame as y equals 1/7x.
Dialogue: 0,0:15:53.92,0:15:54.99,Default,,0000,0000,0000,,Uh-huh.
Dialogue: 0,0:15:54.99,0:15:58.02,Default,,0000,0000,0000,,That should be a\Nfriendlier function
Dialogue: 0,0:15:58.02,0:16:02.65,Default,,0000,0000,0000,,to draw because I'm smart\Nenough to even imagine
Dialogue: 0,0:16:02.65,0:16:05.36,Default,,0000,0000,0000,,what it looks like.
Dialogue: 0,0:16:05.36,0:16:10.51,Default,,0000,0000,0000,,y equals mx is a line that\Npasses through the origin.
Dialogue: 0,0:16:10.51,0:16:12.75,Default,,0000,0000,0000,,It's part of a pencil of planes.
Dialogue: 0,0:16:12.75,0:16:18.06,Default,,0000,0000,0000,,A pencil of planes is infinitely\Nmany-- pencil of lines,
Dialogue: 0,0:16:18.06,0:16:18.56,Default,,0000,0000,0000,,I'm sorry.
Dialogue: 0,0:16:18.56,0:16:21.31,Default,,0000,0000,0000,,Infinitely many lines that all\Npass through the same point.
Dialogue: 0,0:16:21.31,0:16:23.42,Default,,0000,0000,0000,,So they all pass\Nthrough the origin.
Dialogue: 0,0:16:23.42,0:16:29.40,Default,,0000,0000,0000,,For 7, x equals 7 is going\Nto give me y, 1, y equals 1.
Dialogue: 0,0:16:29.40,0:16:33.80,Default,,0000,0000,0000,,So I'm going to erase this\Ndotted line and draw the line.
Dialogue: 0,0:16:33.80,0:16:37.77,Default,,0000,0000,0000,,This is y equals x/7,\Nand we look at it,
Dialogue: 0,0:16:37.77,0:16:41.25,Default,,0000,0000,0000,,and we think how nice\Nit is and how ugly it
Dialogue: 0,0:16:41.25,0:16:42.79,Default,,0000,0000,0000,,is because it's [? fat ?].
Dialogue: 0,0:16:42.79,0:16:43.95,Default,,0000,0000,0000,,It's not a straight line.
Dialogue: 0,0:16:43.95,0:16:47.07,Default,,0000,0000,0000,,
Dialogue: 0,0:16:47.07,0:16:49.02,Default,,0000,0000,0000,,Now it looks straighter.
Dialogue: 0,0:16:49.02,0:16:54.82,Default,,0000,0000,0000,,So simply, I get to 1, y equals\N1 here, which is good for me
Dialogue: 0,0:16:54.82,0:16:58.17,Default,,0000,0000,0000,,because that's\Nexactly what I wanted.
Dialogue: 0,0:16:58.17,0:17:04.59,Default,,0000,0000,0000,,I wanted to draw the horizontal\Nstrips for y between 0 and 1.
Dialogue: 0,0:17:04.59,0:17:07.14,Default,,0000,0000,0000,,I know I'm going\Nvery slow, but that's
Dialogue: 0,0:17:07.14,0:17:09.09,Default,,0000,0000,0000,,kind of the idea\Nbecause-- do you
Dialogue: 0,0:17:09.09,0:17:11.30,Default,,0000,0000,0000,,mind that I'm going so slow?
Dialogue: 0,0:17:11.30,0:17:11.80,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:17:11.80,0:17:17.05,Default,,0000,0000,0000,,This is review for the\Nmidterm slowly, a little bit.
Dialogue: 0,0:17:17.05,0:17:18.47,Default,,0000,0000,0000,,So y between 0 and 1.
Dialogue: 0,0:17:18.47,0:17:20.99,Default,,0000,0000,0000,,I'm drawing the\Nhorizontal strips,
Dialogue: 0,0:17:20.99,0:17:23.79,Default,,0000,0000,0000,,and this is exactly\Nwhat you guys have.
Dialogue: 0,0:17:23.79,0:17:26.85,Default,,0000,0000,0000,,This is the red domain.
Dialogue: 0,0:17:26.85,0:17:28.11,Default,,0000,0000,0000,,Let's call it d.
Dialogue: 0,0:17:28.11,0:17:30.54,Default,,0000,0000,0000,,It's the same domain but\Nwith horizontal strips.
Dialogue: 0,0:17:30.54,0:17:33.58,Default,,0000,0000,0000,,And I'm going to\Ndraw the same domain.
Dialogue: 0,0:17:33.58,0:17:34.71,Default,,0000,0000,0000,,What color do you like?
Dialogue: 0,0:17:34.71,0:17:37.20,Default,,0000,0000,0000,,I like green because it's\Nin contrast with red.
Dialogue: 0,0:17:37.20,0:17:45.76,Default,,0000,0000,0000,,I'm going to use green to\Ndraw the vertical strip domain
Dialogue: 0,0:17:45.76,0:17:46.91,Default,,0000,0000,0000,,and say, all right.
Dialogue: 0,0:17:46.91,0:17:49.46,Default,,0000,0000,0000,,Now I know what I'm\Nsupposed to say,
Dialogue: 0,0:17:49.46,0:17:57.44,Default,,0000,0000,0000,,that d with vertical strips is\Ngoing to be x between-- what?
Dialogue: 0,0:17:57.44,0:17:58.72,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:17:58.72,0:18:03.51,Default,,0000,0000,0000,,First the fixed\Nnumbers, 0 and 7.
Dialogue: 0,0:18:03.51,0:18:04.76,Default,,0000,0000,0000,,And y between--
Dialogue: 0,0:18:04.76,0:18:09.56,Default,,0000,0000,0000,,
Dialogue: 0,0:18:09.56,0:18:12.58,Default,,0000,0000,0000,,STUDENT: 0 and x plus 7.
Dialogue: 0,0:18:12.58,0:18:13.21,Default,,0000,0000,0000,,STUDENT: And 1.
Dialogue: 0,0:18:13.21,0:18:16.48,Default,,0000,0000,0000,,
Dialogue: 0,0:18:16.48,0:18:17.69,Default,,0000,0000,0000,,DR. MAGDALENA TODA: This one.
Dialogue: 0,0:18:17.69,0:18:21.32,Default,,0000,0000,0000,,x/7, 1/7x.
Dialogue: 0,0:18:21.32,0:18:22.64,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:18:22.64,0:18:24.35,Default,,0000,0000,0000,,Is it x/7?
Dialogue: 0,0:18:24.35,0:18:30.86,Default,,0000,0000,0000,,x/7y equals x is the same\Nthing as y equals x/7.
Dialogue: 0,0:18:30.86,0:18:37.36,Default,,0000,0000,0000,,So y equals x/7 is\Nthis problem, which was
Dialogue: 0,0:18:37.36,0:18:43.04,Default,,0000,0000,0000,,the same as x equals 7y before.
Dialogue: 0,0:18:43.04,0:18:43.58,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:18:43.58,0:18:45.84,Default,,0000,0000,0000,,So how do I set up\Nthe new integral?
Dialogue: 0,0:18:45.84,0:18:54.23,Default,,0000,0000,0000,,I'm going to say dydx, and then\Ny will be between 0 and x/7.
Dialogue: 0,0:18:54.23,0:18:56.69,Default,,0000,0000,0000,,And x will be between 0 and 7.
Dialogue: 0,0:18:56.69,0:19:02.70,Default,,0000,0000,0000,,
Dialogue: 0,0:19:02.70,0:19:03.62,Default,,0000,0000,0000,,Is it solved?
Dialogue: 0,0:19:03.62,0:19:04.12,Default,,0000,0000,0000,,No.
Dialogue: 0,0:19:04.12,0:19:08.67,Default,,0000,0000,0000,,But I promise from my heart that\Nif you do that on the midterm,
Dialogue: 0,0:19:08.67,0:19:12.76,Default,,0000,0000,0000,,you'll get 75% on this\Nproblem, even if doesn't say
Dialogue: 0,0:19:12.76,0:19:13.74,Default,,0000,0000,0000,,don't compute it.
Dialogue: 0,0:19:13.74,0:19:15.94,Default,,0000,0000,0000,,If it says, don't compute\Nit or anything like that,
Dialogue: 0,0:19:15.94,0:19:17.85,Default,,0000,0000,0000,,you got 100%.
Dialogue: 0,0:19:17.85,0:19:18.86,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:19:18.86,0:19:20.95,Default,,0000,0000,0000,,So this is the most\Nimportant step.
Dialogue: 0,0:19:20.95,0:19:23.52,Default,,0000,0000,0000,,From this on, I know\Nyou can do it with what
Dialogue: 0,0:19:23.52,0:19:25.59,Default,,0000,0000,0000,,you've learned in Calc 1 and 2.
Dialogue: 0,0:19:25.59,0:19:28.62,Default,,0000,0000,0000,,It's a piece of cake, and you\Nshould do it with no problem.
Dialogue: 0,0:19:28.62,0:19:34.95,Default,,0000,0000,0000,,Now how are we going\Nto handle this fellow?
Dialogue: 0,0:19:34.95,0:19:40.72,Default,,0000,0000,0000,,This fellow says, I have nothing\Nto do with you, Mr. Y. I'm out,
Dialogue: 0,0:19:40.72,0:19:42.82,Default,,0000,0000,0000,,and you're alone.
Dialogue: 0,0:19:42.82,0:19:44.43,Default,,0000,0000,0000,,I don't need you as my friend.
Dialogue: 0,0:19:44.43,0:19:44.97,Default,,0000,0000,0000,,I'm out.
Dialogue: 0,0:19:44.97,0:19:46.22,Default,,0000,0000,0000,,I'm independent.
Dialogue: 0,0:19:46.22,0:19:48.51,Default,,0000,0000,0000,,So Mr. Y starts sulking.
Dialogue: 0,0:19:48.51,0:19:52.62,Default,,0000,0000,0000,,And say I have an integral\Nof 1dy between 0 and x/7.
Dialogue: 0,0:19:52.62,0:19:54.88,Default,,0000,0000,0000,,I'm x/7.
Dialogue: 0,0:19:54.88,0:19:58.63,Default,,0000,0000,0000,,So you are reduced to\Na very simple integral.
Dialogue: 0,0:19:58.63,0:20:02.14,Default,,0000,0000,0000,,That is the integral that you\Nlearned in-- was it Calc 1?
Dialogue: 0,0:20:02.14,0:20:06.21,Default,,0000,0000,0000,,Calc 1, yes, the end of Calc 1.
Dialogue: 0,0:20:06.21,0:20:07.10,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:20:07.10,0:20:09.40,Default,,0000,0000,0000,,So you don't need\Nthe picture anymore.
Dialogue: 0,0:20:09.40,0:20:12.08,Default,,0000,0000,0000,,You've done most of\Nthe work, and you say,
Dialogue: 0,0:20:12.08,0:20:25.26,Default,,0000,0000,0000,,I have an integral from 0 to\N7, x over-- so this guy-- which
Dialogue: 0,0:20:25.26,0:20:26.21,Default,,0000,0000,0000,,one shall I put first?
Dialogue: 0,0:20:26.21,0:20:27.27,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:20:27.27,0:20:30.48,Default,,0000,0000,0000,,e to the x squared\Ngot out first.
Dialogue: 0,0:20:30.48,0:20:31.67,Default,,0000,0000,0000,,He said, I'm out.
Dialogue: 0,0:20:31.67,0:20:35.45,Default,,0000,0000,0000,,And then the integral of\N1dy was y between these two,
Dialogue: 0,0:20:35.45,0:20:39.80,Default,,0000,0000,0000,,so it's x/7 dx.
Dialogue: 0,0:20:39.80,0:20:41.15,Default,,0000,0000,0000,,And this is a 7.
Dialogue: 0,0:20:41.15,0:20:44.30,Default,,0000,0000,0000,,
Dialogue: 0,0:20:44.30,0:20:45.03,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:20:45.03,0:20:45.68,Default,,0000,0000,0000,,We are happy.
Dialogue: 0,0:20:45.68,0:20:47.64,Default,,0000,0000,0000,,So what happens?
Dialogue: 0,0:20:47.64,0:20:51.17,Default,,0000,0000,0000,,1/7 also goes for a walk.
Dialogue: 0,0:20:51.17,0:20:55.10,Default,,0000,0000,0000,,And xdx says, OK, I need\Nto think about who I am.
Dialogue: 0,0:20:55.10,0:20:57.72,Default,,0000,0000,0000,,I have to find my own\Nidentity because I
Dialogue: 0,0:20:57.72,0:20:59.49,Default,,0000,0000,0000,,don't know who I am anymore.
Dialogue: 0,0:20:59.49,0:21:02.35,Default,,0000,0000,0000,,So he says, I need\Na u substitution.
Dialogue: 0,0:21:02.35,0:21:05.76,Default,,0000,0000,0000,,u substitution is\Nu equals x squared.
Dialogue: 0,0:21:05.76,0:21:08.72,Default,,0000,0000,0000,,du equals 2xdx.
Dialogue: 0,0:21:08.72,0:21:13.46,Default,,0000,0000,0000,,So xdx says, I know at least\Nthat I am a differential
Dialogue: 0,0:21:13.46,0:21:18.44,Default,,0000,0000,0000,,form, a 1 form, which is du/2.
Dialogue: 0,0:21:18.44,0:21:22.66,Default,,0000,0000,0000,,And that's exactly what you\Nguys need to change the inputs.
Dialogue: 0,0:21:22.66,0:21:24.40,Default,,0000,0000,0000,,1/7 was a [? custom ?].
Dialogue: 0,0:21:24.40,0:21:25.72,Default,,0000,0000,0000,,He got out of here.
Dialogue: 0,0:21:25.72,0:21:30.67,Default,,0000,0000,0000,,But you have to think,\Nwhen x is 0, what is u?
Dialogue: 0,0:21:30.67,0:21:32.13,Default,,0000,0000,0000,,0.
Dialogue: 0,0:21:32.13,0:21:36.64,Default,,0000,0000,0000,,When x is 7, what is u?
Dialogue: 0,0:21:36.64,0:21:37.65,Default,,0000,0000,0000,,49.
Dialogue: 0,0:21:37.65,0:21:39.23,Default,,0000,0000,0000,,Even my son would know this one.
Dialogue: 0,0:21:39.23,0:21:40.13,Default,,0000,0000,0000,,He would know more.
Dialogue: 0,0:21:40.13,0:21:42.07,Default,,0000,0000,0000,,He would know\Nfractions and stuff.
Dialogue: 0,0:21:42.07,0:21:42.81,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:21:42.81,0:21:45.47,Default,,0000,0000,0000,,So e to the u.
Dialogue: 0,0:21:45.47,0:21:48.18,Default,,0000,0000,0000,,
Dialogue: 0,0:21:48.18,0:21:50.39,Default,,0000,0000,0000,,And the 1/7 was out.
Dialogue: 0,0:21:50.39,0:21:51.75,Default,,0000,0000,0000,,But what is xdx?
Dialogue: 0,0:21:51.75,0:21:53.27,Default,,0000,0000,0000,,du/2.
Dialogue: 0,0:21:53.27,0:21:54.52,Default,,0000,0000,0000,,So I'll say 1/2 du.
Dialogue: 0,0:21:54.52,0:21:58.71,Default,,0000,0000,0000,,
Dialogue: 0,0:21:58.71,0:21:59.59,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:21:59.59,0:22:00.75,Default,,0000,0000,0000,,Could you follow everything?
Dialogue: 0,0:22:00.75,0:22:01.29,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:22:01.29,0:22:03.21,Default,,0000,0000,0000,,It shouldn't be a problem.
Dialogue: 0,0:22:03.21,0:22:05.03,Default,,0000,0000,0000,,Now 1/7 got out.
Dialogue: 0,0:22:05.03,0:22:06.54,Default,,0000,0000,0000,,1/2 gets out.
Dialogue: 0,0:22:06.54,0:22:10.18,Default,,0000,0000,0000,,Everybody gets out.
Dialogue: 0,0:22:10.18,0:22:13.93,Default,,0000,0000,0000,,And the guy in the middle who is\Nleft alone, the integral from e
Dialogue: 0,0:22:13.93,0:22:16.98,Default,,0000,0000,0000,,to the u du-- what is he?
Dialogue: 0,0:22:16.98,0:22:17.81,Default,,0000,0000,0000,,e to the u.
Dialogue: 0,0:22:17.81,0:22:19.83,Default,,0000,0000,0000,,Between what values?
Dialogue: 0,0:22:19.83,0:22:22.90,Default,,0000,0000,0000,,Between 49 and 0.
Dialogue: 0,0:22:22.90,0:22:25.03,Default,,0000,0000,0000,,So I'm going to--\Nshall I write it again?
Dialogue: 0,0:22:25.03,0:22:26.14,Default,,0000,0000,0000,,I'm too lazy for that.
Dialogue: 0,0:22:26.14,0:22:28.51,Default,,0000,0000,0000,,e to the u-- OK, I'll write it.
Dialogue: 0,0:22:28.51,0:22:32.92,Default,,0000,0000,0000,,e to the u between 49 and 0.
Dialogue: 0,0:22:32.92,0:22:40.33,Default,,0000,0000,0000,,So I have 1/14, parentheses,\Ne to the 49 minus e to the 0.
Dialogue: 0,0:22:40.33,0:22:42.30,Default,,0000,0000,0000,,That's a piece of cake.
Dialogue: 0,0:22:42.30,0:22:42.80,Default,,0000,0000,0000,,1.
Dialogue: 0,0:22:42.80,0:22:47.47,Default,,0000,0000,0000,,OK, so presumably if you\Nanswered that in WeBWork,
Dialogue: 0,0:22:47.47,0:22:49.49,Default,,0000,0000,0000,,this the precise answer.
Dialogue: 0,0:22:49.49,0:22:52.64,Default,,0000,0000,0000,,Finding it correctly, you\Nwould get the right answer.
Dialogue: 0,0:22:52.64,0:22:55.71,Default,,0000,0000,0000,,Of course, you could do\Nthat with the calculator.
Dialogue: 0,0:22:55.71,0:22:56.97,Default,,0000,0000,0000,,MATLAB could do it for you.
Dialogue: 0,0:22:56.97,0:22:58.24,Default,,0000,0000,0000,,Maple could do it for you.
Dialogue: 0,0:22:58.24,0:23:00.33,Default,,0000,0000,0000,,Mathematica could do it for you.
Dialogue: 0,0:23:00.33,0:23:04.34,Default,,0000,0000,0000,,But they will come up\Nwith a numerical answer,
Dialogue: 0,0:23:04.34,0:23:06.46,Default,,0000,0000,0000,,an approximation.
Dialogue: 0,0:23:06.46,0:23:08.46,Default,,0000,0000,0000,,And you haven't learned\Nanything in the process.
Dialogue: 0,0:23:08.46,0:23:12.34,Default,,0000,0000,0000,,Somebody just served you\Nthe answer on a plate,
Dialogue: 0,0:23:12.34,0:23:14.48,Default,,0000,0000,0000,,and that's not the idea.
Dialogue: 0,0:23:14.48,0:23:17.52,Default,,0000,0000,0000,,
Dialogue: 0,0:23:17.52,0:23:19.64,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:23:19.64,0:23:26.52,Default,,0000,0000,0000,,I'm saying on the midterm that's\Nbased on the double integral
Dialogue: 0,0:23:26.52,0:23:29.69,Default,,0000,0000,0000,,with switching order integrals,\Nthis is as hard as it can get.
Dialogue: 0,0:23:29.69,0:23:32.65,Default,,0000,0000,0000,,It cannot get worse than that.
Dialogue: 0,0:23:32.65,0:23:38.34,Default,,0000,0000,0000,,So that will tell you about\Nthe level of the midterm that's
Dialogue: 0,0:23:38.34,0:23:42.38,Default,,0000,0000,0000,,coming up on the 2nd of April,\Nnot something to be worried
Dialogue: 0,0:23:42.38,0:23:43.21,Default,,0000,0000,0000,,about.
Dialogue: 0,0:23:43.21,0:23:46.65,Default,,0000,0000,0000,,Do you need to learn a little\Nbit during the spring break?
Dialogue: 0,0:23:46.65,0:23:48.17,Default,,0000,0000,0000,,Maybe a few hours.
Dialogue: 0,0:23:48.17,0:23:51.89,Default,,0000,0000,0000,,But I would not worry my\Nfamily about it and say,
Dialogue: 0,0:23:51.89,0:23:53.12,Default,,0000,0000,0000,,there is this witch.
Dialogue: 0,0:23:53.12,0:23:55.72,Default,,0000,0000,0000,,And I'm going back\Nto [? Lubbock ?],
Dialogue: 0,0:23:55.72,0:23:58.78,Default,,0000,0000,0000,,and I have to take\Nher stinking midterm.
Dialogue: 0,0:23:58.78,0:24:03.43,Default,,0000,0000,0000,,And that stresses me out, so I\Ncannot enjoy my spring break.
Dialogue: 0,0:24:03.43,0:24:05.33,Default,,0000,0000,0000,,By all means, enjoy\Nyour spring break.
Dialogue: 0,0:24:05.33,0:24:09.45,Default,,0000,0000,0000,,And just devote a few\Nhours to your homework.
Dialogue: 0,0:24:09.45,0:24:11.35,Default,,0000,0000,0000,,But don't fret.
Dialogue: 0,0:24:11.35,0:24:14.94,Default,,0000,0000,0000,,Don't be worried\Nabout the coming exam,
Dialogue: 0,0:24:14.94,0:24:16.52,Default,,0000,0000,0000,,because you will be prepared.
Dialogue: 0,0:24:16.52,0:24:18.31,Default,,0000,0000,0000,,And I'm going to do\Nmore review so that you
Dialogue: 0,0:24:18.31,0:24:22.61,Default,,0000,0000,0000,,can be confident about it.
Dialogue: 0,0:24:22.61,0:24:23.37,Default,,0000,0000,0000,,Another one.
Dialogue: 0,0:24:23.37,0:24:25.00,Default,,0000,0000,0000,,Well, they're all easy.
Dialogue: 0,0:24:25.00,0:24:27.76,Default,,0000,0000,0000,,But I just want to help you\Nto the best of my extent.
Dialogue: 0,0:24:27.76,0:24:31.60,Default,,0000,0000,0000,,
Dialogue: 0,0:24:31.60,0:24:33.45,Default,,0000,0000,0000,,One more.
Dialogue: 0,0:24:33.45,0:24:35.42,Default,,0000,0000,0000,,Here also is-- I don't-- OK.
Dialogue: 0,0:24:35.42,0:24:40.73,Default,,0000,0000,0000,,Let's take this one because\Nit's not computational.
Dialogue: 0,0:24:40.73,0:24:41.59,Default,,0000,0000,0000,,And I love it.
Dialogue: 0,0:24:41.59,0:24:42.86,Default,,0000,0000,0000,,It's number 14.
Dialogue: 0,0:24:42.86,0:24:46.55,Default,,0000,0000,0000,,Number 14 and number 15\Nare so much the same type.
Dialogue: 0,0:24:46.55,0:24:48.00,Default,,0000,0000,0000,,And 16.
Dialogue: 0,0:24:48.00,0:24:49.39,Default,,0000,0000,0000,,It's a theoretical problem.
Dialogue: 0,0:24:49.39,0:24:52.95,Default,,0000,0000,0000,,It practically tests if\Nyou understood the idea.
Dialogue: 0,0:24:52.95,0:24:54.77,Default,,0000,0000,0000,,That's why I love this problem.
Dialogue: 0,0:24:54.77,0:24:57.97,Default,,0000,0000,0000,,And it appears\Nobsessively, this problem.
Dialogue: 0,0:24:57.97,0:25:01.31,Default,,0000,0000,0000,,I saw it in-- I've\Nbeen here for 14 years.
Dialogue: 0,0:25:01.31,0:25:05.31,Default,,0000,0000,0000,,I've seen it at least on 10\Ndifferent finals, the same type
Dialogue: 0,0:25:05.31,0:25:07.76,Default,,0000,0000,0000,,of theoretical problem.
Dialogue: 0,0:25:07.76,0:25:15.29,Default,,0000,0000,0000,,So it's number 14\Nover homework four.
Dialogue: 0,0:25:15.29,0:25:19.00,Default,,0000,0000,0000,,Find an equivalent integral\Nwith the order of integration
Dialogue: 0,0:25:19.00,0:25:20.42,Default,,0000,0000,0000,,reversed.
Dialogue: 0,0:25:20.42,0:25:24.10,Default,,0000,0000,0000,,So you need to\Nreverse some integral.
Dialogue: 0,0:25:24.10,0:25:31.33,Default,,0000,0000,0000,,And since you are so savvy about\Nreversing the ordered integral,
Dialogue: 0,0:25:31.33,0:25:35.29,Default,,0000,0000,0000,,you should not have\Na problem with it.
Dialogue: 0,0:25:35.29,0:25:48.39,Default,,0000,0000,0000,,
Dialogue: 0,0:25:48.39,0:25:51.16,Default,,0000,0000,0000,,And WeBWork is\Nasking you to fill
Dialogue: 0,0:25:51.16,0:25:53.28,Default,,0000,0000,0000,,in the following expressions.
Dialogue: 0,0:25:53.28,0:25:55.62,Default,,0000,0000,0000,,You know the type.
Dialogue: 0,0:25:55.62,0:25:59.85,Default,,0000,0000,0000,,f of y, you have to\Ntype in your answer.
Dialogue: 0,0:25:59.85,0:26:02.83,Default,,0000,0000,0000,,And g of y, to type\Nin your answer.
Dialogue: 0,0:26:02.83,0:26:05.82,Default,,0000,0000,0000,,
Dialogue: 0,0:26:05.82,0:26:07.09,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:26:07.09,0:26:11.51,Default,,0000,0000,0000,,So you're thinking, I know\Nhow to do this problem.
Dialogue: 0,0:26:11.51,0:26:15.52,Default,,0000,0000,0000,,It must be the idea as before.
Dialogue: 0,0:26:15.52,0:26:20.10,Default,,0000,0000,0000,,This integral should\Nbe-- according
Dialogue: 0,0:26:20.10,0:26:22.52,Default,,0000,0000,0000,,to the order of\Nintegration, it should
Dialogue: 0,0:26:22.52,0:26:27.73,Default,,0000,0000,0000,,be a vertical strip thing\Nswitching to a horizontal strip
Dialogue: 0,0:26:27.73,0:26:29.08,Default,,0000,0000,0000,,thing.
Dialogue: 0,0:26:29.08,0:26:33.38,Default,,0000,0000,0000,,And once I draw the domain,\NI'm going to know everything.
Dialogue: 0,0:26:33.38,0:26:35.38,Default,,0000,0000,0000,,And the answer is, yes,\Nyou can do this problem
Dialogue: 0,0:26:35.38,0:26:37.10,Default,,0000,0000,0000,,in about 25 seconds.
Dialogue: 0,0:26:37.10,0:26:40.07,Default,,0000,0000,0000,,The moment you've learned\Nit and understood it,
Dialogue: 0,0:26:40.07,0:26:42.04,Default,,0000,0000,0000,,it's going to go very smoothly.
Dialogue: 0,0:26:42.04,0:26:45.36,Default,,0000,0000,0000,,And to convince\Nyou, I'm just going
Dialogue: 0,0:26:45.36,0:26:48.93,Default,,0000,0000,0000,,to go ahead and say, 0 and 1.
Dialogue: 0,0:26:48.93,0:26:51.24,Default,,0000,0000,0000,,And draw, Magdalena.
Dialogue: 0,0:26:51.24,0:26:52.29,Default,,0000,0000,0000,,You know how to draw.
Dialogue: 0,0:26:52.29,0:26:53.29,Default,,0000,0000,0000,,Come on.
Dialogue: 0,0:26:53.29,0:26:53.79,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:26:53.79,0:26:56.29,Default,,0000,0000,0000,,From 1-- 1, 1, right?
Dialogue: 0,0:26:56.29,0:26:59.13,Default,,0000,0000,0000,,Is this the corner-- does\Nit look like a square?
Dialogue: 0,0:26:59.13,0:27:00.01,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:27:00.01,0:27:06.12,Default,,0000,0000,0000,,So the parabola y equals x\Nsquared is the bottom one.
Dialogue: 0,0:27:06.12,0:27:06.74,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,0:27:06.74,0:27:08.89,Default,,0000,0000,0000,,That is the bottom one, guys?
Dialogue: 0,0:27:08.89,0:27:12.20,Default,,0000,0000,0000,,But when you see-- when\Nyou are between 0 and 1,
Dialogue: 0,0:27:12.20,0:27:16.08,Default,,0000,0000,0000,,x squared is a lot less\Nthan the square root of x.
Dialogue: 0,0:27:16.08,0:27:20.21,Default,,0000,0000,0000,,The square root of x is the\Ntop, is the function on top.
Dialogue: 0,0:27:20.21,0:27:24.63,Default,,0000,0000,0000,,And then you say, OK, I\Ngot-- somebody gave me
Dialogue: 0,0:27:24.63,0:27:27.01,Default,,0000,0000,0000,,the vertical strips.
Dialogue: 0,0:27:27.01,0:27:29.62,Default,,0000,0000,0000,,I'll put the [INAUDIBLE],\Nbut I don't need them.
Dialogue: 0,0:27:29.62,0:27:37.18,Default,,0000,0000,0000,,I'll just go ahead\Nand take the purple,
Dialogue: 0,0:27:37.18,0:27:43.37,Default,,0000,0000,0000,,and I'll draw the\Nhorizontal strips.
Dialogue: 0,0:27:43.37,0:27:46.75,Default,,0000,0000,0000,,And you are already\Nthere because I
Dialogue: 0,0:27:46.75,0:27:49.51,Default,,0000,0000,0000,,see the light in your eyes.
Dialogue: 0,0:27:49.51,0:27:52.77,Default,,0000,0000,0000,,So tell me what you\Nhave. y between n--
Dialogue: 0,0:27:52.77,0:27:53.48,Default,,0000,0000,0000,,STUDENT: 0 and 1.
Dialogue: 0,0:27:53.48,0:27:54.65,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 0 and 1.
Dialogue: 0,0:27:54.65,0:27:55.94,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:27:55.94,0:27:58.74,Default,,0000,0000,0000,,And y between what and x?
Dialogue: 0,0:27:58.74,0:28:00.47,Default,,0000,0000,0000,,Oh, sorry, guys.
Dialogue: 0,0:28:00.47,0:28:02.45,Default,,0000,0000,0000,,I need to protect my hand.
Dialogue: 0,0:28:02.45,0:28:04.46,Default,,0000,0000,0000,,That's the secret recipe.
Dialogue: 0,0:28:04.46,0:28:06.98,Default,,0000,0000,0000,,x is between a function of y.
Dialogue: 0,0:28:06.98,0:28:10.07,Default,,0000,0000,0000,,Now what's the\Nhighest function of y?
Dialogue: 0,0:28:10.07,0:28:11.15,Default,,0000,0000,0000,,STUDENT: Square root of y.
Dialogue: 0,0:28:11.15,0:28:12.30,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NSquare root of y.
Dialogue: 0,0:28:12.30,0:28:13.61,Default,,0000,0000,0000,,And who is that fellow?
Dialogue: 0,0:28:13.61,0:28:15.05,Default,,0000,0000,0000,,This one.
Dialogue: 0,0:28:15.05,0:28:17.66,Default,,0000,0000,0000,,x equals square root\Nof y, the green fellow.
Dialogue: 0,0:28:17.66,0:28:21.01,Default,,0000,0000,0000,,I should have written in\Ngreen, but I was too lazy.
Dialogue: 0,0:28:21.01,0:28:26.17,Default,,0000,0000,0000,,And this one is going to\Nbe just x equals y squared.
Dialogue: 0,0:28:26.17,0:28:29.77,Default,,0000,0000,0000,,So between y square\Ndown, down, down, down.
Dialogue: 0,0:28:29.77,0:28:31.32,Default,,0000,0000,0000,,Who is down? f is down.
Dialogue: 0,0:28:31.32,0:28:32.55,Default,,0000,0000,0000,,Right, guys?
Dialogue: 0,0:28:32.55,0:28:34.92,Default,,0000,0000,0000,,The bottom one is f.
Dialogue: 0,0:28:34.92,0:28:37.49,Default,,0000,0000,0000,,The bottom one is y squared.
Dialogue: 0,0:28:37.49,0:28:43.93,Default,,0000,0000,0000,,The upper one is the\Nsquare root of y.
Dialogue: 0,0:28:43.93,0:28:46.41,Default,,0000,0000,0000,,You cannot type that\Nin WeBWork, right?
Dialogue: 0,0:28:46.41,0:28:49.32,Default,,0000,0000,0000,,You type sqrt, what?
Dialogue: 0,0:28:49.32,0:28:50.53,Default,,0000,0000,0000,,y, caret, 2.
Dialogue: 0,0:28:50.53,0:28:52.09,Default,,0000,0000,0000,,And here, what do you have?
Dialogue: 0,0:28:52.09,0:28:53.29,Default,,0000,0000,0000,,0 and 1.
Dialogue: 0,0:28:53.29,0:28:55.13,Default,,0000,0000,0000,,So I talk too much.
Dialogue: 0,0:28:55.13,0:28:58.44,Default,,0000,0000,0000,,But if you were on your\Nown doing this in WeBWork,
Dialogue: 0,0:28:58.44,0:29:01.75,Default,,0000,0000,0000,,it would take you no\Nmore than-- I don't
Dialogue: 0,0:29:01.75,0:29:05.22,Default,,0000,0000,0000,,know-- 60 seconds to type in.
Dialogue: 0,0:29:05.22,0:29:07.44,Default,,0000,0000,0000,,Remember this problem\Nfor the midterm.
Dialogue: 0,0:29:07.44,0:29:08.75,Default,,0000,0000,0000,,It's an important idea.
Dialogue: 0,0:29:08.75,0:29:11.00,Default,,0000,0000,0000,,And you've seen it emphasized.
Dialogue: 0,0:29:11.00,0:29:16.92,Default,,0000,0000,0000,,You will see it emphasized\Nin problems 14, 15, 16.
Dialogue: 0,0:29:16.92,0:29:19.80,Default,,0000,0000,0000,,It's embedded in this\Ntype of exchange,
Dialogue: 0,0:29:19.80,0:29:21.67,Default,,0000,0000,0000,,change the order of\Nintegration type problem.
Dialogue: 0,0:29:21.67,0:29:25.41,Default,,0000,0000,0000,,
Dialogue: 0,0:29:25.41,0:29:26.98,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:29:26.98,0:29:32.02,Default,,0000,0000,0000,,Anything else I would like\Nto show you from-- there
Dialogue: 0,0:29:32.02,0:29:34.13,Default,,0000,0000,0000,,are many things I\Nwould like to show you.
Dialogue: 0,0:29:34.13,0:29:37.29,Default,,0000,0000,0000,,But I better let you\Ndo things on your own.
Dialogue: 0,0:29:37.29,0:29:41.80,Default,,0000,0000,0000,,How about 17, which is a\Nsimilar type of problem,
Dialogue: 0,0:29:41.80,0:29:43.86,Default,,0000,0000,0000,,theoretical, just like this one?
Dialogue: 0,0:29:43.86,0:29:48.74,Default,,0000,0000,0000,,But it's testing\Nif you know the--
Dialogue: 0,0:29:48.74,0:29:53.32,Default,,0000,0000,0000,,if you understood the idea\Nbehind polar integration,
Dialogue: 0,0:29:53.32,0:29:55.93,Default,,0000,0000,0000,,integration in\Npolar coordinates.
Dialogue: 0,0:29:55.93,0:29:57.44,Default,,0000,0000,0000,,Can I erase?
Dialogue: 0,0:29:57.44,0:29:59.88,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:29:59.88,0:30:02.77,Default,,0000,0000,0000,,So let's switch to number\N17 from your homework.
Dialogue: 0,0:30:02.77,0:30:05.54,Default,,0000,0000,0000,,
Dialogue: 0,0:30:05.54,0:30:08.63,Default,,0000,0000,0000,,Write down the problems\Nwe are going over,
Dialogue: 0,0:30:08.63,0:30:10.32,Default,,0000,0000,0000,,so when you do\Nyour homework, you
Dialogue: 0,0:30:10.32,0:30:12.74,Default,,0000,0000,0000,,refer to your lecture notes.
Dialogue: 0,0:30:12.74,0:30:14.42,Default,,0000,0000,0000,,This is not a lecture.
Dialogue: 0,0:30:14.42,0:30:16.19,Default,,0000,0000,0000,,What is this, what\Nyou're doing now?
Dialogue: 0,0:30:16.19,0:30:18.89,Default,,0000,0000,0000,,It's like-- what is this?
Dialogue: 0,0:30:18.89,0:30:22.69,Default,,0000,0000,0000,,An application session,\Na problem session.
Dialogue: 0,0:30:22.69,0:30:24.77,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:30:24.77,0:30:30.32,Default,,0000,0000,0000,,Number 17, homework four.
Dialogue: 0,0:30:30.32,0:30:35.76,Default,,0000,0000,0000,,On this one, unfortunately\NI'm doing just your homework
Dialogue: 0,0:30:35.76,0:30:38.68,Default,,0000,0000,0000,,because there is no data.
Dialogue: 0,0:30:38.68,0:30:45.81,Default,,0000,0000,0000,,So when-- it's the unique\Nproblem you're going to get.
Dialogue: 0,0:30:45.81,0:30:53.13,Default,,0000,0000,0000,,You have a picture, and that\Npicture looks like that.
Dialogue: 0,0:30:53.13,0:30:58.14,Default,,0000,0000,0000,,From here, [INAUDIBLE]\Na half of an annulus.
Dialogue: 0,0:30:58.14,0:31:00.76,Default,,0000,0000,0000,,
Dialogue: 0,0:31:00.76,0:31:02.98,Default,,0000,0000,0000,,You have half of a ring.
Dialogue: 0,0:31:02.98,0:31:07.45,Default,,0000,0000,0000,,And it says, suppose that\Nr is the shaded region
Dialogue: 0,0:31:07.45,0:31:10.38,Default,,0000,0000,0000,,in the figure.
Dialogue: 0,0:31:10.38,0:31:13.15,Default,,0000,0000,0000,,As an iterated integral\Nin polar coordinates,
Dialogue: 0,0:31:13.15,0:31:20.20,Default,,0000,0000,0000,,the double integral\Nover R f of x, y dA
Dialogue: 0,0:31:20.20,0:31:24.49,Default,,0000,0000,0000,,is the integral from A to\NB of the integral from C
Dialogue: 0,0:31:24.49,0:31:37.75,Default,,0000,0000,0000,,to B of f of r, theta times r\Ndrd theta with the following
Dialogue: 0,0:31:37.75,0:31:39.72,Default,,0000,0000,0000,,limits of integration.
Dialogue: 0,0:31:39.72,0:31:44.76,Default,,0000,0000,0000,,A. And WeBWork says, you say it.
Dialogue: 0,0:31:44.76,0:31:45.38,Default,,0000,0000,0000,,You say.
Dialogue: 0,0:31:45.38,0:31:47.22,Default,,0000,0000,0000,,It's playing games with you.
Dialogue: 0,0:31:47.22,0:31:48.22,Default,,0000,0000,0000,,B, you say.
Dialogue: 0,0:31:48.22,0:31:50.04,Default,,0000,0000,0000,,It's a guessing game.
Dialogue: 0,0:31:50.04,0:31:51.11,Default,,0000,0000,0000,,C, you say.
Dialogue: 0,0:31:51.11,0:31:55.11,Default,,0000,0000,0000,,Then D, you say it.
Dialogue: 0,0:31:55.11,0:31:58.39,Default,,0000,0000,0000,,And let's see what you say.
Dialogue: 0,0:31:58.39,0:32:02.08,Default,,0000,0000,0000,,
Dialogue: 0,0:32:02.08,0:32:06.43,Default,,0000,0000,0000,,Well, we say, well,\Nhow am I going to go?
Dialogue: 0,0:32:06.43,0:32:09.28,Default,,0000,0000,0000,,I have to disclose\Nthe graphing paper.
Dialogue: 0,0:32:09.28,0:32:10.56,Default,,0000,0000,0000,,They are so mean.
Dialogue: 0,0:32:10.56,0:32:14.60,Default,,0000,0000,0000,,They don't show you\Nthe actual numbers.
Dialogue: 0,0:32:14.60,0:32:16.54,Default,,0000,0000,0000,,They only give you\Ngraphing paper.
Dialogue: 0,0:32:16.54,0:32:18.95,Default,,0000,0000,0000,,I'm not good at graphing, OK?
Dialogue: 0,0:32:18.95,0:32:22.80,Default,,0000,0000,0000,,So you will have to\Nguess what this says.
Dialogue: 0,0:32:22.80,0:32:24.14,Default,,0000,0000,0000,,That should be good enough.
Dialogue: 0,0:32:24.14,0:32:24.86,Default,,0000,0000,0000,,Perfect.
Dialogue: 0,0:32:24.86,0:32:29.10,Default,,0000,0000,0000,,So the unit supposedly\Nis this much.
Dialogue: 0,0:32:29.10,0:32:31.62,Default,,0000,0000,0000,,1 inch, whatever.
Dialogue: 0,0:32:31.62,0:32:33.22,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,0:32:33.22,0:32:36.27,Default,,0000,0000,0000,,So is it hard?
Dialogue: 0,0:32:36.27,0:32:37.33,Default,,0000,0000,0000,,It's a piece of cake.
Dialogue: 0,0:32:37.33,0:32:39.02,Default,,0000,0000,0000,,It's a 10 second problem.
Dialogue: 0,0:32:39.02,0:32:41.39,Default,,0000,0000,0000,,It's a good problem for the\Nmidterm because it's fast.
Dialogue: 0,0:32:41.39,0:32:48.75,Default,,0000,0000,0000,,
Dialogue: 0,0:32:48.75,0:32:53.40,Default,,0000,0000,0000,,Theta is a wonderful angle.
Dialogue: 0,0:32:53.40,0:32:56.32,Default,,0000,0000,0000,,
Dialogue: 0,0:32:56.32,0:32:58.49,Default,,0000,0000,0000,,It is nice to look at.
Dialogue: 0,0:32:58.49,0:33:02.22,Default,,0000,0000,0000,,And they really don't\Nput numbers here?
Dialogue: 0,0:33:02.22,0:33:03.95,Default,,0000,0000,0000,,They do.
Dialogue: 0,0:33:03.95,0:33:06.88,Default,,0000,0000,0000,,They do on the margin\Nof the graphing paper.
Dialogue: 0,0:33:06.88,0:33:08.59,Default,,0000,0000,0000,,They have a scale.
Dialogue: 0,0:33:08.59,0:33:09.49,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:33:09.49,0:33:11.01,Default,,0000,0000,0000,,So come on.
Dialogue: 0,0:33:11.01,0:33:11.66,Default,,0000,0000,0000,,This is easy.
Dialogue: 0,0:33:11.66,0:33:14.88,Default,,0000,0000,0000,,You guys are too smart\Nfor this problem.
Dialogue: 0,0:33:14.88,0:33:17.62,Default,,0000,0000,0000,,From what to what?
Dialogue: 0,0:33:17.62,0:33:18.50,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:33:18.50,0:33:19.54,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Nope.
Dialogue: 0,0:33:19.54,0:33:20.69,Default,,0000,0000,0000,,No, that's a problem.
Dialogue: 0,0:33:20.69,0:33:23.10,Default,,0000,0000,0000,,So when we measure\Nthe angle theta,
Dialogue: 0,0:33:23.10,0:33:24.60,Default,,0000,0000,0000,,where do we start measuring?
Dialogue: 0,0:33:24.60,0:33:25.28,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:33:25.28,0:33:26.53,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Over here.
Dialogue: 0,0:33:26.53,0:33:29.08,Default,,0000,0000,0000,,So we go down there\Nclockwise because that's
Dialogue: 0,0:33:29.08,0:33:31.45,Default,,0000,0000,0000,,how we mix in the bowl,\Ncounter-clockwise.
Dialogue: 0,0:33:31.45,0:33:38.42,Default,,0000,0000,0000,,So 0-- so this is\Ngoing to be pi.
Dialogue: 0,0:33:38.42,0:33:38.92,Default,,0000,0000,0000,,Pi.
Dialogue: 0,0:33:38.92,0:33:42.92,Default,,0000,0000,0000,,
Dialogue: 0,0:33:42.92,0:33:44.96,Default,,0000,0000,0000,,And what is the end?
Dialogue: 0,0:33:44.96,0:33:45.65,Default,,0000,0000,0000,,STUDENT: 2 pi.
Dialogue: 0,0:33:45.65,0:33:46.78,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 2 pi.
Dialogue: 0,0:33:46.78,0:33:47.28,Default,,0000,0000,0000,,2 pi.
Dialogue: 0,0:33:47.28,0:33:49.75,Default,,0000,0000,0000,,
Dialogue: 0,0:33:49.75,0:33:52.56,Default,,0000,0000,0000,,Don't type-- oh, I mean, you\Ncannot type the symbol part,
Dialogue: 0,0:33:52.56,0:33:53.54,Default,,0000,0000,0000,,right?
Dialogue: 0,0:33:53.54,0:33:58.09,Default,,0000,0000,0000,,And then what do you\Ntype, in terms of C and D?
Dialogue: 0,0:33:58.09,0:33:58.93,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:33:58.93,0:33:59.97,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Nope.
Dialogue: 0,0:33:59.97,0:34:00.47,Default,,0000,0000,0000,,No, no.
Dialogue: 0,0:34:00.47,0:34:03.19,Default,,0000,0000,0000,,The radius is positive only.
Dialogue: 0,0:34:03.19,0:34:05.74,Default,,0000,0000,0000,,STUDENT: 0 to 1.
Dialogue: 0,0:34:05.74,0:34:06.86,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 1 to 2.
Dialogue: 0,0:34:06.86,0:34:07.77,Default,,0000,0000,0000,,Why 1 to 2?
Dialogue: 0,0:34:07.77,0:34:08.60,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:34:08.60,0:34:13.15,Default,,0000,0000,0000,,Because the shaded area\Nrepresents the half of a donut.
Dialogue: 0,0:34:13.15,0:34:14.43,Default,,0000,0000,0000,,You have nothing inside.
Dialogue: 0,0:34:14.43,0:34:17.58,Default,,0000,0000,0000,,There is a whole in\Nhere, in the donut.
Dialogue: 0,0:34:17.58,0:34:20.08,Default,,0000,0000,0000,,So between 0 and 1,\Nyou have nothing.
Dialogue: 0,0:34:20.08,0:34:23.13,Default,,0000,0000,0000,,And the radius-- take\Na point in your domain.
Dialogue: 0,0:34:23.13,0:34:23.92,Default,,0000,0000,0000,,It's here.
Dialogue: 0,0:34:23.92,0:34:26.15,Default,,0000,0000,0000,,The radius you\Nhave, the red radius
Dialogue: 0,0:34:26.15,0:34:32.31,Default,,0000,0000,0000,,you guys see on the picture is\Na value that's between 1 and 2,
Dialogue: 0,0:34:32.31,0:34:34.70,Default,,0000,0000,0000,,between 1 and 2.
Dialogue: 0,0:34:34.70,0:34:35.88,Default,,0000,0000,0000,,And that's it.
Dialogue: 0,0:34:35.88,0:34:37.52,Default,,0000,0000,0000,,That was a 10 second problem.
Dialogue: 0,0:34:37.52,0:34:38.44,Default,,0000,0000,0000,,So promise me.
Dialogue: 0,0:34:38.44,0:34:40.55,Default,,0000,0000,0000,,You are going to do\Nthe homework and stuff.
Dialogue: 0,0:34:40.55,0:34:44.13,Default,,0000,0000,0000,,You have two or three like that.
Dialogue: 0,0:34:44.13,0:34:46.03,Default,,0000,0000,0000,,If you see this on\Nthe midterm, are you
Dialogue: 0,0:34:46.03,0:34:50.26,Default,,0000,0000,0000,,going to remember the procedure,\Nthe idea of the problem?
Dialogue: 0,0:34:50.26,0:34:52.86,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:34:52.86,0:34:55.37,Default,,0000,0000,0000,,I'm going to also think\Nof writing a sample.
Dialogue: 0,0:34:55.37,0:34:57.51,Default,,0000,0000,0000,,I promised Stacy I'm\Ngoing to do that.
Dialogue: 0,0:34:57.51,0:34:58.72,Default,,0000,0000,0000,,And I did not forget.
Dialogue: 0,0:34:58.72,0:34:59.91,Default,,0000,0000,0000,,It's going to happen.
Dialogue: 0,0:34:59.91,0:35:02.30,Default,,0000,0000,0000,,After spring break, you're\Ngoing to get a review
Dialogue: 0,0:35:02.30,0:35:04.96,Default,,0000,0000,0000,,sheet for the midterm.
Dialogue: 0,0:35:04.96,0:35:08.54,Default,,0000,0000,0000,,I promised you a sample, right?
Dialogue: 0,0:35:08.54,0:35:09.04,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:35:09.04,0:35:14.53,Default,,0000,0000,0000,,
Dialogue: 0,0:35:14.53,0:35:18.32,Default,,0000,0000,0000,,Shall I do more or not?
Dialogue: 0,0:35:18.32,0:35:20.68,Default,,0000,0000,0000,,Yes?
Dialogue: 0,0:35:20.68,0:35:22.51,Default,,0000,0000,0000,,You know what I'm\Nafraid of, really?
Dialogue: 0,0:35:22.51,0:35:28.71,Default,,0000,0000,0000,,I think you will be able to do\Nfine with most of the problems
Dialogue: 0,0:35:28.71,0:35:32.15,Default,,0000,0000,0000,,you have here.
Dialogue: 0,0:35:32.15,0:35:38.93,Default,,0000,0000,0000,,I'm more worried about\Ngeometric representations
Dialogue: 0,0:35:38.93,0:35:44.41,Default,,0000,0000,0000,,in 3D of quadrics that\Nyou guys became familiar
Dialogue: 0,0:35:44.41,0:35:47.68,Default,,0000,0000,0000,,with only now,\Nonly this semester.
Dialogue: 0,0:35:47.68,0:35:51.33,Default,,0000,0000,0000,,And you have a grasp of them.
Dialogue: 0,0:35:51.33,0:35:52.23,Default,,0000,0000,0000,,You've seen them.
Dialogue: 0,0:35:52.23,0:35:56.85,Default,,0000,0000,0000,,But you're still not\Nvery friendly with them,
Dialogue: 0,0:35:56.85,0:35:59.10,Default,,0000,0000,0000,,and you don't\Nquite like to draw.
Dialogue: 0,0:35:59.10,0:36:03.27,Default,,0000,0000,0000,,So let's see if we can learn\Nhow to draw one of them together
Dialogue: 0,0:36:03.27,0:36:07.62,Default,,0000,0000,0000,,and see if it's a big\Ndeal or not because it's
Dialogue: 0,0:36:07.62,0:36:10.13,Default,,0000,0000,0000,,pretty as a picture.
Dialogue: 0,0:36:10.13,0:36:11.89,Default,,0000,0000,0000,,And when we set it\Nup as an integral,
Dialogue: 0,0:36:11.89,0:36:17.15,Default,,0000,0000,0000,,it should be done wisely.
Dialogue: 0,0:36:17.15,0:36:18.64,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,0:36:18.64,0:36:21.63,Default,,0000,0000,0000,,
Dialogue: 0,0:36:21.63,0:36:25.48,Default,,0000,0000,0000,,We have to do a good job\Nfrom the moment we draw.
Dialogue: 0,0:36:25.48,0:36:30.13,Default,,0000,0000,0000,,And if we don't do that,\Nwe don't have much chance.
Dialogue: 0,0:36:30.13,0:36:36.11,Default,,0000,0000,0000,,The problem is going to\Nchange the data a little bit
Dialogue: 0,0:36:36.11,0:36:39.09,Default,,0000,0000,0000,,to numbers that I like.
Dialogue: 0,0:36:39.09,0:36:39.59,Default,,0000,0000,0000,,29.
Dialogue: 0,0:36:39.59,0:36:45.01,Default,,0000,0000,0000,,
Dialogue: 0,0:36:45.01,0:36:46.93,Default,,0000,0000,0000,,You have a solid.
Dialogue: 0,0:36:46.93,0:36:49.22,Default,,0000,0000,0000,,And I say solid gold, 24 k.
Dialogue: 0,0:36:49.22,0:36:50.94,Default,,0000,0000,0000,,I don't know what.
Dialogue: 0,0:36:50.94,0:36:54.00,Default,,0000,0000,0000,,That is between two paraboloids.
Dialogue: 0,0:36:54.00,0:36:56.79,Default,,0000,0000,0000,,And those paraboloids\Nare given, and I'd
Dialogue: 0,0:36:56.79,0:36:59.78,Default,,0000,0000,0000,,like you to tell me\Nwhat they look like.
Dialogue: 0,0:36:59.78,0:37:05.61,Default,,0000,0000,0000,,One paraboloid is y-- no.
Dialogue: 0,0:37:05.61,0:37:08.59,Default,,0000,0000,0000,,
Dialogue: 0,0:37:08.59,0:37:09.09,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:37:09.09,0:37:13.75,Default,,0000,0000,0000,,One paraboloid is\Ny-- I'll change it.
Dialogue: 0,0:37:13.75,0:37:14.56,Default,,0000,0000,0000,,z.
Dialogue: 0,0:37:14.56,0:37:16.65,Default,,0000,0000,0000,,So I can change your\Nproblem, and then you
Dialogue: 0,0:37:16.65,0:37:19.44,Default,,0000,0000,0000,,will figure it out by yourself.
Dialogue: 0,0:37:19.44,0:37:21.19,Default,,0000,0000,0000,,z equals x squared\Nplus y squared.
Dialogue: 0,0:37:21.19,0:37:23.75,Default,,0000,0000,0000,,They give you y equals x\Nsquared plus d squared.
Dialogue: 0,0:37:23.75,0:37:26.73,Default,,0000,0000,0000,,So you have to change\Ncompletely the configuration
Dialogue: 0,0:37:26.73,0:37:28.90,Default,,0000,0000,0000,,of your frame.
Dialogue: 0,0:37:28.90,0:37:33.14,Default,,0000,0000,0000,,And then z equals 8 minus\Nx squared minus y squared.
Dialogue: 0,0:37:33.14,0:37:39.28,Default,,0000,0000,0000,,I'm I'm changing problem 29,\Nbut it's practically the same.
Dialogue: 0,0:37:39.28,0:37:43.76,Default,,0000,0000,0000,,Find the volume of the solid\Nenclosed by the two paraboloids
Dialogue: 0,0:37:43.76,0:37:45.37,Default,,0000,0000,0000,,and write down the answer.
Dialogue: 0,0:37:45.37,0:37:49.24,Default,,0000,0000,0000,,
Dialogue: 0,0:37:49.24,0:37:55.50,Default,,0000,0000,0000,,Find the volume of\Nthe solid enclosed
Dialogue: 0,0:37:55.50,0:37:58.16,Default,,0000,0000,0000,,by the two paraboloids.
Dialogue: 0,0:37:58.16,0:37:59.41,Default,,0000,0000,0000,,You go, oh, my god.
Dialogue: 0,0:37:59.41,0:38:03.06,Default,,0000,0000,0000,,How am I going to do that?
Dialogue: 0,0:38:03.06,0:38:04.18,Default,,0000,0000,0000,,STUDENT: Draw the pictures.
Dialogue: 0,0:38:04.18,0:38:04.73,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NDraw the pictures.
Dialogue: 0,0:38:04.73,0:38:05.86,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:38:05.86,0:38:08.98,Default,,0000,0000,0000,,So he's teaching\Nmy sensing to me
Dialogue: 0,0:38:08.98,0:38:10.94,Default,,0000,0000,0000,,and says, OK, go ahead\Nand draw the picture.
Dialogue: 0,0:38:10.94,0:38:13.10,Default,,0000,0000,0000,,Don't be lazy, because\Nif you don't, it's
Dialogue: 0,0:38:13.10,0:38:14.79,Default,,0000,0000,0000,,never going to happen.
Dialogue: 0,0:38:14.79,0:38:17.24,Default,,0000,0000,0000,,You're never going\Nto see the domain
Dialogue: 0,0:38:17.24,0:38:19.66,Default,,0000,0000,0000,,if you don't draw the pictures.
Dialogue: 0,0:38:19.66,0:38:24.30,Default,,0000,0000,0000,,So the first one will\Nbe the shell of the egg.
Dialogue: 0,0:38:24.30,0:38:25.46,Default,,0000,0000,0000,,Easter is coming.
Dialogue: 0,0:38:25.46,0:38:29.88,Default,,0000,0000,0000,,So that's something\Nlike the shell.
Dialogue: 0,0:38:29.88,0:38:33.72,Default,,0000,0000,0000,,It's a terrible shell,\Na paraboloid, circular
Dialogue: 0,0:38:33.72,0:38:35.44,Default,,0000,0000,0000,,paraboloid.
Dialogue: 0,0:38:35.44,0:38:40.74,Default,,0000,0000,0000,,And that is called z equals\Nx squared plus y squared.
Dialogue: 0,0:38:40.74,0:38:49.62,Default,,0000,0000,0000,,
Dialogue: 0,0:38:49.62,0:38:50.12,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:38:50.12,0:38:53.37,Default,,0000,0000,0000,,
Dialogue: 0,0:38:53.37,0:38:56.21,Default,,0000,0000,0000,,This guy keeps going.
Dialogue: 0,0:38:56.21,0:38:58.13,Default,,0000,0000,0000,,But there will be\Nanother paraboloid
Dialogue: 0,0:38:58.13,0:39:04.99,Default,,0000,0000,0000,,that has the shape of exactly\Nthe same thing upside down.
Dialogue: 0,0:39:04.99,0:39:06.61,Default,,0000,0000,0000,,STUDENT: Where's 8?
Dialogue: 0,0:39:06.61,0:39:08.15,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Where is 8?
Dialogue: 0,0:39:08.15,0:39:09.96,Default,,0000,0000,0000,,The 8 is far away.
Dialogue: 0,0:39:09.96,0:39:10.86,Default,,0000,0000,0000,,STUDENT: It's on--
Dialogue: 0,0:39:10.86,0:39:12.07,Default,,0000,0000,0000,,DR. MAGDALENA TODA: I'll try.
Dialogue: 0,0:39:12.07,0:39:16.60,Default,,0000,0000,0000,,
Dialogue: 0,0:39:16.60,0:39:19.14,Default,,0000,0000,0000,,STUDENT: Did they tell you\Nthat a had to be positive?
Dialogue: 0,0:39:19.14,0:39:20.14,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Huh?
Dialogue: 0,0:39:20.14,0:39:22.44,Default,,0000,0000,0000,,STUDENT: Did they tell\Nyou a had to be positive?
Dialogue: 0,0:39:22.44,0:39:23.60,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Which a?
Dialogue: 0,0:39:23.60,0:39:25.93,Default,,0000,0000,0000,,STUDENT: That a or whatever.
Dialogue: 0,0:39:25.93,0:39:27.08,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 8.
Dialogue: 0,0:39:27.08,0:39:27.71,Default,,0000,0000,0000,,STUDENT: Oh, 8.
Dialogue: 0,0:39:27.71,0:39:28.53,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 8.
Dialogue: 0,0:39:28.53,0:39:29.35,Default,,0000,0000,0000,,STUDENT: Oh, that's\Nwhy I'm confused.
Dialogue: 0,0:39:29.35,0:39:31.06,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NHow do I know it's 8?
Dialogue: 0,0:39:31.06,0:39:33.95,Default,,0000,0000,0000,,Because when I put\Nx equals 7 equals 0,
Dialogue: 0,0:39:33.95,0:39:36.62,Default,,0000,0000,0000,,I get z equals 8\Nfor this paraboloid.
Dialogue: 0,0:39:36.62,0:39:40.70,Default,,0000,0000,0000,,This is the red paraboloid.
Dialogue: 0,0:39:40.70,0:39:43.22,Default,,0000,0000,0000,,The problem-- my\Nquestion is, OK, it's
Dialogue: 0,0:39:43.22,0:39:46.19,Default,,0000,0000,0000,,like it is two\Neggshells that are
Dialogue: 0,0:39:46.19,0:39:48.56,Default,,0000,0000,0000,,connecting, exactly this egg.
Dialogue: 0,0:39:48.56,0:39:52.57,Default,,0000,0000,0000,,But the bound-- the--\Nhow do you call that?
Dialogue: 0,0:39:52.57,0:39:55.85,Default,,0000,0000,0000,,Boundary, the thing where\Nthey glue it together.
Dialogue: 0,0:39:55.85,0:39:58.15,Default,,0000,0000,0000,,What is the equation\Nof this circle?
Dialogue: 0,0:39:58.15,0:39:59.47,Default,,0000,0000,0000,,This is the question.
Dialogue: 0,0:39:59.47,0:40:01.44,Default,,0000,0000,0000,,Where do they intersect?
Dialogue: 0,0:40:01.44,0:40:04.74,Default,,0000,0000,0000,,How do you find out where\Ntwo surfaces intersect?
Dialogue: 0,0:40:04.74,0:40:06.16,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:40:06.16,0:40:08.08,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NSolve a system.
Dialogue: 0,0:40:08.08,0:40:12.80,Default,,0000,0000,0000,,Make a system of two equations\Nand solve the system.
Dialogue: 0,0:40:12.80,0:40:15.19,Default,,0000,0000,0000,,You have to intersect them.
Dialogue: 0,0:40:15.19,0:40:21.20,Default,,0000,0000,0000,,So whoever x, y, z will be, they\Nhave to satisfy both equations.
Dialogue: 0,0:40:21.20,0:40:22.20,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,0:40:22.20,0:40:25.23,Default,,0000,0000,0000,,So we have to look for the\Nsolutions of both equations
Dialogue: 0,0:40:25.23,0:40:32.33,Default,,0000,0000,0000,,at the same time, which\Nmeans that I'm going
Dialogue: 0,0:40:32.33,0:40:35.68,Default,,0000,0000,0000,,to say these are equal, right?
Dialogue: 0,0:40:35.68,0:40:37.93,Default,,0000,0000,0000,,Let's write that down.
Dialogue: 0,0:40:37.93,0:40:41.45,Default,,0000,0000,0000,,x squared plus y\Nsquared equals 8 minus x
Dialogue: 0,0:40:41.45,0:40:43.26,Default,,0000,0000,0000,,squared minus y squared.
Dialogue: 0,0:40:43.26,0:40:48.31,Default,,0000,0000,0000,,Then z is whatever.
Dialogue: 0,0:40:48.31,0:40:52.01,Default,,0000,0000,0000,,What is this equation?
Dialogue: 0,0:40:52.01,0:40:54.08,Default,,0000,0000,0000,,We'll find out who\Nz is in a second.
Dialogue: 0,0:40:54.08,0:40:57.30,Default,,0000,0000,0000,,
Dialogue: 0,0:40:57.30,0:41:00.20,Default,,0000,0000,0000,,z has to be x squared\Nplus y squared.
Dialogue: 0,0:41:00.20,0:41:04.16,Default,,0000,0000,0000,,If we find out who the sum\Nof the squares will be,
Dialogue: 0,0:41:04.16,0:41:07.49,Default,,0000,0000,0000,,we'll find out the altitude z.
Dialogue: 0,0:41:07.49,0:41:08.85,Default,,0000,0000,0000,,z equals what number?
Dialogue: 0,0:41:08.85,0:41:10.08,Default,,0000,0000,0000,,This is the whole idea.
Dialogue: 0,0:41:10.08,0:41:11.35,Default,,0000,0000,0000,,So x squared.
Dialogue: 0,0:41:11.35,0:41:13.62,Default,,0000,0000,0000,,I move everything to\Nthe left hand side.
Dialogue: 0,0:41:13.62,0:41:16.69,Default,,0000,0000,0000,,So I have 2x squared\Nplus 2y squared equals 8.
Dialogue: 0,0:41:16.69,0:41:23.37,Default,,0000,0000,0000,,
Dialogue: 0,0:41:23.37,0:41:29.02,Default,,0000,0000,0000,,And then I have z equals\Nx squared plus y squared.
Dialogue: 0,0:41:29.02,0:41:33.21,Default,,0000,0000,0000,,And then that's if and only\Nif x squared plus y squared
Dialogue: 0,0:41:33.21,0:41:35.97,Default,,0000,0000,0000,,equals 4.
Dialogue: 0,0:41:35.97,0:41:37.01,Default,,0000,0000,0000,,STUDENT: Then z equals 4.
Dialogue: 0,0:41:37.01,0:41:39.07,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NSo z equals 4.
Dialogue: 0,0:41:39.07,0:41:45.08,Default,,0000,0000,0000,,So z equals 4 is exactly what\NI guessed because come on.
Dialogue: 0,0:41:45.08,0:41:48.08,Default,,0000,0000,0000,,The two eggshells\Nhave to be equal.
Dialogue: 0,0:41:48.08,0:41:52.02,Default,,0000,0000,0000,,So this should be in the\Nmiddle between 0 and 8.
Dialogue: 0,0:41:52.02,0:41:54.43,Default,,0000,0000,0000,,So I knew it was z equals 4.
Dialogue: 0,0:41:54.43,0:41:56.28,Default,,0000,0000,0000,,But I had to check\Nit mathematically.
Dialogue: 0,0:41:56.28,0:41:59.22,Default,,0000,0000,0000,,So z equals 4, and x\Nsquared plus y squared
Dialogue: 0,0:41:59.22,0:42:02.45,Default,,0000,0000,0000,,equals 4 is the boundary.
Dialogue: 0,0:42:02.45,0:42:05.13,Default,,0000,0000,0000,,Let's make it purple\Nbecause it's the same
Dialogue: 0,0:42:05.13,0:42:06.87,Default,,0000,0000,0000,,as the purple equation there.
Dialogue: 0,0:42:06.87,0:42:09.46,Default,,0000,0000,0000,,
Dialogue: 0,0:42:09.46,0:42:15.92,Default,,0000,0000,0000,,So the domain has to be the\Nprojection of this purple--
Dialogue: 0,0:42:15.92,0:42:19.97,Default,,0000,0000,0000,,it looks like a sci-fi thing.
Dialogue: 0,0:42:19.97,0:42:21.70,Default,,0000,0000,0000,,You have some hologram.
Dialogue: 0,0:42:21.70,0:42:23.32,Default,,0000,0000,0000,,I don't know what it is.
Dialogue: 0,0:42:23.32,0:42:25.21,Default,,0000,0000,0000,,It's all in your imagination.
Dialogue: 0,0:42:25.21,0:42:28.40,Default,,0000,0000,0000,,You want to know the domain\ND. Could somebody tell me
Dialogue: 0,0:42:28.40,0:42:30.73,Default,,0000,0000,0000,,what the domain D will be?
Dialogue: 0,0:42:30.73,0:42:36.52,Default,,0000,0000,0000,,It will be those x's and y's\Non the floor with the quality
Dialogue: 0,0:42:36.52,0:42:42.01,Default,,0000,0000,0000,,that x squared plus y squared\Nwill be between 0 and--
Dialogue: 0,0:42:42.01,0:42:42.51,Default,,0000,0000,0000,,STUDENT: 4
Dialogue: 0,0:42:42.51,0:42:44.39,Default,,0000,0000,0000,,DR. MAGDALENA TODA: --4.
Dialogue: 0,0:42:44.39,0:42:47.40,Default,,0000,0000,0000,,So I can do everything\Nin polar coordinates.
Dialogue: 0,0:42:47.40,0:42:50.71,Default,,0000,0000,0000,,This is the same thing as\Nsaying rho, theta-- r, theta.
Dialogue: 0,0:42:50.71,0:42:51.57,Default,,0000,0000,0000,,Not rho.
Dialogue: 0,0:42:51.57,0:42:52.54,Default,,0000,0000,0000,,Rho is Greek.
Dialogue: 0,0:42:52.54,0:42:53.53,Default,,0000,0000,0000,,It's all Greek to me.
Dialogue: 0,0:42:53.53,0:42:57.25,Default,,0000,0000,0000,,So rho is sometimes\Nused by people
Dialogue: 0,0:42:57.25,0:43:01.89,Default,,0000,0000,0000,,for the polar coordinates,\Nrho and theta.
Dialogue: 0,0:43:01.89,0:43:05.14,Default,,0000,0000,0000,,But we use r.
Dialogue: 0,0:43:05.14,0:43:08.71,Default,,0000,0000,0000,,r squared between 0 and 4.
Dialogue: 0,0:43:08.71,0:43:10.00,Default,,0000,0000,0000,,You'll say, Magdalena, come on.
Dialogue: 0,0:43:10.00,0:43:10.60,Default,,0000,0000,0000,,That's silly.
Dialogue: 0,0:43:10.60,0:43:14.34,Default,,0000,0000,0000,,Why didn't you write\Nr between 0 and 2?
Dialogue: 0,0:43:14.34,0:43:15.61,Default,,0000,0000,0000,,I will.
Dialogue: 0,0:43:15.61,0:43:16.11,Default,,0000,0000,0000,,I will.
Dialogue: 0,0:43:16.11,0:43:16.70,Default,,0000,0000,0000,,I will.
Dialogue: 0,0:43:16.70,0:43:18.46,Default,,0000,0000,0000,,This is 2, right?
Dialogue: 0,0:43:18.46,0:43:20.71,Default,,0000,0000,0000,,So r between 0 and 2.
Dialogue: 0,0:43:20.71,0:43:23.00,Default,,0000,0000,0000,,I erase this.
Dialogue: 0,0:43:23.00,0:43:27.20,Default,,0000,0000,0000,,And theta is between 0 and 2 pi.
Dialogue: 0,0:43:27.20,0:43:28.04,Default,,0000,0000,0000,,And I'm done.
Dialogue: 0,0:43:28.04,0:43:28.66,Default,,0000,0000,0000,,Why 0 and 2 pi?
Dialogue: 0,0:43:28.66,0:43:30.10,Default,,0000,0000,0000,,Because we have the whole egg.
Dialogue: 0,0:43:30.10,0:43:32.99,Default,,0000,0000,0000,,I mean, I could\Ncut the egg in half
Dialogue: 0,0:43:32.99,0:43:37.20,Default,,0000,0000,0000,,and say 0 to pi or something,\Ninvent a different problem.
Dialogue: 0,0:43:37.20,0:43:39.72,Default,,0000,0000,0000,,But for the time\Nbeing, I'm rotating
Dialogue: 0,0:43:39.72,0:43:44.92,Default,,0000,0000,0000,,a full rotation of 2 pi to\Ncreate the egg all around.
Dialogue: 0,0:43:44.92,0:43:48.96,Default,,0000,0000,0000,,So finally, what is\Nthe volume of-- suppose
Dialogue: 0,0:43:48.96,0:43:55.98,Default,,0000,0000,0000,,this is like in the story\Nwith the golden eggs.
Dialogue: 0,0:43:55.98,0:43:58.15,Default,,0000,0000,0000,,They are solid gold eggs.
Dialogue: 0,0:43:58.15,0:43:59.70,Default,,0000,0000,0000,,Wouldn't that be wonderful?
Dialogue: 0,0:43:59.70,0:44:03.36,Default,,0000,0000,0000,,We want to know the\Nvolume of this golden egg.
Dialogue: 0,0:44:03.36,0:44:08.11,Default,,0000,0000,0000,,What's inside the solid egg, not\Nthe shell, not just the shell
Dialogue: 0,0:44:08.11,0:44:09.24,Default,,0000,0000,0000,,made of gold.
Dialogue: 0,0:44:09.24,0:44:12.12,Default,,0000,0000,0000,,The whole thing is made of gold.
Dialogue: 0,0:44:12.12,0:44:14.89,Default,,0000,0000,0000,,And who's coming tomorrow\Nto the-- sorry, guys.
Dialogue: 0,0:44:14.89,0:44:16.34,Default,,0000,0000,0000,,Mathematician talking.
Dialogue: 0,0:44:16.34,0:44:18.06,Default,,0000,0000,0000,,Switching from\Nanother-- who's coming
Dialogue: 0,0:44:18.06,0:44:20.31,Default,,0000,0000,0000,,tomorrow to the honors society?
Dialogue: 0,0:44:20.31,0:44:23.03,Default,,0000,0000,0000,,Do you-- did you decide?
Dialogue: 0,0:44:23.03,0:44:23.94,Default,,0000,0000,0000,,You have.
Dialogue: 0,0:44:23.94,0:44:26.12,Default,,0000,0000,0000,,And Rachel comes.
Dialogue: 0,0:44:26.12,0:44:27.35,Default,,0000,0000,0000,,Are you coming?
Dialogue: 0,0:44:27.35,0:44:28.26,Default,,0000,0000,0000,,No, no, no, no.
Dialogue: 0,0:44:28.26,0:44:29.64,Default,,0000,0000,0000,,Tomorrow night.
Dialogue: 0,0:44:29.64,0:44:30.26,Default,,0000,0000,0000,,Tomorrow night.
Dialogue: 0,0:44:30.26,0:44:31.55,Default,,0000,0000,0000,,Tomorrow.
Dialogue: 0,0:44:31.55,0:44:33.50,Default,,0000,0000,0000,,What time does that--
Dialogue: 0,0:44:33.50,0:44:34.13,Default,,0000,0000,0000,,STUDENT: 3:00.
Dialogue: 0,0:44:34.13,0:44:35.50,Default,,0000,0000,0000,,DR. MAGDALENA\NTODA: At 3 o'clock.
Dialogue: 0,0:44:35.50,0:44:36.91,Default,,0000,0000,0000,,At 3 o'clock.
Dialogue: 0,0:44:36.91,0:44:37.88,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:44:37.88,0:44:41.00,Default,,0000,0000,0000,,So if you want, I can\Npay your membership.
Dialogue: 0,0:44:41.00,0:44:44.58,Default,,0000,0000,0000,,And then you'll be members.
Dialogue: 0,0:44:44.58,0:44:46.97,Default,,0000,0000,0000,,I saw one of the certificates.
Dialogue: 0,0:44:46.97,0:44:48.47,Default,,0000,0000,0000,,It was really beautiful.
Dialogue: 0,0:44:48.47,0:44:49.85,Default,,0000,0000,0000,,That one [INAUDIBLE].
Dialogue: 0,0:44:49.85,0:44:53.18,Default,,0000,0000,0000,,It was really-- some\Nparents frame these things.
Dialogue: 0,0:44:53.18,0:44:54.67,Default,,0000,0000,0000,,My parents don't care.
Dialogue: 0,0:44:54.67,0:44:56.27,Default,,0000,0000,0000,,But I wish they cared.
Dialogue: 0,0:44:56.27,0:45:01.08,Default,,0000,0000,0000,,So the more certificates you\Nget, and the older you get,
Dialogue: 0,0:45:01.08,0:45:04.03,Default,,0000,0000,0000,,the nicer it is to put them,\Nframe them and put them
Dialogue: 0,0:45:04.03,0:45:06.06,Default,,0000,0000,0000,,on the wall of\Nfame of the family.
Dialogue: 0,0:45:06.06,0:45:09.45,Default,,0000,0000,0000,,This certificate, the KME\None, looks so much better
Dialogue: 0,0:45:09.45,0:45:15.41,Default,,0000,0000,0000,,than my own diplomas, the PhD\Ndiplomas, the math diplomas.
Dialogue: 0,0:45:15.41,0:45:20.11,Default,,0000,0000,0000,,And it's huge, and it\Nhas a golden silver seal
Dialogue: 0,0:45:20.11,0:45:22.10,Default,,0000,0000,0000,,will all the stuff.
Dialogue: 0,0:45:22.10,0:45:24.73,Default,,0000,0000,0000,,And it's really nice.
Dialogue: 0,0:45:24.73,0:45:25.64,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:45:25.64,0:45:29.39,Default,,0000,0000,0000,,Now coming back to this thing.
Dialogue: 0,0:45:29.39,0:45:32.23,Default,,0000,0000,0000,,
Dialogue: 0,0:45:32.23,0:45:34.44,Default,,0000,0000,0000,,STUDENT: Can we multiply by 2?
Dialogue: 0,0:45:34.44,0:45:35.07,Default,,0000,0000,0000,,Just find the--
Dialogue: 0,0:45:35.07,0:45:36.24,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Exactly.
Dialogue: 0,0:45:36.24,0:45:37.33,Default,,0000,0000,0000,,That's what we will do.
Dialogue: 0,0:45:37.33,0:45:43.53,Default,,0000,0000,0000,,We could set up the integral\Nfrom whatever it is.
Dialogue: 0,0:45:43.53,0:45:47.77,Default,,0000,0000,0000,,My one function to\Nanother function.
Dialogue: 0,0:45:47.77,0:45:51.07,Default,,0000,0000,0000,,But the simplest way\Nto compute the volume
Dialogue: 0,0:45:51.07,0:45:53.70,Default,,0000,0000,0000,,would be to say\Nthere are two types.
Dialogue: 0,0:45:53.70,0:45:57.59,Default,,0000,0000,0000,,And set up the\Nintegral for this one,
Dialogue: 0,0:45:57.59,0:46:01.84,Default,,0000,0000,0000,,for example or the other one.
Dialogue: 0,0:46:01.84,0:46:04.68,Default,,0000,0000,0000,,It doesn't matter which one.
Dialogue: 0,0:46:04.68,0:46:06.43,Default,,0000,0000,0000,,It doesn't really\Nmatter which one.
Dialogue: 0,0:46:06.43,0:46:06.96,Default,,0000,0000,0000,,Which one we would prefer?
Dialogue: 0,0:46:06.96,0:46:07.62,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:46:07.62,0:46:10.60,Default,,0000,0000,0000,,Maybe you like the bottom\Npart of the [INAUDIBLE].
Dialogue: 0,0:46:10.60,0:46:12.01,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:46:12.01,0:46:14.28,Default,,0000,0000,0000,,Do you guys understand\Nwhat I'm talking about?
Dialogue: 0,0:46:14.28,0:46:15.70,Default,,0000,0000,0000,,STUDENT: If we\Njust-- I don't know
Dialogue: 0,0:46:15.70,0:46:24.74,Default,,0000,0000,0000,,where to find B. Find the\Narea left, like indented?
Dialogue: 0,0:46:24.74,0:46:28.27,Default,,0000,0000,0000,,Because if you did it at the\Nbottom, the domain is zero.
Dialogue: 0,0:46:28.27,0:46:31.10,Default,,0000,0000,0000,,Then you have-- wouldn't\Nit find the stuff that
Dialogue: 0,0:46:31.10,0:46:34.48,Default,,0000,0000,0000,,was not cupped out, the edges?
Dialogue: 0,0:46:34.48,0:46:35.98,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NIsn't it exactly
Dialogue: 0,0:46:35.98,0:46:39.24,Default,,0000,0000,0000,,the same volume up and down?
Dialogue: 0,0:46:39.24,0:46:39.84,Default,,0000,0000,0000,,STUDENT: Yes.
Dialogue: 0,0:46:39.84,0:46:42.05,Default,,0000,0000,0000,,DR. MAGDALENA TODA: It's\Nthe same volume up and down.
Dialogue: 0,0:46:42.05,0:46:44.83,Default,,0000,0000,0000,,So it's enough for\Nme to take the volume
Dialogue: 0,0:46:44.83,0:46:47.88,Default,,0000,0000,0000,,of the lower part and w.
Dialogue: 0,0:46:47.88,0:46:51.61,Default,,0000,0000,0000,,Can you help me set\Nup the lower part?
Dialogue: 0,0:46:51.61,0:46:53.23,Default,,0000,0000,0000,,So I'm going to have two types.
Dialogue: 0,0:46:53.23,0:46:55.68,Default,,0000,0000,0000,,Can I do that directly\Nin polar coordinates?
Dialogue: 0,0:46:55.68,0:46:57.36,Default,,0000,0000,0000,,That's the thing.
Dialogue: 0,0:46:57.36,0:46:59.35,Default,,0000,0000,0000,,1 is 1.
Dialogue: 0,0:46:59.35,0:47:00.62,Default,,0000,0000,0000,,r is r.
Dialogue: 0,0:47:00.62,0:47:03.44,Default,,0000,0000,0000,,r is going to be-- this is\Nthe Jacobian r drd theta.
Dialogue: 0,0:47:03.44,0:47:10.20,Default,,0000,0000,0000,,
Dialogue: 0,0:47:10.20,0:47:12.97,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:47:12.97,0:47:21.10,Default,,0000,0000,0000,,But now let me ask you, how\Ndo we compute-- I'm sorry.
Dialogue: 0,0:47:21.10,0:47:31.85,Default,,0000,0000,0000,,This is the function f of x, y.
Dialogue: 0,0:47:31.85,0:47:34.43,Default,,0000,0000,0000,,
Dialogue: 0,0:47:34.43,0:47:37.77,Default,,0000,0000,0000,,Yeah, it's a little\Nbit more complicated.
Dialogue: 0,0:47:37.77,0:47:40.78,Default,,0000,0000,0000,,So you have to subtract\Nfrom one the other one.
Dialogue: 0,0:47:40.78,0:47:53.98,Default,,0000,0000,0000,,
Dialogue: 0,0:47:53.98,0:47:57.13,Default,,0000,0000,0000,,So I'm referring to the domain\Nas being only the planar
Dialogue: 0,0:47:57.13,0:47:57.63,Default,,0000,0000,0000,,domain.
Dialogue: 0,0:47:57.63,0:48:02.34,Default,,0000,0000,0000,,
Dialogue: 0,0:48:02.34,0:48:07.35,Default,,0000,0000,0000,,And I have first a graph\Nand then another graph.
Dialogue: 0,0:48:07.35,0:48:12.10,Default,,0000,0000,0000,,So when I want to compute,\Nforget about this part.
Dialogue: 0,0:48:12.10,0:48:18.00,Default,,0000,0000,0000,,I want to compute the\Nvolume of this, the volume
Dialogue: 0,0:48:18.00,0:48:22.33,Default,,0000,0000,0000,,of this egg, the inside.
Dialogue: 0,0:48:22.33,0:48:26.36,Default,,0000,0000,0000,,I have to say, OK, integral over\Nthe d of the function that's
Dialogue: 0,0:48:26.36,0:48:27.20,Default,,0000,0000,0000,,on top.
Dialogue: 0,0:48:27.20,0:48:31.64,Default,,0000,0000,0000,,The function that's on top\Nis the z equals f of x, y.
Dialogue: 0,0:48:31.64,0:48:34.79,Default,,0000,0000,0000,,And the function that's\Non the bottom for this egg
Dialogue: 0,0:48:34.79,0:48:37.11,Default,,0000,0000,0000,,is just this.
Dialogue: 0,0:48:37.11,0:48:40.63,Default,,0000,0000,0000,,So this is just\Na flat altitude g
Dialogue: 0,0:48:40.63,0:48:43.13,Default,,0000,0000,0000,,of x, y equals-- what is that?
Dialogue: 0,0:48:43.13,0:48:45.92,Default,,0000,0000,0000,,4.
Dialogue: 0,0:48:45.92,0:48:53.95,Default,,0000,0000,0000,,So I have to subtract the two\Nbecause I have first this body.
Dialogue: 0,0:48:53.95,0:48:57.58,Default,,0000,0000,0000,,If this would not exist, how\Nwould I get the purple part?
Dialogue: 0,0:48:57.58,0:49:03.03,Default,,0000,0000,0000,,I would say for the function f,\Nthe protection on the ground,
Dialogue: 0,0:49:03.03,0:49:08.69,Default,,0000,0000,0000,,I have this whole body\Nthat looks like a crayon.
Dialogue: 0,0:49:08.69,0:49:10.85,Default,,0000,0000,0000,,A whole body that\Nlooks like crayon.
Dialogue: 0,0:49:10.85,0:49:13.14,Default,,0000,0000,0000,,This is the first integral.
Dialogue: 0,0:49:13.14,0:49:18.32,Default,,0000,0000,0000,,I minus the cylinder\Nthat's dotted with floating
Dialogue: 0,0:49:18.32,0:49:20.64,Default,,0000,0000,0000,,points, which is this part.
Dialogue: 0,0:49:20.64,0:49:22.88,Default,,0000,0000,0000,,So it's V1 minus V2.
Dialogue: 0,0:49:22.88,0:49:27.14,Default,,0000,0000,0000,,V1 is the volume of the whole\Nbody that looks like a crayon.
Dialogue: 0,0:49:27.14,0:49:32.28,Default,,0000,0000,0000,,V2 is just the volume of the\Ncylinder under the crayon.
Dialogue: 0,0:49:32.28,0:49:35.64,Default,,0000,0000,0000,,We want-- minus\NV2 is exactly half
Dialogue: 0,0:49:35.64,0:49:41.74,Default,,0000,0000,0000,,of the egg, the volume of the\Nhalf of the egg, give or take.
Dialogue: 0,0:49:41.74,0:49:43.11,Default,,0000,0000,0000,,So is this hard?
Dialogue: 0,0:49:43.11,0:49:44.97,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,0:49:44.97,0:49:49.98,Default,,0000,0000,0000,,f of x, y-- can you guys\Ntell me who that is?
Dialogue: 0,0:49:49.98,0:49:54.14,Default,,0000,0000,0000,,A minus x squared\Nminus y squared.
Dialogue: 0,0:49:54.14,0:49:55.22,Default,,0000,0000,0000,,And who is g?
Dialogue: 0,0:49:55.22,0:49:58.54,Default,,0000,0000,0000,,
Dialogue: 0,0:49:58.54,0:49:59.16,Default,,0000,0000,0000,,4.
Dialogue: 0,0:49:59.16,0:50:01.96,Default,,0000,0000,0000,,Just the altitude, 4.
Dialogue: 0,0:50:01.96,0:50:02.59,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:50:02.59,0:50:05.41,Default,,0000,0000,0000,,So I'm going to go\Nahead and say, OK,
Dialogue: 0,0:50:05.41,0:50:10.61,Default,,0000,0000,0000,,I have to integrate 2 double\Nintegral over D. 8 minus 4
Dialogue: 0,0:50:10.61,0:50:17.91,Default,,0000,0000,0000,,is 4 minus x squared\Nminus y squared dA.
Dialogue: 0,0:50:17.91,0:50:23.76,Default,,0000,0000,0000,,dA is the area element dxdy.
Dialogue: 0,0:50:23.76,0:50:27.52,Default,,0000,0000,0000,,Now switch to polar.
Dialogue: 0,0:50:27.52,0:50:28.88,Default,,0000,0000,0000,,How do you switch to polar?
Dialogue: 0,0:50:28.88,0:50:32.76,Default,,0000,0000,0000,,
Dialogue: 0,0:50:32.76,0:50:34.78,Default,,0000,0000,0000,,You can also set this\Nup as a triple integral.
Dialogue: 0,0:50:34.78,0:50:36.93,Default,,0000,0000,0000,,And that's what I\Nwanted to do at first.
Dialogue: 0,0:50:36.93,0:50:39.85,Default,,0000,0000,0000,,But then I realized that you\Ndon't know triple integrals,
Dialogue: 0,0:50:39.85,0:50:42.27,Default,,0000,0000,0000,,so I set it up as\Na double integral.
Dialogue: 0,0:50:42.27,0:50:44.38,Default,,0000,0000,0000,,For a triple integral,\Nyou have three snakes.
Dialogue: 0,0:50:44.38,0:50:46.67,Default,,0000,0000,0000,,And you integrate the\Nelement 1, and that's
Dialogue: 0,0:50:46.67,0:50:47.63,Default,,0000,0000,0000,,going to be the volume.
Dialogue: 0,0:50:47.63,0:50:51.95,Default,,0000,0000,0000,,And I'll teach you in\Nthe next two sessions.
Dialogue: 0,0:50:51.95,0:50:55.63,Default,,0000,0000,0000,,2 times the double integral.
Dialogue: 0,0:50:55.63,0:50:56.97,Default,,0000,0000,0000,,Who is this nice fellow?
Dialogue: 0,0:50:56.97,0:50:59.42,Default,,0000,0000,0000,,Look how nice and sassy he is.
Dialogue: 0,0:50:59.42,0:51:08.80,Default,,0000,0000,0000,,4 minus r squared times--\Nnever forget the r drd theta.
Dialogue: 0,0:51:08.80,0:51:15.40,Default,,0000,0000,0000,,Theta goes between 0\Nand 2 pi and r between--
Dialogue: 0,0:51:15.40,0:51:16.56,Default,,0000,0000,0000,,STUDENT: 0 and 2.
Dialogue: 0,0:51:16.56,0:51:17.88,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 0 and--
Dialogue: 0,0:51:17.88,0:51:18.38,Default,,0000,0000,0000,,STUDENT: 2.
Dialogue: 0,0:51:18.38,0:51:19.01,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 2.
Dialogue: 0,0:51:19.01,0:51:19.51,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:51:19.51,0:51:24.82,Default,,0000,0000,0000,,Because when I had 4 here,\Nthat's the radius squared.
Dialogue: 0,0:51:24.82,0:51:27.05,Default,,0000,0000,0000,,So r is 2.
Dialogue: 0,0:51:27.05,0:51:28.21,Default,,0000,0000,0000,,Look at this integral.
Dialogue: 0,0:51:28.21,0:51:31.16,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:51:31.16,0:51:32.55,Default,,0000,0000,0000,,Not so hard.
Dialogue: 0,0:51:32.55,0:51:33.70,Default,,0000,0000,0000,,Not so hard at all.
Dialogue: 0,0:51:33.70,0:51:36.69,Default,,0000,0000,0000,,So what would you\Ndo if you were me?
Dialogue: 0,0:51:36.69,0:51:38.43,Default,,0000,0000,0000,,Would you do a u substitution?
Dialogue: 0,0:51:38.43,0:51:40.77,Default,,0000,0000,0000,,Do you need a u\Nsubstitution necessarily?
Dialogue: 0,0:51:40.77,0:51:42.07,Default,,0000,0000,0000,,You don't need it.
Dialogue: 0,0:51:42.07,0:51:47.62,Default,,0000,0000,0000,,So just say 4r minus r cubed.
Dialogue: 0,0:51:47.62,0:51:49.87,Default,,0000,0000,0000,,Now what do you see again?
Dialogue: 0,0:51:49.87,0:51:52.36,Default,,0000,0000,0000,,Theta is missing\Nfrom the picture.
Dialogue: 0,0:51:52.36,0:51:54.02,Default,,0000,0000,0000,,Theta says, I'm out of here.
Dialogue: 0,0:51:54.02,0:51:54.66,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,0:51:54.66,0:52:00.14,Default,,0000,0000,0000,,So you get 2 times the integral\Nfrom 0 to 2 pi of nothing--
Dialogue: 0,0:52:00.14,0:52:04.02,Default,,0000,0000,0000,,well, of 1d theta, not of\Nnothing-- times the integral
Dialogue: 0,0:52:04.02,0:52:15.90,Default,,0000,0000,0000,,from 0 to 2 of 4r\Nminus r cubed dr.
Dialogue: 0,0:52:15.90,0:52:21.78,Default,,0000,0000,0000,,4r minus r cubed dr, the\Nintegral from 0 to 2.
Dialogue: 0,0:52:21.78,0:52:22.28,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:52:22.28,0:52:31.70,Default,,0000,0000,0000,,
Dialogue: 0,0:52:31.70,0:52:32.65,Default,,0000,0000,0000,,Who's going to help me?
Dialogue: 0,0:52:32.65,0:52:35.37,Default,,0000,0000,0000,,
Dialogue: 0,0:52:35.37,0:52:39.33,Default,,0000,0000,0000,,I give you how\Nmuch money-- money.
Dialogue: 0,0:52:39.33,0:52:43.42,Default,,0000,0000,0000,,Time shall I give\Nyou to do this one?
Dialogue: 0,0:52:43.42,0:52:46.49,Default,,0000,0000,0000,,And I need three people\Nto respond and get
Dialogue: 0,0:52:46.49,0:52:47.55,Default,,0000,0000,0000,,the same answer.
Dialogue: 0,0:52:47.55,0:52:53.91,Default,,0000,0000,0000,,So [INAUDIBLE] 2r squared\Nminus r to the 4 over 4
Dialogue: 0,0:52:53.91,0:52:56.05,Default,,0000,0000,0000,,between 0 and 2.
Dialogue: 0,0:52:56.05,0:52:59.62,Default,,0000,0000,0000,,Can you do it please?
Dialogue: 0,0:52:59.62,0:53:01.91,Default,,0000,0000,0000,,STUDENT: 16 [INAUDIBLE].
Dialogue: 0,0:53:01.91,0:53:03.62,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NHow much did you get?
Dialogue: 0,0:53:03.62,0:53:04.23,Default,,0000,0000,0000,,STUDENT: 4.
Dialogue: 0,0:53:04.23,0:53:05.48,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NHow much did you get?
Dialogue: 0,0:53:05.48,0:53:06.37,Default,,0000,0000,0000,,STUDENT: For just this one?
Dialogue: 0,0:53:06.37,0:53:07.74,Default,,0000,0000,0000,,DR. MAGDALENA\NTODA: For all this.
Dialogue: 0,0:53:07.74,0:53:10.18,Default,,0000,0000,0000,,
Dialogue: 0,0:53:10.18,0:53:10.68,Default,,0000,0000,0000,,STUDENT: 4.
Dialogue: 0,0:53:10.68,0:53:18.09,Default,,0000,0000,0000,,
Dialogue: 0,0:53:18.09,0:53:20.12,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYes, it is, right?
Dialogue: 0,0:53:20.12,0:53:21.08,Default,,0000,0000,0000,,Are you with me?
Dialogue: 0,0:53:21.08,0:53:23.01,Default,,0000,0000,0000,,You have 2r squared\Nwhen you integrate
Dialogue: 0,0:53:23.01,0:53:26.43,Default,,0000,0000,0000,,minus r to the 4 over\N4 between 0 and 2.
Dialogue: 0,0:53:26.43,0:53:30.68,Default,,0000,0000,0000,,That means 2 times 4 minus 16/4.
Dialogue: 0,0:53:30.68,0:53:32.65,Default,,0000,0000,0000,,8 minus 4 is 4.
Dialogue: 0,0:53:32.65,0:53:37.90,Default,,0000,0000,0000,,So with 4 for this guy, 2 pi\Nfor this guy, and one 2 outside,
Dialogue: 0,0:53:37.90,0:53:42.84,Default,,0000,0000,0000,,you have 16 pi.
Dialogue: 0,0:53:42.84,0:53:46.77,Default,,0000,0000,0000,,And that was-- I remember\Nit as if it was yesterday.
Dialogue: 0,0:53:46.77,0:53:55.20,Default,,0000,0000,0000,,That was on a final\Ntwo or three years ago.
Dialogue: 0,0:53:55.20,0:53:57.73,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:53:57.73,0:54:03.99,Default,,0000,0000,0000,,So you've seen many\Nof these problems now.
Dialogue: 0,0:54:03.99,0:54:07.31,Default,,0000,0000,0000,,It shouldn't be complicated\Nto start your homework.
Dialogue: 0,0:54:07.31,0:54:08.41,Default,,0000,0000,0000,,Go ahead.
Dialogue: 0,0:54:08.41,0:54:10.59,Default,,0000,0000,0000,,If you want, go ahead and\Nstart with the problems
Dialogue: 0,0:54:10.59,0:54:12.98,Default,,0000,0000,0000,,that we did today.
Dialogue: 0,0:54:12.98,0:54:16.00,Default,,0000,0000,0000,,And when you see numbers\Nchanged or something,
Dialogue: 0,0:54:16.00,0:54:17.92,Default,,0000,0000,0000,,go ahead and work the\Nproblem the same way.
Dialogue: 0,0:54:17.92,0:54:20.30,Default,,0000,0000,0000,,Make sure you understood it.
Dialogue: 0,0:54:20.30,0:54:22.16,Default,,0000,0000,0000,,I'm going to do more.
Dialogue: 0,0:54:22.16,0:54:23.41,Default,,0000,0000,0000,,Is this useful for you?
Dialogue: 0,0:54:23.41,0:54:25.37,Default,,0000,0000,0000,,I mean-- OK.
Dialogue: 0,0:54:25.37,0:54:27.17,Default,,0000,0000,0000,,So you agree that\Nevery now and then,
Dialogue: 0,0:54:27.17,0:54:31.04,Default,,0000,0000,0000,,we do homework in the classroom?
Dialogue: 0,0:54:31.04,0:54:33.37,Default,,0000,0000,0000,,Homework like problems\Nin the classroom.
Dialogue: 0,0:54:33.37,0:54:35.76,Default,,0000,0000,0000,,In the homework, you\Nmay have different data,
Dialogue: 0,0:54:35.76,0:54:38.83,Default,,0000,0000,0000,,but it's the same\Ntype of problem.
Dialogue: 0,0:54:38.83,0:54:39.33,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:54:39.33,0:54:45.29,Default,,0000,0000,0000,,
Dialogue: 0,0:54:45.29,0:54:49.76,Default,,0000,0000,0000,,I'm going to remind you\Nof some Calc 2 notions
Dialogue: 0,0:54:49.76,0:54:57.15,Default,,0000,0000,0000,,because today I will\Ncover the surface area.
Dialogue: 0,0:54:57.15,0:54:57.93,Default,,0000,0000,0000,,STUDENT: Dr. Toda?
Dialogue: 0,0:54:57.93,0:54:58.30,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Yes, sir?
Dialogue: 0,0:54:58.30,0:54:59.72,Default,,0000,0000,0000,,STUDENT: I have a question\Non the last problem.
Dialogue: 0,0:54:59.72,0:55:00.63,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Yes, sir?
Dialogue: 0,0:55:00.63,0:55:02.92,Default,,0000,0000,0000,,STUDENT: If we had seen\Nsomething like that on the exam
Dialogue: 0,0:55:02.92,0:55:06.79,Default,,0000,0000,0000,,and had done it using the fact\Nthat it's a solid revolution--
Dialogue: 0,0:55:06.79,0:55:08.88,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYeah, you can do that.
Dialogue: 0,0:55:08.88,0:55:11.01,Default,,0000,0000,0000,,There are at least four\Nmethods to do this problem.
Dialogue: 0,0:55:11.01,0:55:12.43,Default,,0000,0000,0000,,One would be with\Ntriple integral.
Dialogue: 0,0:55:12.43,0:55:14.27,Default,,0000,0000,0000,,One would be with\Na double integral
Dialogue: 0,0:55:14.27,0:55:17.08,Default,,0000,0000,0000,,of a function on top\Nminus the function below.
Dialogue: 0,0:55:17.08,0:55:21.04,Default,,0000,0000,0000,,One would be with solid of\Nrevolution like in Calc 2,
Dialogue: 0,0:55:21.04,0:55:24.49,Default,,0000,0000,0000,,where your axis is the z axis.
Dialogue: 0,0:55:24.49,0:55:26.37,Default,,0000,0000,0000,,I don't care how you\Nsolve the problem.
Dialogue: 0,0:55:26.37,0:55:28.60,Default,,0000,0000,0000,,Again, if I were\Nthe CEO of a company
Dialogue: 0,0:55:28.60,0:55:31.13,Default,,0000,0000,0000,,or the boss of a\Nfirm or something,
Dialogue: 0,0:55:31.13,0:55:36.57,Default,,0000,0000,0000,,I would care for my employees to\Nbe solving problems the fastest
Dialogue: 0,0:55:36.57,0:55:37.53,Default,,0000,0000,0000,,possible way.
Dialogue: 0,0:55:37.53,0:55:39.54,Default,,0000,0000,0000,,As long as the\Nanswer is correct,
Dialogue: 0,0:55:39.54,0:55:40.99,Default,,0000,0000,0000,,I don't care how you do it.
Dialogue: 0,0:55:40.99,0:55:42.42,Default,,0000,0000,0000,,STUDENT: Thank you, Doctor.
Dialogue: 0,0:55:42.42,0:55:43.78,Default,,0000,0000,0000,,DR. MAGDALENA TODA: So go ahead.
Dialogue: 0,0:55:43.78,0:55:44.28,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:55:44.28,0:55:47.02,Default,,0000,0000,0000,,
Dialogue: 0,0:55:47.02,0:56:00.36,Default,,0000,0000,0000,,Oh, and by the way, I want to\Ngive you another example where
Dialogue: 0,0:56:00.36,0:56:03.46,Default,,0000,0000,0000,,the students were able\Nto very beautifully cheat
Dialogue: 0,0:56:03.46,0:56:06.13,Default,,0000,0000,0000,,and get the right answer.
Dialogue: 0,0:56:06.13,0:56:08.63,Default,,0000,0000,0000,,That was funny.
Dialogue: 0,0:56:08.63,0:56:12.47,Default,,0000,0000,0000,,But that is again\Na Calc 3 problem
Dialogue: 0,0:56:12.47,0:56:17.42,Default,,0000,0000,0000,,in an elementary way\Nthat can be solved
Dialogue: 0,0:56:17.42,0:56:22.67,Default,,0000,0000,0000,,with the notions you have from\NK-12, if you mastered them
Dialogue: 0,0:56:22.67,0:56:24.64,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,0:56:24.64,0:56:28.97,Default,,0000,0000,0000,,So you are given x\Nplus y plus z equals 1.
Dialogue: 0,0:56:28.97,0:56:31.28,Default,,0000,0000,0000,,Before I do the\Nsurface integral--
Dialogue: 0,0:56:31.28,0:56:34.07,Default,,0000,0000,0000,,I could do the surface\Nintegral for such a problem.
Dialogue: 0,0:56:34.07,0:56:49.49,Default,,0000,0000,0000,,This is a plane that intersects\Nthe coordinate planes
Dialogue: 0,0:56:49.49,0:56:55.11,Default,,0000,0000,0000,,and forms a\Ntetrahedron with them.
Dialogue: 0,0:56:55.11,0:57:04.80,Default,,0000,0000,0000,,
Dialogue: 0,0:57:04.80,0:57:07.02,Default,,0000,0000,0000,,Find the volume of\Nthat tetrahedron.
Dialogue: 0,0:57:07.02,0:57:18.79,Default,,0000,0000,0000,,
Dialogue: 0,0:57:18.79,0:57:25.55,Default,,0000,0000,0000,,Now I say, with Calc 3,\Nbecause the course coordinator
Dialogue: 0,0:57:25.55,0:57:30.09,Default,,0000,0000,0000,,several years ago did not\Nspecify with what you learned.
Dialogue: 0,0:57:30.09,0:57:33.19,Default,,0000,0000,0000,,Set up a double integral\Nor set up-- he simply
Dialogue: 0,0:57:33.19,0:57:35.62,Default,,0000,0000,0000,,said, find the volume.
Dialogue: 0,0:57:35.62,0:57:39.72,Default,,0000,0000,0000,,So the students-- what's\Nthe simplest way to do it?
Dialogue: 0,0:57:39.72,0:57:41.10,Default,,0000,0000,0000,,STUDENT: That's\Njust half a cube.
Dialogue: 0,0:57:41.10,0:57:44.96,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NJust draw the thingy.
Dialogue: 0,0:57:44.96,0:57:46.05,Default,,0000,0000,0000,,And they were smart.
Dialogue: 0,0:57:46.05,0:57:47.43,Default,,0000,0000,0000,,They knew how to draw it.
Dialogue: 0,0:57:47.43,0:57:51.01,Default,,0000,0000,0000,,The knew what the vertices were.
Dialogue: 0,0:57:51.01,0:57:52.86,Default,,0000,0000,0000,,The plane looks like this.
Dialogue: 0,0:57:52.86,0:57:56.98,Default,,0000,0000,0000,,If you shade it, you see\Nthat it's x plus y plus z.
Dialogue: 0,0:57:56.98,0:58:00.02,Default,,0000,0000,0000,,And I'm going to try\Nand write with my hands.
Dialogue: 0,0:58:00.02,0:58:01.07,Default,,0000,0000,0000,,It's very hard.
Dialogue: 0,0:58:01.07,0:58:04.07,Default,,0000,0000,0000,,But it comes from 0, 0, 1 point.
Dialogue: 0,0:58:04.07,0:58:06.01,Default,,0000,0000,0000,,This is the 0, 0, 1.
Dialogue: 0,0:58:06.01,0:58:07.87,Default,,0000,0000,0000,,And it comes like that.
Dialogue: 0,0:58:07.87,0:58:11.20,Default,,0000,0000,0000,,And it hits the floor over here.
Dialogue: 0,0:58:11.20,0:58:16.14,Default,,0000,0000,0000,,And these points are 1,\N0, 0; 0, 1, 0; and 0, 0,
Dialogue: 0,0:58:16.14,0:58:19.82,Default,,0000,0000,0000,,1 on the vertices\Nof a tetrahedron,
Dialogue: 0,0:58:19.82,0:58:22.11,Default,,0000,0000,0000,,including the origin.
Dialogue: 0,0:58:22.11,0:58:24.62,Default,,0000,0000,0000,,How do I know those are\Nexactly the vertices
Dialogue: 0,0:58:24.62,0:58:27.10,Default,,0000,0000,0000,,of the tetrahedron?
Dialogue: 0,0:58:27.10,0:58:30.42,Default,,0000,0000,0000,,Because they verify x\Nplus y plus z equals 1.
Dialogue: 0,0:58:30.42,0:58:33.01,Default,,0000,0000,0000,,As long as the point\Nverifies the equation,
Dialogue: 0,0:58:33.01,0:58:35.27,Default,,0000,0000,0000,,it is in the plane.
Dialogue: 0,0:58:35.27,0:58:38.04,Default,,0000,0000,0000,,For example, another point\Nthat's not in the picture
Dialogue: 0,0:58:38.04,0:58:40.34,Default,,0000,0000,0000,,would be 1/3 plus 1/3 plus 1/3.
Dialogue: 0,0:58:40.34,0:58:41.85,Default,,0000,0000,0000,,1/3 and 1/3 is 1/3.
Dialogue: 0,0:58:41.85,0:58:46.59,Default,,0000,0000,0000,,Anything that verifies the\Nequation is in the plane.
Dialogue: 0,0:58:46.59,0:58:48.63,Default,,0000,0000,0000,,So the tetrahedron has a name.
Dialogue: 0,0:58:48.63,0:58:56.93,Default,,0000,0000,0000,,It's called-- let's call\Nthis A, B, C, and O. OABC.
Dialogue: 0,0:58:56.93,0:58:57.84,Default,,0000,0000,0000,,It's a tetrahedron.
Dialogue: 0,0:58:57.84,0:59:00.30,Default,,0000,0000,0000,,It's a pyramid.
Dialogue: 0,0:59:00.30,0:59:06.80,Default,,0000,0000,0000,,So how does the smart\Nstudent who was not given
Dialogue: 0,0:59:06.80,0:59:09.40,Default,,0000,0000,0000,,a specific method solve that?
Dialogue: 0,0:59:09.40,0:59:10.52,Default,,0000,0000,0000,,They did that on the final.
Dialogue: 0,0:59:10.52,0:59:12.05,Default,,0000,0000,0000,,I'm so proud of them.
Dialogue: 0,0:59:12.05,0:59:12.91,Default,,0000,0000,0000,,I said, come on now.
Dialogue: 0,0:59:12.91,0:59:14.33,Default,,0000,0000,0000,,The final is two\Nhours and a half.
Dialogue: 0,0:59:14.33,0:59:15.79,Default,,0000,0000,0000,,You don't know what to do first.
Dialogue: 0,0:59:15.79,0:59:22.15,Default,,0000,0000,0000,,So they said-- they\Ndid the base multiplied
Dialogue: 0,0:59:22.15,0:59:25.04,Default,,0000,0000,0000,,by the height divided by 3.
Dialogue: 0,0:59:25.04,0:59:29.69,Default,,0000,0000,0000,,So you get 1 times 1.
Dialogue: 0,0:59:29.69,0:59:31.95,Default,,0000,0000,0000,,So practically, divided by 2.
Dialogue: 0,0:59:31.95,0:59:32.79,Default,,0000,0000,0000,,1/2.
Dialogue: 0,0:59:32.79,0:59:35.87,Default,,0000,0000,0000,,You don't even have\Nto do the-- even
Dialogue: 0,0:59:35.87,0:59:40.79,Default,,0000,0000,0000,,my son would know that this\Nis half of a square, a 1
Dialogue: 0,0:59:40.79,0:59:42.30,Default,,0000,0000,0000,,by 1 square.
Dialogue: 0,0:59:42.30,0:59:46.37,Default,,0000,0000,0000,,So it's half the area of the\Nbase times the height, which
Dialogue: 0,0:59:46.37,0:59:49.41,Default,,0000,0000,0000,,is 1, divided by 3 is 1/6.
Dialogue: 0,0:59:49.41,0:59:52.18,Default,,0000,0000,0000,,And goodbye and see you later.
Dialogue: 0,0:59:52.18,0:59:56.00,Default,,0000,0000,0000,,But if you wanted-- if\Nthe author of the problem
Dialogue: 0,0:59:56.00,0:59:58.37,Default,,0000,0000,0000,,would indicate, do\Nthat with Calculus 3,
Dialogue: 0,0:59:58.37,1:00:01.41,Default,,0000,0000,0000,,then that's another\Nstory because you
Dialogue: 0,1:00:01.41,1:00:06.48,Default,,0000,0000,0000,,have to realize what the domain\Nwould be, the planar domain.
Dialogue: 0,1:00:06.48,1:00:09.35,Default,,0000,0000,0000,,You practically have a surface.
Dialogue: 0,1:00:09.35,1:00:11.73,Default,,0000,0000,0000,,The green-shaded\Nequilateral triangle
Dialogue: 0,1:00:11.73,1:00:16.25,Default,,0000,0000,0000,,is your surface, which-- let's\Ncall it c of f from surface.
Dialogue: 0,1:00:16.25,1:00:20.21,Default,,0000,0000,0000,,But this would be\Nz equals f of x, y.
Dialogue: 0,1:00:20.21,1:00:22.01,Default,,0000,0000,0000,,How do you get to that?
Dialogue: 0,1:00:22.01,1:00:23.55,Default,,0000,0000,0000,,You get it from here.
Dialogue: 0,1:00:23.55,1:00:27.61,Default,,0000,0000,0000,,The explicit equation\Nis-- [INAUDIBLE].
Dialogue: 0,1:00:27.61,1:00:29.30,Default,,0000,0000,0000,,1 minus x minus y.
Dialogue: 0,1:00:29.30,1:00:33.05,Default,,0000,0000,0000,,That is the surface,\Nthe green surface.
Dialogue: 0,1:00:33.05,1:00:35.15,Default,,0000,0000,0000,,And the domain--\Nlet's draw that in.
Dialogue: 0,1:00:35.15,1:00:39.28,Default,,0000,0000,0000,,Do you prefer red or purple?
Dialogue: 0,1:00:39.28,1:00:39.90,Default,,0000,0000,0000,,You don't care?
Dialogue: 0,1:00:39.90,1:00:44.50,Default,,0000,0000,0000,,
Dialogue: 0,1:00:44.50,1:00:46.88,Default,,0000,0000,0000,,OK, I'll take red.
Dialogue: 0,1:00:46.88,1:00:47.38,Default,,0000,0000,0000,,Red.
Dialogue: 0,1:00:47.38,1:00:51.37,Default,,0000,0000,0000,,
Dialogue: 0,1:00:51.37,1:00:52.43,Default,,0000,0000,0000,,Red.
Dialogue: 0,1:00:52.43,1:00:56.97,Default,,0000,0000,0000,,That's the domain D. So you'll\Nhave to set up I, integral.
Dialogue: 0,1:00:56.97,1:01:04.64,Default,,0000,0000,0000,,I for an I. And volume, double\Nintegral over D of f of x, y,
Dialogue: 0,1:01:04.64,1:01:07.95,Default,,0000,0000,0000,,whatever that is, dA.
Dialogue: 0,1:01:07.95,1:01:11.25,Default,,0000,0000,0000,,That's going to be-- who is D?
Dialogue: 0,1:01:11.25,1:01:13.72,Default,,0000,0000,0000,,Somebody help me, OK?
Dialogue: 0,1:01:13.72,1:01:15.28,Default,,0000,0000,0000,,That's not easy.
Dialogue: 0,1:01:15.28,1:01:19.56,Default,,0000,0000,0000,,So to draw the domain D, I have\Nto have a little bit of skill,
Dialogue: 0,1:01:19.56,1:01:23.24,Default,,0000,0000,0000,,if I don't have any skill, I\Ndon't belong in this class.
Dialogue: 0,1:01:23.24,1:01:24.55,Default,,0000,0000,0000,,What do I have to draw?
Dialogue: 0,1:01:24.55,1:01:26.34,Default,,0000,0000,0000,,Guys, tell me what to do.
Dialogue: 0,1:01:26.34,1:01:29.28,Default,,0000,0000,0000,,0, x, and y.
Dialogue: 0,1:01:29.28,1:01:37.03,Default,,0000,0000,0000,,To draw z, 0, z equals 0 gives\Nme x plus y equals 1, right?
Dialogue: 0,1:01:37.03,1:01:38.91,Default,,0000,0000,0000,,So this is the floor.
Dialogue: 0,1:01:38.91,1:01:42.10,Default,,0000,0000,0000,,Guys, this is the floor.
Dialogue: 0,1:01:42.10,1:01:43.78,Default,,0000,0000,0000,,So why don't I shade it?
Dialogue: 0,1:01:43.78,1:01:45.95,Default,,0000,0000,0000,,Because I'm not sure\Nwhich one I want.
Dialogue: 0,1:01:45.95,1:01:48.46,Default,,0000,0000,0000,,Do I want vertical strips\Nor horizontal strips?
Dialogue: 0,1:01:48.46,1:01:49.27,Default,,0000,0000,0000,,You're the boss.
Dialogue: 0,1:01:49.27,1:01:51.57,Default,,0000,0000,0000,,You tell me what I want.
Dialogue: 0,1:01:51.57,1:01:54.97,Default,,0000,0000,0000,,So do you want vertical strips?
Dialogue: 0,1:01:54.97,1:01:58.37,Default,,0000,0000,0000,,
Dialogue: 0,1:01:58.37,1:02:01.30,Default,,0000,0000,0000,,Let's draw vertical strips.
Dialogue: 0,1:02:01.30,1:02:03.46,Default,,0000,0000,0000,,So how do I\Nrepresent this domain
Dialogue: 0,1:02:03.46,1:02:05.55,Default,,0000,0000,0000,,from the vertical strips?
Dialogue: 0,1:02:05.55,1:02:09.47,Default,,0000,0000,0000,,x is between 0 and 1.
Dialogue: 0,1:02:09.47,1:02:13.61,Default,,0000,0000,0000,,These are fixed\Nvariable values of x
Dialogue: 0,1:02:13.61,1:02:15.46,Default,,0000,0000,0000,,between fixed values 0 and 1.
Dialogue: 0,1:02:15.46,1:02:21.13,Default,,0000,0000,0000,,For any such blue choice\Nof a point, I have a strip,
Dialogue: 0,1:02:21.13,1:02:27.89,Default,,0000,0000,0000,,a vertical strip that goes\Nfrom y equals 0 down to--
Dialogue: 0,1:02:27.89,1:02:28.68,Default,,0000,0000,0000,,STUDENT: 1 minus x.
Dialogue: 0,1:02:28.68,1:02:31.28,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\N--1 minus x up.
Dialogue: 0,1:02:31.28,1:02:33.24,Default,,0000,0000,0000,,Excellent, excellent.
Dialogue: 0,1:02:33.24,1:02:37.38,Default,,0000,0000,0000,,This is exactly-- Roberto, you\Nwere the one who said that?
Dialogue: 0,1:02:37.38,1:02:38.25,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:02:38.25,1:02:39.22,Default,,0000,0000,0000,,So this is the domain.
Dialogue: 0,1:02:39.22,1:02:41.63,Default,,0000,0000,0000,,So how do we write it down?
Dialogue: 0,1:02:41.63,1:02:43.30,Default,,0000,0000,0000,,0 to 1.
Dialogue: 0,1:02:43.30,1:02:46.14,Default,,0000,0000,0000,,0 to 1 minus x.
Dialogue: 0,1:02:46.14,1:02:47.35,Default,,0000,0000,0000,,That is what I want to write.
Dialogue: 0,1:02:47.35,1:02:48.66,Default,,0000,0000,0000,,No polar coordinates.
Dialogue: 0,1:02:48.66,1:02:49.29,Default,,0000,0000,0000,,Goodbye.
Dialogue: 0,1:02:49.29,1:02:50.86,Default,,0000,0000,0000,,There is no problem.
Dialogue: 0,1:02:50.86,1:02:52.70,Default,,0000,0000,0000,,This is all a typical\NCartesian problem.
Dialogue: 0,1:02:52.70,1:02:55.62,Default,,0000,0000,0000,,
Dialogue: 0,1:02:55.62,1:02:59.10,Default,,0000,0000,0000,,f-- f.
Dialogue: 0,1:02:59.10,1:03:03.57,Default,,0000,0000,0000,,f is 1 minus x minus\Ny, thank you very much.
Dialogue: 0,1:03:03.57,1:03:07.59,Default,,0000,0000,0000,,This is f dydx.
Dialogue: 0,1:03:07.59,1:03:12.12,Default,,0000,0000,0000,,
Dialogue: 0,1:03:12.12,1:03:16.81,Default,,0000,0000,0000,,Homework, get 1/6.
Dialogue: 0,1:03:16.81,1:03:20.13,Default,,0000,0000,0000,,
Dialogue: 0,1:03:20.13,1:03:25.64,Default,,0000,0000,0000,,So trying to do\Nthat and get a 1/6.
Dialogue: 0,1:03:25.64,1:03:28.66,Default,,0000,0000,0000,,And of course in the\Nexam-- oh, in the exam,
Dialogue: 0,1:03:28.66,1:03:30.56,Default,,0000,0000,0000,,you will cheat big time, right?
Dialogue: 0,1:03:30.56,1:03:31.57,Default,,0000,0000,0000,,What would you do?
Dialogue: 0,1:03:31.57,1:03:37.00,Default,,0000,0000,0000,,You would set it up and forget\Nabout computing it, integrating
Dialogue: 0,1:03:37.00,1:03:40.07,Default,,0000,0000,0000,,one at a time, doing this.
Dialogue: 0,1:03:40.07,1:03:42.26,Default,,0000,0000,0000,,And you would put equals 1/6.
Dialogue: 0,1:03:42.26,1:03:43.93,Default,,0000,0000,0000,,Thank you very much.
Dialogue: 0,1:03:43.93,1:03:44.53,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:03:44.53,1:03:47.14,Default,,0000,0000,0000,,Can I check that you\Ndidn't do the work?
Dialogue: 0,1:03:47.14,1:03:49.52,Default,,0000,0000,0000,,No.
Dialogue: 0,1:03:49.52,1:03:50.23,Default,,0000,0000,0000,,You trapped me.
Dialogue: 0,1:03:50.23,1:03:50.73,Default,,0000,0000,0000,,You got me.
Dialogue: 0,1:03:50.73,1:03:55.29,Default,,0000,0000,0000,,I have no-- I mean, I need\Nto say, is this correct?
Dialogue: 0,1:03:55.29,1:03:55.79,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:03:55.79,1:03:56.71,Default,,0000,0000,0000,,Is the answer correct?
Dialogue: 0,1:03:56.71,1:03:57.60,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:03:57.60,1:03:59.87,Default,,0000,0000,0000,,Do they get full credit?
Dialogue: 0,1:03:59.87,1:04:00.78,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:04:00.78,1:04:04.67,Default,,0000,0000,0000,,So it's a sneaky problem.
Dialogue: 0,1:04:04.67,1:04:06.48,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:04:06.48,1:04:11.73,Default,,0000,0000,0000,,Now finally, let's plunge\Ninto 12.4, which is-- can you
Dialogue: 0,1:04:11.73,1:04:14.05,Default,,0000,0000,0000,,remember this problem for 12.4?
Dialogue: 0,1:04:14.05,1:04:16.15,Default,,0000,0000,0000,,I want to draw this again.
Dialogue: 0,1:04:16.15,1:04:20.06,Default,,0000,0000,0000,,So I'll try not to\Nerase the picture.
Dialogue: 0,1:04:20.06,1:04:22.87,Default,,0000,0000,0000,,I'll erase all the\Ndata I have here.
Dialogue: 0,1:04:22.87,1:04:26.93,Default,,0000,0000,0000,,And I'll keep the future because\NI don't want to draw it again.
Dialogue: 0,1:04:26.93,1:04:30.47,Default,,0000,0000,0000,,
Dialogue: 0,1:04:30.47,1:04:39.74,Default,,0000,0000,0000,,When we were small-- I\Nmean small in Calc 1 and 2,
Dialogue: 0,1:04:39.74,1:04:46.93,Default,,0000,0000,0000,,they gave us a function\Ny equals f of x.
Dialogue: 0,1:04:46.93,1:04:50.22,Default,,0000,0000,0000,,That is smooth, at least C1.
Dialogue: 0,1:04:50.22,1:04:58.31,Default,,0000,0000,0000,,C1 means differentiable, and\Nthe derivative is continuous.
Dialogue: 0,1:04:58.31,1:05:01.19,Default,,0000,0000,0000,,
Dialogue: 0,1:05:01.19,1:05:05.42,Default,,0000,0000,0000,,And we said, OK, between\Nx equals a and x equals b,
Dialogue: 0,1:05:05.42,1:05:10.04,Default,,0000,0000,0000,,I want you-- you, any\Nstudent-- to compute
Dialogue: 0,1:05:10.04,1:05:12.78,Default,,0000,0000,0000,,the length of the arch.
Dialogue: 0,1:05:12.78,1:05:14.76,Default,,0000,0000,0000,,Length of the arch.
Dialogue: 0,1:05:14.76,1:05:17.49,Default,,0000,0000,0000,,And how did we do\Nthat in Calc 2?
Dialogue: 0,1:05:17.49,1:05:21.12,Default,,0000,0000,0000,,I have colleagues who\Ndrive me crazy by refusing
Dialogue: 0,1:05:21.12,1:05:24.46,Default,,0000,0000,0000,,to teach that in Calc 2.
Dialogue: 0,1:05:24.46,1:05:25.74,Default,,0000,0000,0000,,Well, I disagree.
Dialogue: 0,1:05:25.74,1:05:28.76,Default,,0000,0000,0000,,Of course, I can teach\Nit only in Calc 3,
Dialogue: 0,1:05:28.76,1:05:32.33,Default,,0000,0000,0000,,and I can do it with\Nparametrization, which we did,
Dialogue: 0,1:05:32.33,1:05:36.46,Default,,0000,0000,0000,,and then come back to the case\Nyou have, y equals f of x,
Dialogue: 0,1:05:36.46,1:05:38.91,Default,,0000,0000,0000,,and get the formula.
Dialogue: 0,1:05:38.91,1:05:42.31,Default,,0000,0000,0000,,But it should be taught at\Nboth levels, in both courses.
Dialogue: 0,1:05:42.31,1:05:47.45,Default,,0000,0000,0000,,So when you have a\Ngeneral parametrization
Dialogue: 0,1:05:47.45,1:05:50.83,Default,,0000,0000,0000,,rt equals x of ty\Nof t [INAUDIBLE],
Dialogue: 0,1:05:50.83,1:05:56.83,Default,,0000,0000,0000,,this is a parametrized\Ncurve that's in C1, in time.
Dialogue: 0,1:05:56.83,1:06:01.07,Default,,0000,0000,0000,,What is the length of the\Narch between time t0 and time
Dialogue: 0,1:06:01.07,1:06:02.42,Default,,0000,0000,0000,,t1 [INAUDIBLE]?
Dialogue: 0,1:06:02.42,1:06:04.96,Default,,0000,0000,0000,,
Dialogue: 0,1:06:04.96,1:06:08.59,Default,,0000,0000,0000,,The integral from t0 to t1 or\Nthe speed because the space
Dialogue: 0,1:06:08.59,1:06:10.65,Default,,0000,0000,0000,,is the integral\Nof speed in time.
Dialogue: 0,1:06:10.65,1:06:13.82,Default,,0000,0000,0000,,
Dialogue: 0,1:06:13.82,1:06:17.12,Default,,0000,0000,0000,,So in terms of speed,\Nremember that we
Dialogue: 0,1:06:17.12,1:06:24.34,Default,,0000,0000,0000,,put square root of x prime of\Nt squared plus y prime of t
Dialogue: 0,1:06:24.34,1:06:26.48,Default,,0000,0000,0000,,squared dt.
Dialogue: 0,1:06:26.48,1:06:27.24,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,1:06:27.24,1:06:28.31,Default,,0000,0000,0000,,Somebody tell me.
Dialogue: 0,1:06:28.31,1:06:30.87,Default,,0000,0000,0000,,That was the speed.
Dialogue: 0,1:06:30.87,1:06:35.87,Default,,0000,0000,0000,,That was the magnitude\Nof the velocity vector.
Dialogue: 0,1:06:35.87,1:06:39.24,Default,,0000,0000,0000,,And we've done that, and we\Ndiscovered that in Calculus 3.
Dialogue: 0,1:06:39.24,1:06:44.25,Default,,0000,0000,0000,,In Calculus 2, they\Ntaught this for what case?
Dialogue: 0,1:06:44.25,1:06:48.09,Default,,0000,0000,0000,,The case when x is t--\Nsay it again, I will now.
Dialogue: 0,1:06:48.09,1:06:53.36,Default,,0000,0000,0000,,The case when x is\Nt, and y is f of t,
Dialogue: 0,1:06:53.36,1:06:56.91,Default,,0000,0000,0000,,which is f of x--\Nand in that case,
Dialogue: 0,1:06:56.91,1:06:59.45,Default,,0000,0000,0000,,as I told you before,\Nthe length will
Dialogue: 0,1:06:59.45,1:07:05.38,Default,,0000,0000,0000,,be the integral from A to B.\NWhatever, it's the same thing.
Dialogue: 0,1:07:05.38,1:07:07.14,Default,,0000,0000,0000,,A to B.
Dialogue: 0,1:07:07.14,1:07:11.80,Default,,0000,0000,0000,,Square root-- since x\Nis t, x prime of t is 1.
Dialogue: 0,1:07:11.80,1:07:16.20,Default,,0000,0000,0000,,So you get 1 plus-- y is just f.
Dialogue: 0,1:07:16.20,1:07:16.92,Default,,0000,0000,0000,,y is f.
Dialogue: 0,1:07:16.92,1:07:23.48,Default,,0000,0000,0000,,So you have f prime\Nof x squared dx.
Dialogue: 0,1:07:23.48,1:07:30.03,Default,,0000,0000,0000,,So the length of this arch--\Nlet me draw the arch in green,
Dialogue: 0,1:07:30.03,1:07:32.02,Default,,0000,0000,0000,,so it's beautiful.
Dialogue: 0,1:07:32.02,1:07:37.24,Default,,0000,0000,0000,,The length of this green\Narch will be the length of r.
Dialogue: 0,1:07:37.24,1:07:41.92,Default,,0000,0000,0000,,The integral from A to B square\Nroot of 1 plus f prime 1x
Dialogue: 0,1:07:41.92,1:07:43.94,Default,,0000,0000,0000,,squared dx.
Dialogue: 0,1:07:43.94,1:07:46.31,Default,,0000,0000,0000,,Now there is a beautiful,\Nbeautiful generalization
Dialogue: 0,1:07:46.31,1:07:56.83,Default,,0000,0000,0000,,of that for-- generalization\Nfor extension gives you
Dialogue: 0,1:07:56.83,1:08:10.90,Default,,0000,0000,0000,,the surface area of a graph z\Nequals f of x, y over domain D.
Dialogue: 0,1:08:10.90,1:08:11.93,Default,,0000,0000,0000,,Say what?
Dialogue: 0,1:08:11.93,1:08:16.25,Default,,0000,0000,0000,,OK, it's exactly the\Nsame formula generalized.
Dialogue: 0,1:08:16.25,1:08:18.78,Default,,0000,0000,0000,,And I would like you to guess.
Dialogue: 0,1:08:18.78,1:08:21.06,Default,,0000,0000,0000,,So I'd like you to\Nexperimentally get
Dialogue: 0,1:08:21.06,1:08:21.86,Default,,0000,0000,0000,,to the formula.
Dialogue: 0,1:08:21.86,1:08:23.77,Default,,0000,0000,0000,,It can be proved.
Dialogue: 0,1:08:23.77,1:08:28.73,Default,,0000,0000,0000,,It can be proved by taking\Nthe equivalence of some sort
Dialogue: 0,1:08:28.73,1:08:32.63,Default,,0000,0000,0000,,of Riemann summation\Nand passing to the limit
Dialogue: 0,1:08:32.63,1:08:34.04,Default,,0000,0000,0000,,and get the formula.
Dialogue: 0,1:08:34.04,1:08:41.95,Default,,0000,0000,0000,,But I would like you\Nto imagine you have--
Dialogue: 0,1:08:41.95,1:08:45.12,Default,,0000,0000,0000,,so you have z equals f of x, y.
Dialogue: 0,1:08:45.12,1:08:54.59,Default,,0000,0000,0000,,That projects exactly over\ND. The area of the surface--
Dialogue: 0,1:08:54.59,1:08:56.36,Default,,0000,0000,0000,,let's call it A of s.
Dialogue: 0,1:08:56.36,1:09:02.97,Default,,0000,0000,0000,,
Dialogue: 0,1:09:02.97,1:09:08.71,Default,,0000,0000,0000,,The area of the\Nsurface will be--
Dialogue: 0,1:09:08.71,1:09:10.00,Default,,0000,0000,0000,,what do you think it will be?
Dialogue: 0,1:09:10.00,1:09:12.25,Default,,0000,0000,0000,,You are smart people.
Dialogue: 0,1:09:12.25,1:09:17.04,Default,,0000,0000,0000,,It will be double integral\Ninstead of one integral
Dialogue: 0,1:09:17.04,1:09:19.38,Default,,0000,0000,0000,,over-- what do you think?
Dialogue: 0,1:09:19.38,1:09:20.05,Default,,0000,0000,0000,,Over the domain.
Dialogue: 0,1:09:20.05,1:09:22.97,Default,,0000,0000,0000,,
Dialogue: 0,1:09:22.97,1:09:26.23,Default,,0000,0000,0000,,What's the simplest\Nway to generalize this
Dialogue: 0,1:09:26.23,1:09:29.70,Default,,0000,0000,0000,,through probably [INAUDIBLE]?
Dialogue: 0,1:09:29.70,1:09:30.96,Default,,0000,0000,0000,,Another square root.
Dialogue: 0,1:09:30.96,1:09:33.77,Default,,0000,0000,0000,,
Dialogue: 0,1:09:33.77,1:09:37.17,Default,,0000,0000,0000,,We don't have just one\Nderivative, f prime of x.
Dialogue: 0,1:09:37.17,1:09:40.48,Default,,0000,0000,0000,,We are going to have two\Nderivatives, f sub x and f sub
Dialogue: 0,1:09:40.48,1:09:41.06,Default,,0000,0000,0000,,y.
Dialogue: 0,1:09:41.06,1:09:43.87,Default,,0000,0000,0000,,So what do you think the\Nsimplest generalization
Dialogue: 0,1:09:43.87,1:09:44.53,Default,,0000,0000,0000,,will look like?
Dialogue: 0,1:09:44.53,1:09:46.18,Default,,0000,0000,0000,,STUDENT: 1 plus [INAUDIBLE].
Dialogue: 0,1:09:46.18,1:09:51.74,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 1 plus\Nf sub x squared plus f sub y
Dialogue: 0,1:09:51.74,1:09:53.68,Default,,0000,0000,0000,,squared dx.
Dialogue: 0,1:09:53.68,1:09:55.22,Default,,0000,0000,0000,,That's it.
Dialogue: 0,1:09:55.22,1:09:57.19,Default,,0000,0000,0000,,So you say, oh, I'm a genius.
Dialogue: 0,1:09:57.19,1:09:57.92,Default,,0000,0000,0000,,I discovered it.
Dialogue: 0,1:09:57.92,1:09:59.20,Default,,0000,0000,0000,,Yes, you are.
Dialogue: 0,1:09:59.20,1:10:00.86,Default,,0000,0000,0000,,I mean, in a sense that-- no.
Dialogue: 0,1:10:00.86,1:10:03.19,Default,,0000,0000,0000,,In the sense that we\Nall have that kind
Dialogue: 0,1:10:03.19,1:10:06.97,Default,,0000,0000,0000,,of mathematical intuition,\Ncreativity that you come up
Dialogue: 0,1:10:06.97,1:10:08.40,Default,,0000,0000,0000,,with something.
Dialogue: 0,1:10:08.40,1:10:10.23,Default,,0000,0000,0000,,And you say, OK, can I verify?
Dialogue: 0,1:10:10.23,1:10:10.90,Default,,0000,0000,0000,,Can I prove it?
Dialogue: 0,1:10:10.90,1:10:11.40,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:10:11.40,1:10:13.45,Default,,0000,0000,0000,,Can you discover\Nthings on your own?
Dialogue: 0,1:10:13.45,1:10:15.06,Default,,0000,0000,0000,,Yes, you can.
Dialogue: 0,1:10:15.06,1:10:18.12,Default,,0000,0000,0000,,Actually, that's how all\Nthe mathematical minds came.
Dialogue: 0,1:10:18.12,1:10:20.58,Default,,0000,0000,0000,,They came up to it\Nwith a conjecture based
Dialogue: 0,1:10:20.58,1:10:24.77,Default,,0000,0000,0000,,on some prior experiences, some\Nprior observations and said,
Dialogue: 0,1:10:24.77,1:10:26.32,Default,,0000,0000,0000,,I think it's going\Nto look like that.
Dialogue: 0,1:10:26.32,1:10:30.09,Default,,0000,0000,0000,,I bet you like 90% that it's\Ngoing to look like that.
Dialogue: 0,1:10:30.09,1:10:32.04,Default,,0000,0000,0000,,But then it took them\Ntime to prove it.
Dialogue: 0,1:10:32.04,1:10:36.65,Default,,0000,0000,0000,,But they were convinced that\Nthis is what it's going to be.
Dialogue: 0,1:10:36.65,1:10:37.16,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:10:37.16,1:10:43.69,Default,,0000,0000,0000,,So if you want the area\Nof the patch of a surface,
Dialogue: 0,1:10:43.69,1:10:47.90,Default,,0000,0000,0000,,that's going to be 4.1, and\Nthat's page-- god knows.
Dialogue: 0,1:10:47.90,1:10:48.62,Default,,0000,0000,0000,,Wait a second.
Dialogue: 0,1:10:48.62,1:10:49.12,Default,,0000,0000,0000,,Wait.
Dialogue: 0,1:10:49.12,1:10:50.33,Default,,0000,0000,0000,,Bare with me.
Dialogue: 0,1:10:50.33,1:11:01.83,Default,,0000,0000,0000,,It starts at page 951,\Nand it ends at page 957.
Dialogue: 0,1:11:01.83,1:11:05.47,Default,,0000,0000,0000,,It's only seven pages, OK?
Dialogue: 0,1:11:05.47,1:11:06.70,Default,,0000,0000,0000,,So it's not hard.
Dialogue: 0,1:11:06.70,1:11:08.30,Default,,0000,0000,0000,,You have several examples.
Dialogue: 0,1:11:08.30,1:11:10.30,Default,,0000,0000,0000,,I'm going to work\Non an example that
Dialogue: 0,1:11:10.30,1:11:12.17,Default,,0000,0000,0000,,is straight out of the book.
Dialogue: 0,1:11:12.17,1:11:13.73,Default,,0000,0000,0000,,And guess what?
Dialogue: 0,1:11:13.73,1:11:16.34,Default,,0000,0000,0000,,You see, that's why I like\Nthis problem, because it's
Dialogue: 0,1:11:16.34,1:11:21.56,Default,,0000,0000,0000,,in-- example one is exactly the\None that I came up with today
Dialogue: 0,1:11:21.56,1:11:26.27,Default,,0000,0000,0000,,and says, find the\Nsame tetrahedron thing.
Dialogue: 0,1:11:26.27,1:11:29.07,Default,,0000,0000,0000,,Find the surface area of\Nthe portion of the plane
Dialogue: 0,1:11:29.07,1:11:39.00,Default,,0000,0000,0000,,x plus y plus z equals\N1, which was that, which
Dialogue: 0,1:11:39.00,1:11:42.31,Default,,0000,0000,0000,,lies in the first octant\Nwhere-- what does it mean,
Dialogue: 0,1:11:42.31,1:11:42.92,Default,,0000,0000,0000,,first octant?
Dialogue: 0,1:11:42.92,1:11:46.45,Default,,0000,0000,0000,,It means that x is\Npositive. y is positive.
Dialogue: 0,1:11:46.45,1:11:49.17,Default,,0000,0000,0000,,z is positive for z.
Dialogue: 0,1:11:49.17,1:11:52.42,Default,,0000,0000,0000,,
Dialogue: 0,1:11:52.42,1:11:53.02,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:11:53.02,1:11:53.90,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,1:11:53.90,1:11:54.59,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:11:54.59,1:11:56.11,Default,,0000,0000,0000,,Let's find out.
Dialogue: 0,1:11:56.11,1:11:59.04,Default,,0000,0000,0000,,
Dialogue: 0,1:11:59.04,1:12:05.38,Default,,0000,0000,0000,,So if we were to apply this\Nformula, how would we do that?
Dialogue: 0,1:12:05.38,1:12:07.27,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:12:07.27,1:12:09.74,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:12:09.74,1:12:12.13,Default,,0000,0000,0000,,We have to recollect\Nwho everybody
Dialogue: 0,1:12:12.13,1:12:15.60,Default,,0000,0000,0000,,is from scratch, one at a time.
Dialogue: 0,1:12:15.60,1:12:19.58,Default,,0000,0000,0000,,
Dialogue: 0,1:12:19.58,1:12:20.39,Default,,0000,0000,0000,,Hmm?
Dialogue: 0,1:12:20.39,1:12:24.10,Default,,0000,0000,0000,,STUDENT: Could we just\Nuse our K-12 knowledge?
Dialogue: 0,1:12:24.10,1:12:26.27,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Well,\Nyou can do that very well.
Dialogue: 0,1:12:26.27,1:12:28.95,Default,,0000,0000,0000,,But let's do it\Nfirst-- you're sneaky.
Dialogue: 0,1:12:28.95,1:12:31.06,Default,,0000,0000,0000,,Let's do it first as Calc 3.
Dialogue: 0,1:12:31.06,1:12:36.60,Default,,0000,0000,0000,,And then let's see who comes\Nup with the fastest solution
Dialogue: 0,1:12:36.60,1:12:37.64,Default,,0000,0000,0000,,in terms of surface area.
Dialogue: 0,1:12:37.64,1:12:42.54,Default,,0000,0000,0000,,By the way, this individual--\Nthis whole fat, sausage
Dialogue: 0,1:12:42.54,1:12:45.54,Default,,0000,0000,0000,,kind of thing is ds.
Dialogue: 0,1:12:45.54,1:12:50.90,Default,,0000,0000,0000,,This expression is called\Nthe surface element.
Dialogue: 0,1:12:50.90,1:12:53.66,Default,,0000,0000,0000,,Make a distinction\Nbetween dA, which is
Dialogue: 0,1:12:53.66,1:12:56.13,Default,,0000,0000,0000,,called area element in plane.
Dialogue: 0,1:12:56.13,1:12:59.51,Default,,0000,0000,0000,,
Dialogue: 0,1:12:59.51,1:13:08.24,Default,,0000,0000,0000,,ds is the surface element on the\Nsurface, on the surface on top.
Dialogue: 0,1:13:08.24,1:13:17.67,Default,,0000,0000,0000,,So practically, guys, you have\Nsome [? healy ?] part, which
Dialogue: 0,1:13:17.67,1:13:22.72,Default,,0000,0000,0000,,projects on a domain in plane.
Dialogue: 0,1:13:22.72,1:13:27.21,Default,,0000,0000,0000,,The dA is the infinite\Ndecimal area of this thingy.
Dialogue: 0,1:13:27.21,1:13:30.46,Default,,0000,0000,0000,,And ds is the infinite\Ndecimal area of that.
Dialogue: 0,1:13:30.46,1:13:31.67,Default,,0000,0000,0000,,What do you mean by that?
Dialogue: 0,1:13:31.67,1:13:33.00,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:13:33.00,1:13:36.57,Default,,0000,0000,0000,,Imagine this grid of pixels\Nthat becomes smaller and smaller
Dialogue: 0,1:13:36.57,1:13:37.41,Default,,0000,0000,0000,,and smaller.
Dialogue: 0,1:13:37.41,1:13:38.45,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:13:38.45,1:13:41.48,Default,,0000,0000,0000,,Take one pixel already and\Nmake it infinitesimally small.
Dialogue: 0,1:13:41.48,1:13:46.40,Default,,0000,0000,0000,,That's going to be\Nda dxdy, dx times dy.
Dialogue: 0,1:13:46.40,1:13:51.83,Default,,0000,0000,0000,,What is the corresponding\Npixel on the round surface?
Dialogue: 0,1:13:51.83,1:13:52.80,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:13:52.80,1:13:56.50,Default,,0000,0000,0000,,It's still going to\Nbe given by two lines,
Dialogue: 0,1:13:56.50,1:14:01.72,Default,,0000,0000,0000,,and two lines form a\Ncurvilinear domain.
Dialogue: 0,1:14:01.72,1:14:05.50,Default,,0000,0000,0000,,And that curvilinear tiny-- do\Nyou see how small it is that?
Dialogue: 0,1:14:05.50,1:14:07.75,Default,,0000,0000,0000,,I bet the video cannot see it.
Dialogue: 0,1:14:07.75,1:14:09.00,Default,,0000,0000,0000,,But you can see it.
Dialogue: 0,1:14:09.00,1:14:11.78,Default,,0000,0000,0000,,So this tiny infinitesimally\Nsmall element
Dialogue: 0,1:14:11.78,1:14:15.71,Default,,0000,0000,0000,,on the surface-- this is ds.
Dialogue: 0,1:14:15.71,1:14:17.49,Default,,0000,0000,0000,,This is ds.
Dialogue: 0,1:14:17.49,1:14:18.39,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:14:18.39,1:14:28.20,Default,,0000,0000,0000,,So if it were between a plane\Nand a tiny square, dxdy dA
Dialogue: 0,1:14:28.20,1:14:31.75,Default,,0000,0000,0000,,and the ds here, it would\Nbe easy between a plane
Dialogue: 0,1:14:31.75,1:14:34.83,Default,,0000,0000,0000,,and a floor because\Nyou can do some trick,
Dialogue: 0,1:14:34.83,1:14:37.93,Default,,0000,0000,0000,,like a projection\Nwith cosine and stuff.
Dialogue: 0,1:14:37.93,1:14:39.42,Default,,0000,0000,0000,,But in general,\Nit's not so easy,
Dialogue: 0,1:14:39.42,1:14:42.49,Default,,0000,0000,0000,,because you can have\Na round patch that's
Dialogue: 0,1:14:42.49,1:14:44.46,Default,,0000,0000,0000,,sitting above a domain.
Dialogue: 0,1:14:44.46,1:14:47.75,Default,,0000,0000,0000,,And it's just-- you\Nhave to do integration.
Dialogue: 0,1:14:47.75,1:14:50.69,Default,,0000,0000,0000,,You have no other choice.
Dialogue: 0,1:14:50.69,1:14:52.47,Default,,0000,0000,0000,,Let's compute it both ways.
Dialogue: 0,1:14:52.47,1:14:53.82,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,1:14:53.82,1:15:00.45,Default,,0000,0000,0000,,A of s will be integral over\Ndomain D. What in the world
Dialogue: 0,1:15:00.45,1:15:02.04,Default,,0000,0000,0000,,was the domain D?
Dialogue: 0,1:15:02.04,1:15:05.02,Default,,0000,0000,0000,,The domain D was the\Ndomain on the floor.
Dialogue: 0,1:15:05.02,1:15:08.10,Default,,0000,0000,0000,,And you told me what that\Nis, but I forgot, guys.
Dialogue: 0,1:15:08.10,1:15:12.86,Default,,0000,0000,0000,,x is between 0 and 1.
Dialogue: 0,1:15:12.86,1:15:14.11,Default,,0000,0000,0000,,Did you say so?
Dialogue: 0,1:15:14.11,1:15:16.48,Default,,0000,0000,0000,,And y was between what and what?
Dialogue: 0,1:15:16.48,1:15:19.40,Default,,0000,0000,0000,,Can you remind me?
Dialogue: 0,1:15:19.40,1:15:20.78,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:15:20.78,1:15:22.54,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NBetween 0 and--
Dialogue: 0,1:15:22.54,1:15:23.41,Default,,0000,0000,0000,,STUDENT: 1 minus x.
Dialogue: 0,1:15:23.41,1:15:25.44,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\N1 minus x, excellent.
Dialogue: 0,1:15:25.44,1:15:29.16,Default,,0000,0000,0000,,So this is the\Nmeaning of domain D.
Dialogue: 0,1:15:29.16,1:15:34.58,Default,,0000,0000,0000,,And the square root\Nof-- who is f of x, y?
Dialogue: 0,1:15:34.58,1:15:37.16,Default,,0000,0000,0000,,It's 1 minus x minus what?
Dialogue: 0,1:15:37.16,1:15:37.66,Default,,0000,0000,0000,,Oh, right.
Dialogue: 0,1:15:37.66,1:15:41.71,Default,,0000,0000,0000,,So you guys have to help\Nme compute this animal.
Dialogue: 0,1:15:41.71,1:15:45.87,Default,,0000,0000,0000,,f sub x is negative 1.
Dialogue: 0,1:15:45.87,1:15:47.38,Default,,0000,0000,0000,,Attention, please.
Dialogue: 0,1:15:47.38,1:15:50.62,Default,,0000,0000,0000,,f sub y is negative 1.
Dialogue: 0,1:15:50.62,1:15:51.19,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:15:51.19,1:15:56.56,Default,,0000,0000,0000,,So I have to say 1 plus negative\N1 squared plus negative 1
Dialogue: 0,1:15:56.56,1:15:59.60,Default,,0000,0000,0000,,squared dA.
Dialogue: 0,1:15:59.60,1:16:00.87,Default,,0000,0000,0000,,Gosh, I'm lucky.
Dialogue: 0,1:16:00.87,1:16:01.45,Default,,0000,0000,0000,,Look.
Dialogue: 0,1:16:01.45,1:16:03.44,Default,,0000,0000,0000,,I mean, I'm not\Njust lucky, but that
Dialogue: 0,1:16:03.44,1:16:06.06,Default,,0000,0000,0000,,has to be-- it has to\Nbe something beautiful
Dialogue: 0,1:16:06.06,1:16:09.06,Default,,0000,0000,0000,,because otherwise the\Nelementary formula will not
Dialogue: 0,1:16:09.06,1:16:10.36,Default,,0000,0000,0000,,be so beautiful.
Dialogue: 0,1:16:10.36,1:16:13.69,Default,,0000,0000,0000,,This is root 3, and\Nit brings it back.
Dialogue: 0,1:16:13.69,1:16:18.65,Default,,0000,0000,0000,,Root 3 pulls out\Nof the whole thing.
Dialogue: 0,1:16:18.65,1:16:20.85,Default,,0000,0000,0000,,So you have root 3.
Dialogue: 0,1:16:20.85,1:16:25.10,Default,,0000,0000,0000,,What is double integral--\NOK, let's compute.
Dialogue: 0,1:16:25.10,1:16:29.97,Default,,0000,0000,0000,,So first you go dy and dx.
Dialogue: 0,1:16:29.97,1:16:31.71,Default,,0000,0000,0000,,x, again, you gave\Nit to me, guys.
Dialogue: 0,1:16:31.71,1:16:33.29,Default,,0000,0000,0000,,0 to 1.
Dialogue: 0,1:16:33.29,1:16:37.72,Default,,0000,0000,0000,,y between 0 and 1 minus x.
Dialogue: 0,1:16:37.72,1:16:38.85,Default,,0000,0000,0000,,Great.
Dialogue: 0,1:16:38.85,1:16:40.60,Default,,0000,0000,0000,,We are almost there.
Dialogue: 0,1:16:40.60,1:16:41.43,Default,,0000,0000,0000,,We are almost there.
Dialogue: 0,1:16:41.43,1:16:42.91,Default,,0000,0000,0000,,I just need your\Nhelp a little bit.
Dialogue: 0,1:16:42.91,1:16:45.03,Default,,0000,0000,0000,,The square root of 3 goes out.
Dialogue: 0,1:16:45.03,1:16:46.77,Default,,0000,0000,0000,,The integral from 0 to 1.
Dialogue: 0,1:16:46.77,1:16:49.77,Default,,0000,0000,0000,,What is the integral of 1dy?
Dialogue: 0,1:16:49.77,1:16:56.71,Default,,0000,0000,0000,,It's y, y between 1 minus x\Non top and 0 on the bottom.
Dialogue: 0,1:16:56.71,1:16:58.48,Default,,0000,0000,0000,,That means 1 minus x.
Dialogue: 0,1:16:58.48,1:17:02.32,Default,,0000,0000,0000,,If I make a mistake, just shout.
Dialogue: 0,1:17:02.32,1:17:05.02,Default,,0000,0000,0000,,dx.
Dialogue: 0,1:17:05.02,1:17:11.25,Default,,0000,0000,0000,,The square root of 3 times\Nthe integral of 1 minus x.
Dialogue: 0,1:17:11.25,1:17:12.94,Default,,0000,0000,0000,,STUDENT: x minus\Nthe square root.
Dialogue: 0,1:17:12.94,1:17:14.98,Default,,0000,0000,0000,,DR. MAGDALENA TODA: x\Nminus the square root of 2.
Dialogue: 0,1:17:14.98,1:17:18.22,Default,,0000,0000,0000,,Let me write it separately\Nbecause I should do that fast,
Dialogue: 0,1:17:18.22,1:17:19.40,Default,,0000,0000,0000,,right?
Dialogue: 0,1:17:19.40,1:17:20.15,Default,,0000,0000,0000,,Between 0 and 1.
Dialogue: 0,1:17:20.15,1:17:20.98,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:17:20.98,1:17:23.85,Default,,0000,0000,0000,,
Dialogue: 0,1:17:23.85,1:17:24.53,Default,,0000,0000,0000,,1/2.
Dialogue: 0,1:17:24.53,1:17:25.76,Default,,0000,0000,0000,,That's a piece of cake.
Dialogue: 0,1:17:25.76,1:17:27.46,Default,,0000,0000,0000,,This is 1/2.
Dialogue: 0,1:17:27.46,1:17:30.39,Default,,0000,0000,0000,,So 1/2 is this thing.
Dialogue: 0,1:17:30.39,1:17:32.05,Default,,0000,0000,0000,,And root 3 over 2.
Dialogue: 0,1:17:32.05,1:17:34.49,Default,,0000,0000,0000,,And now Alex says,\Ncongratulations
Dialogue: 0,1:17:34.49,1:17:36.35,Default,,0000,0000,0000,,on your root 3\Nover 2, but I could
Dialogue: 0,1:17:36.35,1:17:39.01,Default,,0000,0000,0000,,have told you that much faster.
Dialogue: 0,1:17:39.01,1:17:41.68,Default,,0000,0000,0000,,So the question\Nis, how could Alex
Dialogue: 0,1:17:41.68,1:17:46.89,Default,,0000,0000,0000,,have shown us this root\N3 over 2 much faster?
Dialogue: 0,1:17:46.89,1:17:48.55,Default,,0000,0000,0000,,STUDENT: Well, it's\Njust a triangle.
Dialogue: 0,1:17:48.55,1:17:50.26,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NIt's just a triangle.
Dialogue: 0,1:17:50.26,1:17:51.66,Default,,0000,0000,0000,,It's not just a triangle.
Dialogue: 0,1:17:51.66,1:17:56.96,Default,,0000,0000,0000,,It's a beautiful triangle\Nthat's an equilateral triangle.
Dialogue: 0,1:17:56.96,1:18:01.33,Default,,0000,0000,0000,,And in school, they used\Nto teach more trigonometry.
Dialogue: 0,1:18:01.33,1:18:03.18,Default,,0000,0000,0000,,And they don't.
Dialogue: 0,1:18:03.18,1:18:04.82,Default,,0000,0000,0000,,And if I had the\Nchoice-- I'm not
Dialogue: 0,1:18:04.82,1:18:10.01,Default,,0000,0000,0000,,involved in the K-12 curriculum\Nstandards for any state.
Dialogue: 0,1:18:10.01,1:18:13.69,Default,,0000,0000,0000,,But if I had the choice, I\Nwould say, teach the kids
Dialogue: 0,1:18:13.69,1:18:15.71,Default,,0000,0000,0000,,a little bit more\Ngeometry in school
Dialogue: 0,1:18:15.71,1:18:18.56,Default,,0000,0000,0000,,because they don't know\Nanything in terms of geometry.
Dialogue: 0,1:18:18.56,1:18:22.34,Default,,0000,0000,0000,,So there were\Ntriumphs in the past,
Dialogue: 0,1:18:22.34,1:18:25.61,Default,,0000,0000,0000,,and your parents may\Nknow those better.
Dialogue: 0,1:18:25.61,1:18:29.83,Default,,0000,0000,0000,,But If somebody gave you\Nan equilateral triangle
Dialogue: 0,1:18:29.83,1:18:34.68,Default,,0000,0000,0000,,of a certain side, you\Nwould be able to tell them,
Dialogue: 0,1:18:34.68,1:18:36.94,Default,,0000,0000,0000,,I know your area.
Dialogue: 0,1:18:36.94,1:18:38.01,Default,,0000,0000,0000,,I know the area.
Dialogue: 0,1:18:38.01,1:18:44.62,Default,,0000,0000,0000,,I know the area being l squared,\Nthe square root of 3 over 4.
Dialogue: 0,1:18:44.62,1:18:46.96,Default,,0000,0000,0000,,Your parents may know that.
Dialogue: 0,1:18:46.96,1:18:47.92,Default,,0000,0000,0000,,Aaron, ask your dad.
Dialogue: 0,1:18:47.92,1:18:49.37,Default,,0000,0000,0000,,He will know.
Dialogue: 0,1:18:49.37,1:18:52.69,Default,,0000,0000,0000,,But we don't teach\Nthat in school anymore.
Dialogue: 0,1:18:52.69,1:18:56.05,Default,,0000,0000,0000,,The smart kids do this\Nby themselves how?
Dialogue: 0,1:18:56.05,1:18:58.13,Default,,0000,0000,0000,,Can you show me how?
Dialogue: 0,1:18:58.13,1:19:02.28,Default,,0000,0000,0000,,They draw the\Nperpendicular bisector.
Dialogue: 0,1:19:02.28,1:19:04.87,Default,,0000,0000,0000,,And there is a\Ntheorem actually--
Dialogue: 0,1:19:04.87,1:19:06.84,Default,,0000,0000,0000,,but we never prove\Nthat in school--
Dialogue: 0,1:19:06.84,1:19:10.72,Default,,0000,0000,0000,,that if we draw that\Nperpendicular bisector,
Dialogue: 0,1:19:10.72,1:19:14.82,Default,,0000,0000,0000,,then the two triangles\Nare congruent.
Dialogue: 0,1:19:14.82,1:19:22.89,Default,,0000,0000,0000,,And as a consequence,\Nwell, that is l/2, l/2.
Dialogue: 0,1:19:22.89,1:19:24.34,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:19:24.34,1:19:28.81,Default,,0000,0000,0000,,So if you draw just\Nthe perpendicular,
Dialogue: 0,1:19:28.81,1:19:36.53,Default,,0000,0000,0000,,you can prove it using some\Ncongruence of triangles
Dialogue: 0,1:19:36.53,1:19:39.20,Default,,0000,0000,0000,,that what you get here\Nis also the median.
Dialogue: 0,1:19:39.20,1:19:41.08,Default,,0000,0000,0000,,So it's going to keep\Nright in the middle
Dialogue: 0,1:19:41.08,1:19:42.67,Default,,0000,0000,0000,,of the opposite side.
Dialogue: 0,1:19:42.67,1:19:45.10,Default,,0000,0000,0000,,So you l/2, l/2.
Dialogue: 0,1:19:45.10,1:19:45.60,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:19:45.60,1:19:47.14,Default,,0000,0000,0000,,That's what I wanted to say.
Dialogue: 0,1:19:47.14,1:19:49.10,Default,,0000,0000,0000,,And then using the\NPythagorean theorem,
Dialogue: 0,1:19:49.10,1:19:50.48,Default,,0000,0000,0000,,you're going to get the height.
Dialogue: 0,1:19:50.48,1:19:55.45,Default,,0000,0000,0000,,So the height will be the square\Nroot of l squared minus l/2
Dialogue: 0,1:19:55.45,1:20:00.40,Default,,0000,0000,0000,,squared, which is the\Nsquare root of l squared
Dialogue: 0,1:20:00.40,1:20:05.85,Default,,0000,0000,0000,,minus l squared over 4, which\Nis the square root of 3l squared
Dialogue: 0,1:20:05.85,1:20:11.83,Default,,0000,0000,0000,,over 4, which simplified\Nwill be l root 3 over 2.
Dialogue: 0,1:20:11.83,1:20:18.66,Default,,0000,0000,0000,,l root 3 over 2 is\Nexactly the height.
Dialogue: 0,1:20:18.66,1:20:24.54,Default,,0000,0000,0000,,And then the area will be\Nheight times the base over 2
Dialogue: 0,1:20:24.54,1:20:25.94,Default,,0000,0000,0000,,for any triangle.
Dialogue: 0,1:20:25.94,1:20:33.86,Default,,0000,0000,0000,,So I have the height times\Nthe base over 2, which
Dialogue: 0,1:20:33.86,1:20:38.55,Default,,0000,0000,0000,,is root 3l squared over 4.
Dialogue: 0,1:20:38.55,1:20:42.37,Default,,0000,0000,0000,,So many people when\Nthey were young
Dialogue: 0,1:20:42.37,1:20:44.68,Default,,0000,0000,0000,,had to learn it in\Nseventh grade by heart.
Dialogue: 0,1:20:44.68,1:20:48.07,Default,,0000,0000,0000,,Now in our case, it\Nshould be a piece of cake.
Dialogue: 0,1:20:48.07,1:20:48.57,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:20:48.57,1:20:53.61,Default,,0000,0000,0000,,Because we know who l is.
Dialogue: 0,1:20:53.61,1:20:56.54,Default,,0000,0000,0000,,l is going to be the hypotenuse.
Dialogue: 0,1:20:56.54,1:21:03.01,Default,,0000,0000,0000,,We have here a 1 and\Na 1, a 1 and a 1.
Dialogue: 0,1:21:03.01,1:21:06.95,Default,,0000,0000,0000,,So this is going to be the\Nhypotenuse, square root of 2.
Dialogue: 0,1:21:06.95,1:21:09.58,Default,,0000,0000,0000,,So if I apply this\Nformula, which
Dialogue: 0,1:21:09.58,1:21:13.13,Default,,0000,0000,0000,,is the area of the\Nequilateral triangle,
Dialogue: 0,1:21:13.13,1:21:23.21,Default,,0000,0000,0000,,that says root 2 squared root\N3 over 4 equals 2 root 3 over 4
Dialogue: 0,1:21:23.21,1:21:24.75,Default,,0000,0000,0000,,equals root 3 over 2.
Dialogue: 0,1:21:24.75,1:21:28.40,Default,,0000,0000,0000,,
Dialogue: 0,1:21:28.40,1:21:32.96,Default,,0000,0000,0000,,So can you do that?
Dialogue: 0,1:21:32.96,1:21:35.11,Default,,0000,0000,0000,,Are you allowed to do that?
Dialogue: 0,1:21:35.11,1:21:38.91,Default,,0000,0000,0000,,Well, we never formulated\Nit actually saying
Dialogue: 0,1:21:38.91,1:21:46.16,Default,,0000,0000,0000,,compute the surface of\Nthis patch of a plane using
Dialogue: 0,1:21:46.16,1:21:47.90,Default,,0000,0000,0000,,the surface integral.
Dialogue: 0,1:21:47.90,1:21:49.10,Default,,0000,0000,0000,,We didn't say that.
Dialogue: 0,1:21:49.10,1:21:53.41,Default,,0000,0000,0000,,We said, compute it,\Nperiod We didn't care how.
Dialogue: 0,1:21:53.41,1:21:55.72,Default,,0000,0000,0000,,If you can do it\Nby another method,
Dialogue: 0,1:21:55.72,1:21:58.85,Default,,0000,0000,0000,,whether to stick to that\Nmethod, elementary method
Dialogue: 0,1:21:58.85,1:22:01.27,Default,,0000,0000,0000,,or to just check\Nyour work and say,
Dialogue: 0,1:22:01.27,1:22:03.42,Default,,0000,0000,0000,,is it really a square\Nroot of 3 over 2?
Dialogue: 0,1:22:03.42,1:22:06.86,Default,,0000,0000,0000,,You are allowed to do that.
Dialogue: 0,1:22:06.86,1:22:07.93,Default,,0000,0000,0000,,Questions?
Dialogue: 0,1:22:07.93,1:22:10.49,Default,,0000,0000,0000,,STUDENT: So would the\Nlength be square root
Dialogue: 0,1:22:10.49,1:22:12.58,Default,,0000,0000,0000,,of 2 squared, which is 2.
Dialogue: 0,1:22:12.58,1:22:16.77,Default,,0000,0000,0000,,2 divided by 4 is [INAUDIBLE]\Nsquare root of 3 over 2.
Dialogue: 0,1:22:16.77,1:22:17.94,Default,,0000,0000,0000,,I'm just talking-- oh, yeah.
Dialogue: 0,1:22:17.94,1:22:20.23,Default,,0000,0000,0000,,DR. MAGDALENA TODA: You are\Njust repeating what I have.
Dialogue: 0,1:22:20.23,1:22:23.20,Default,,0000,0000,0000,,So the answer i got\Nlike this is elementary.
Dialogue: 0,1:22:23.20,1:22:27.93,Default,,0000,0000,0000,,And the answer I got\Nlike this is with Calc 3.
Dialogue: 0,1:22:27.93,1:22:30.97,Default,,0000,0000,0000,,It's the same answer, which\Ngives me the reassurance
Dialogue: 0,1:22:30.97,1:22:32.39,Default,,0000,0000,0000,,I wasn't speaking nonsense.
Dialogue: 0,1:22:32.39,1:22:38.13,Default,,0000,0000,0000,,I did it in two different ways,\Nand I got the same answer.
Dialogue: 0,1:22:38.13,1:22:41.68,Default,,0000,0000,0000,,Let's do one or\Ntwo more examples
Dialogue: 0,1:22:41.68,1:22:48.91,Default,,0000,0000,0000,,of surface integrals, surface\Nareas and surface integrals.
Dialogue: 0,1:22:48.91,1:22:49.93,Default,,0000,0000,0000,,It's not hard.
Dialogue: 0,1:22:49.93,1:22:52.27,Default,,0000,0000,0000,,It's actually quite fun.
Dialogue: 0,1:22:52.27,1:22:54.53,Default,,0000,0000,0000,,Some of them are\Nharder than others.
Dialogue: 0,1:22:54.53,1:22:55.98,Default,,0000,0000,0000,,Let's see what we can do.
Dialogue: 0,1:22:55.98,1:23:01.52,Default,,0000,0000,0000,,
Dialogue: 0,1:23:01.52,1:23:02.02,Default,,0000,0000,0000,,Oh, yeah.
Dialogue: 0,1:23:02.02,1:23:03.83,Default,,0000,0000,0000,,I like this one very much.
Dialogue: 0,1:23:03.83,1:23:08.63,Default,,0000,0000,0000,,
Dialogue: 0,1:23:08.63,1:23:13.76,Default,,0000,0000,0000,,I remember we gave it several\Ntimes on the final exams.
Dialogue: 0,1:23:13.76,1:23:16.50,Default,,0000,0000,0000,,So let's go ahead\Nand do one like that
Dialogue: 0,1:23:16.50,1:23:19.51,Default,,0000,0000,0000,,because you've\Nseen-- why don't we
Dialogue: 0,1:23:19.51,1:23:23.07,Default,,0000,0000,0000,,pick the one I picked before\Nwith the eggshells for Easter,
Dialogue: 0,1:23:23.07,1:23:24.23,Default,,0000,0000,0000,,like Easter eggs?
Dialogue: 0,1:23:24.23,1:23:30.04,Default,,0000,0000,0000,,What was the paraboloid I\Nhad on top, the one on top?
Dialogue: 0,1:23:30.04,1:23:30.87,Default,,0000,0000,0000,,STUDENT: 8 minus x--
Dialogue: 0,1:23:30.87,1:23:32.96,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 8 minus\Nx squared [INAUDIBLE].
Dialogue: 0,1:23:32.96,1:23:34.46,Default,,0000,0000,0000,,That's what I'm going to pick.
Dialogue: 0,1:23:34.46,1:23:40.32,Default,,0000,0000,0000,,And I'll say, as Easter\Nis coming, a word problem.
Dialogue: 0,1:23:40.32,1:23:47.19,Default,,0000,0000,0000,,We want to compute the\Nsurface of an egg that
Dialogue: 0,1:23:47.19,1:23:56.34,Default,,0000,0000,0000,,is created by intersecting\Nthe two paraboloids 8 minus x
Dialogue: 0,1:23:56.34,1:23:59.21,Default,,0000,0000,0000,,squared minus y squared and\Nx squared plus y squared.
Dialogue: 0,1:23:59.21,1:24:00.16,Default,,0000,0000,0000,,So let's see.
Dialogue: 0,1:24:00.16,1:24:04.47,Default,,0000,0000,0000,,
Dialogue: 0,1:24:04.47,1:24:05.33,Default,,0000,0000,0000,,No.
Dialogue: 0,1:24:05.33,1:24:06.70,Default,,0000,0000,0000,,Not y, Magdalena.
Dialogue: 0,1:24:06.70,1:24:16.76,Default,,0000,0000,0000,,
Dialogue: 0,1:24:16.76,1:24:20.59,Default,,0000,0000,0000,,Intersect, make\Nthe egg intersect.
Dialogue: 0,1:24:20.59,1:24:21.49,Default,,0000,0000,0000,,Create the eggshells.
Dialogue: 0,1:24:21.49,1:24:25.34,Default,,0000,0000,0000,,
Dialogue: 0,1:24:25.34,1:24:26.76,Default,,0000,0000,0000,,Shells.
Dialogue: 0,1:24:26.76,1:24:29.40,Default,,0000,0000,0000,,Compute the area.
Dialogue: 0,1:24:29.40,1:24:32.01,Default,,0000,0000,0000,,
Dialogue: 0,1:24:32.01,1:24:33.53,Default,,0000,0000,0000,,And you say, wait a minute.
Dialogue: 0,1:24:33.53,1:24:37.39,Default,,0000,0000,0000,,The two eggshells were equal.
Dialogue: 0,1:24:37.39,1:24:39.24,Default,,0000,0000,0000,,Yes, I know.
Dialogue: 0,1:24:39.24,1:24:42.49,Default,,0000,0000,0000,,I know that the two\Neggshells were equal.
Dialogue: 0,1:24:42.49,1:24:44.51,Default,,0000,0000,0000,,But they don't look\Nequal in my picture.
Dialogue: 0,1:24:44.51,1:24:45.49,Default,,0000,0000,0000,,I'll try better.
Dialogue: 0,1:24:45.49,1:24:48.78,Default,,0000,0000,0000,,
Dialogue: 0,1:24:48.78,1:24:50.71,Default,,0000,0000,0000,,Assume they are parabolas.
Dialogue: 0,1:24:50.71,1:24:56.04,Default,,0000,0000,0000,,
Dialogue: 0,1:24:56.04,1:24:58.02,Default,,0000,0000,0000,,Assume this was z equals 4.
Dialogue: 0,1:24:58.02,1:25:01.11,Default,,0000,0000,0000,,This was 8 minus x\Nsquared minus y squared.
Dialogue: 0,1:25:01.11,1:25:04.55,Default,,0000,0000,0000,,This was x squared\Nplus y squared.
Dialogue: 0,1:25:04.55,1:25:08.72,Default,,0000,0000,0000,,How do we compute-- just\Nlike Matthew observed, 8
Dialogue: 0,1:25:08.72,1:25:09.71,Default,,0000,0000,0000,,for the volume.
Dialogue: 0,1:25:09.71,1:25:14.04,Default,,0000,0000,0000,,I only need half of the\N8 multiplied by the 2.
Dialogue: 0,1:25:14.04,1:25:14.74,Default,,0000,0000,0000,,The same thing.
Dialogue: 0,1:25:14.74,1:25:22.25,Default,,0000,0000,0000,,I'm going to take one of\Nthe two shells, this one.
Dialogue: 0,1:25:22.25,1:25:29.14,Default,,0000,0000,0000,,And the surface of the egg will\Nbe twice times the surface S1.
Dialogue: 0,1:25:29.14,1:25:31.70,Default,,0000,0000,0000,,All I have to\Ncompute is S1, right?
Dialogue: 0,1:25:31.70,1:25:33.83,Default,,0000,0000,0000,,It shouldn't be a big problem.
Dialogue: 0,1:25:33.83,1:25:35.84,Default,,0000,0000,0000,,I mean, what do I\Nneed for that S1?
Dialogue: 0,1:25:35.84,1:25:42.59,Default,,0000,0000,0000,,I only need the shadow of\Nit and the expression of it.
Dialogue: 0,1:25:42.59,1:25:49.06,Default,,0000,0000,0000,,The shadow of it and the--\Nthe shadow of it is this.
Dialogue: 0,1:25:49.06,1:25:52.13,Default,,0000,0000,0000,,The shadow of this is this.
Dialogue: 0,1:25:52.13,1:25:53.70,Default,,0000,0000,0000,,And the expression-- hmm.
Dialogue: 0,1:25:53.70,1:25:54.64,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,1:25:54.64,1:25:57.64,Default,,0000,0000,0000,,
Dialogue: 0,1:25:57.64,1:26:01.90,Default,,0000,0000,0000,,So I'm going to-- I'm\Ngoing to start asking
Dialogue: 0,1:26:01.90,1:26:05.26,Default,,0000,0000,0000,,you to tell me what to write.
Dialogue: 0,1:26:05.26,1:26:09.38,Default,,0000,0000,0000,,
Dialogue: 0,1:26:09.38,1:26:11.36,Default,,0000,0000,0000,,What?
Dialogue: 0,1:26:11.36,1:26:13.23,Default,,0000,0000,0000,,STUDENT: Square root of 1 plus--
Dialogue: 0,1:26:13.23,1:26:14.18,Default,,0000,0000,0000,,DR. MAGDALENA TODA: No.
Dialogue: 0,1:26:14.18,1:26:15.60,Default,,0000,0000,0000,,First I will write\Nthe definition.
Dialogue: 0,1:26:15.60,1:26:19.56,Default,,0000,0000,0000,,Double integral over D,\Nsquare root of, as you said--
Dialogue: 0,1:26:19.56,1:26:20.47,Default,,0000,0000,0000,,say it again.
Dialogue: 0,1:26:20.47,1:26:26.16,Default,,0000,0000,0000,,STUDENT: 1 plus f\Nof x squared plus f
Dialogue: 0,1:26:26.16,1:26:31.81,Default,,0000,0000,0000,,of y squared [INAUDIBLE] dA.
Dialogue: 0,1:26:31.81,1:26:34.49,Default,,0000,0000,0000,,
Dialogue: 0,1:26:34.49,1:26:35.74,Default,,0000,0000,0000,,DR. MAGDALENA TODA: All right.
Dialogue: 0,1:26:35.74,1:26:40.12,Default,,0000,0000,0000,,So this is ds, and I'm\Nintegrating over the domain.
Dialogue: 0,1:26:40.12,1:26:41.62,Default,,0000,0000,0000,,Should this be hard?
Dialogue: 0,1:26:41.62,1:26:44.49,Default,,0000,0000,0000,,No, it shouldn't be hard.
Dialogue: 0,1:26:44.49,1:26:48.11,Default,,0000,0000,0000,,But I'm going to get\Nsomething a little bit ugly.
Dialogue: 0,1:26:48.11,1:26:53.08,Default,,0000,0000,0000,,And it doesn't matter, because\Nwe will do it with no problem.
Dialogue: 0,1:26:53.08,1:26:58.02,Default,,0000,0000,0000,,I'm going to say, integral\Nover-- now the domain
Dialogue: 0,1:26:58.02,1:27:00.46,Default,,0000,0000,0000,,D-- I know what it is\Nbecause the domain D will
Dialogue: 0,1:27:00.46,1:27:06.75,Default,,0000,0000,0000,,be given by x squared\Nplus y squared less than
Dialogue: 0,1:27:06.75,1:27:08.18,Default,,0000,0000,0000,,or equal to 4.
Dialogue: 0,1:27:08.18,1:27:13.57,Default,,0000,0000,0000,,So I would know how to\Ndeal with that later on.
Dialogue: 0,1:27:13.57,1:27:18.17,Default,,0000,0000,0000,,Now what scares me off\Na little bit-- and look
Dialogue: 0,1:27:18.17,1:27:19.94,Default,,0000,0000,0000,,what's going to happen.
Dialogue: 0,1:27:19.94,1:27:30.22,Default,,0000,0000,0000,,When I compute f sub x and f sub\Ny, those will be really easy.
Dialogue: 0,1:27:30.22,1:27:35.17,Default,,0000,0000,0000,,But when I plug\Neverything in here,
Dialogue: 0,1:27:35.17,1:27:39.06,Default,,0000,0000,0000,,it's going to be\Na little bit hard.
Dialogue: 0,1:27:39.06,1:27:41.49,Default,,0000,0000,0000,,Never mind, I'm\Ngoing to have to have
Dialogue: 0,1:27:41.49,1:27:45.51,Default,,0000,0000,0000,,to battle the problem\Nwith polar coordinates.
Dialogue: 0,1:27:45.51,1:27:50.75,Default,,0000,0000,0000,,That's why polar coordinates\Nexist, to help us.
Dialogue: 0,1:27:50.75,1:27:54.62,Default,,0000,0000,0000,,So f sub x is minus 2x, right?
Dialogue: 0,1:27:54.62,1:27:56.48,Default,,0000,0000,0000,,f sub y is minus 2y.
Dialogue: 0,1:27:56.48,1:27:59.71,Default,,0000,0000,0000,,
Dialogue: 0,1:27:59.71,1:28:00.34,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:28:00.34,1:28:03.82,Default,,0000,0000,0000,,So what am I going\Nto write over here?
Dialogue: 0,1:28:03.82,1:28:13.94,Default,,0000,0000,0000,,Minus 2x squared plus\Nminus 2y squared dx.
Dialogue: 0,1:28:13.94,1:28:15.51,Default,,0000,0000,0000,,I don't have much room.
Dialogue: 0,1:28:15.51,1:28:17.70,Default,,0000,0000,0000,,But that would mean dxdy.
Dialogue: 0,1:28:17.70,1:28:19.87,Default,,0000,0000,0000,,Am I happy with that?
Dialogue: 0,1:28:19.87,1:28:20.90,Default,,0000,0000,0000,,No.
Dialogue: 0,1:28:20.90,1:28:23.59,Default,,0000,0000,0000,,I'm not happy with\Nit, because here it's
Dialogue: 0,1:28:23.59,1:28:31.57,Default,,0000,0000,0000,,going to be x squared plus\Ny squared between 0 and 4.
Dialogue: 0,1:28:31.57,1:28:35.78,Default,,0000,0000,0000,,
Dialogue: 0,1:28:35.78,1:28:39.39,Default,,0000,0000,0000,,And I'm not happy with it,\Nbecause it looks like a mess.
Dialogue: 0,1:28:39.39,1:28:46.56,Default,,0000,0000,0000,,And I have to find this area\Nintegral with a simple method,
Dialogue: 0,1:28:46.56,1:28:49.48,Default,,0000,0000,0000,,something nicer.
Dialogue: 0,1:28:49.48,1:28:53.78,Default,,0000,0000,0000,,Now the question is,\Ndoes my elementary math
Dialogue: 0,1:28:53.78,1:28:57.08,Default,,0000,0000,0000,,help me find the\Narea of the egg?
Dialogue: 0,1:28:57.08,1:28:59.34,Default,,0000,0000,0000,,Unfortunately, no.
Dialogue: 0,1:28:59.34,1:29:03.08,Default,,0000,0000,0000,,So from this point on, it's\Ngoodbye elementary geometry.
Dialogue: 0,1:29:03.08,1:29:04.88,Default,,0000,0000,0000,,STUDENT: Unless you\Nknow the radius.
Dialogue: 0,1:29:04.88,1:29:07.25,Default,,0000,0000,0000,,DR. MAGDALENA TODA: But they\Nare not spheres or anything.
Dialogue: 0,1:29:07.25,1:29:09.42,Default,,0000,0000,0000,,I can approximate the\Neggs with spheres,
Dialogue: 0,1:29:09.42,1:29:14.44,Default,,0000,0000,0000,,but I cannot do anything with\Nthose paraboloids [INAUDIBLE].
Dialogue: 0,1:29:14.44,1:29:17.49,Default,,0000,0000,0000,,STUDENT: I know the\Nfunction of the top.
Dialogue: 0,1:29:17.49,1:29:19.25,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYeah, yeah, yeah.
Dialogue: 0,1:29:19.25,1:29:20.00,Default,,0000,0000,0000,,You can.
Dialogue: 0,1:29:20.00,1:29:22.16,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nthe integration f prime.
Dialogue: 0,1:29:22.16,1:29:24.12,Default,,0000,0000,0000,,DR. MAGDALENA TODA: But\Nit's still integration.
Dialogue: 0,1:29:24.12,1:29:28.26,Default,,0000,0000,0000,,So can I pretend like I'm a\Nsmart sixth grader, and I can--
Dialogue: 0,1:29:28.26,1:29:32.26,Default,,0000,0000,0000,,how can I measure that if I'm\Nin sixth grade or seventh grade?
Dialogue: 0,1:29:32.26,1:29:35.67,Default,,0000,0000,0000,,With some sort of graphic paper,\Ndo some sort of approximation
Dialogue: 0,1:29:35.67,1:29:37.75,Default,,0000,0000,0000,,of the area of the egg.
Dialogue: 0,1:29:37.75,1:29:41.13,Default,,0000,0000,0000,,It's a school project that's\Nnot worth anything because I
Dialogue: 0,1:29:41.13,1:29:43.73,Default,,0000,0000,0000,,think not even at a science\Nfair, could I do it.
Dialogue: 0,1:29:43.73,1:29:47.90,Default,,0000,0000,0000,,STUDENT: Unless--\Nin the same radius,
Dialogue: 0,1:29:47.90,1:29:51.31,Default,,0000,0000,0000,,I can draw the sphere in.
Dialogue: 0,1:29:51.31,1:29:54.24,Default,,0000,0000,0000,,Then if I apply the\Ndistance between the sphere
Dialogue: 0,1:29:54.24,1:29:57.40,Default,,0000,0000,0000,,and the [INAUDIBLE] the\Ndistance between [INAUDIBLE]
Dialogue: 0,1:29:57.40,1:29:59.04,Default,,0000,0000,0000,,and take it all from there.
Dialogue: 0,1:29:59.04,1:30:02.68,Default,,0000,0000,0000,,But then the function\Nactually will look easier
Dialogue: 0,1:30:02.68,1:30:06.77,Default,,0000,0000,0000,,because it will go from the\Ny axis up to the A axis,
Dialogue: 0,1:30:06.77,1:30:08.40,Default,,0000,0000,0000,,and they meet each other.
Dialogue: 0,1:30:08.40,1:30:10.19,Default,,0000,0000,0000,,So I took up the\Narea and took up
Dialogue: 0,1:30:10.19,1:30:11.44,Default,,0000,0000,0000,,the other area to [INAUDIBLE].
Dialogue: 0,1:30:11.44,1:30:13.16,Default,,0000,0000,0000,,
Dialogue: 0,1:30:13.16,1:30:14.20,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Yeah.
Dialogue: 0,1:30:14.20,1:30:17.33,Default,,0000,0000,0000,,Well, wouldn't that\Nsurface of the egg still
Dialogue: 0,1:30:17.33,1:30:20.59,Default,,0000,0000,0000,,be an approximation\Nof the actual answer?
Dialogue: 0,1:30:20.59,1:30:23.71,Default,,0000,0000,0000,,Anyway, let's come\Nback to the egg.
Dialogue: 0,1:30:23.71,1:30:25.45,Default,,0000,0000,0000,,The egg, the egg.
Dialogue: 0,1:30:25.45,1:30:27.00,Default,,0000,0000,0000,,The egg is [INAUDIBLE].
Dialogue: 0,1:30:27.00,1:30:27.72,Default,,0000,0000,0000,,It's nice.
Dialogue: 0,1:30:27.72,1:30:30.53,Default,,0000,0000,0000,,1 plus 4 x squared\Nplus y squared.
Dialogue: 0,1:30:30.53,1:30:33.87,Default,,0000,0000,0000,,Look at the beauty of the\Nsymmetry of polynomials.
Dialogue: 0,1:30:33.87,1:30:36.54,Default,,0000,0000,0000,,x squared plus y squared says,\NI'm a symmetric polynomial.
Dialogue: 0,1:30:36.54,1:30:39.87,Default,,0000,0000,0000,,You're my friend\Nbecause I'm r squared,
Dialogue: 0,1:30:39.87,1:30:42.20,Default,,0000,0000,0000,,and I know what I'm going to do.
Dialogue: 0,1:30:42.20,1:30:43.93,Default,,0000,0000,0000,,So how do we compute?
Dialogue: 0,1:30:43.93,1:30:46.99,Default,,0000,0000,0000,,What kind of integral\Ndo we need to compute?
Dialogue: 0,1:30:46.99,1:30:53.57,Default,,0000,0000,0000,,So S1 will be the integral of\Nintegral of the square root
Dialogue: 0,1:30:53.57,1:30:58.12,Default,,0000,0000,0000,,of 1 plus 4r squared.
Dialogue: 0,1:30:58.12,1:31:02.17,Default,,0000,0000,0000,,Don't forget the dA\Ncontains the Jacobian.
Dialogue: 0,1:31:02.17,1:31:04.52,Default,,0000,0000,0000,,So don't write drd theta.
Dialogue: 0,1:31:04.52,1:31:06.80,Default,,0000,0000,0000,,I had a student who wrote that.
Dialogue: 0,1:31:06.80,1:31:10.02,Default,,0000,0000,0000,,That is worth\Nexactly zero points.
Dialogue: 0,1:31:10.02,1:31:13.20,Default,,0000,0000,0000,,So say, times r.
Dialogue: 0,1:31:13.20,1:31:19.15,Default,,0000,0000,0000,,r between-- oh, my god,\Nthe poor egg-- 0 to 2.
Dialogue: 0,1:31:19.15,1:31:24.03,Default,,0000,0000,0000,,And theta between 0 to 2 pi.
Dialogue: 0,1:31:24.03,1:31:24.84,Default,,0000,0000,0000,,And come on.
Dialogue: 0,1:31:24.84,1:31:27.68,Default,,0000,0000,0000,,We've done that in Calc 2.
Dialogue: 0,1:31:27.68,1:31:33.30,Default,,0000,0000,0000,,I mean, it's not so hard.
Dialogue: 0,1:31:33.30,1:31:35.77,Default,,0000,0000,0000,,So u substitution.
Dialogue: 0,1:31:35.77,1:31:40.32,Default,,0000,0000,0000,,u is 4r squared plus 1.
Dialogue: 0,1:31:40.32,1:31:41.42,Default,,0000,0000,0000,,That's our only hope.
Dialogue: 0,1:31:41.42,1:31:43.47,Default,,0000,0000,0000,,We have no other hope.
Dialogue: 0,1:31:43.47,1:31:48.41,Default,,0000,0000,0000,,du is going to be 8rdr.
Dialogue: 0,1:31:48.41,1:31:51.01,Default,,0000,0000,0000,,And rdr is a married couple.
Dialogue: 0,1:31:51.01,1:31:52.22,Default,,0000,0000,0000,,They stick together.
Dialogue: 0,1:31:52.22,1:31:53.35,Default,,0000,0000,0000,,Where is the purple?
Dialogue: 0,1:31:53.35,1:31:54.78,Default,,0000,0000,0000,,The purple is here.
Dialogue: 0,1:31:54.78,1:31:58.52,Default,,0000,0000,0000,,rdr, rdr.
Dialogue: 0,1:31:58.52,1:31:59.39,Default,,0000,0000,0000,,rdr is du/8.
Dialogue: 0,1:31:59.39,1:32:03.21,Default,,0000,0000,0000,,
Dialogue: 0,1:32:03.21,1:32:05.91,Default,,0000,0000,0000,,This fellow's name is u.
Dialogue: 0,1:32:05.91,1:32:07.27,Default,,0000,0000,0000,,He is u.
Dialogue: 0,1:32:07.27,1:32:09.14,Default,,0000,0000,0000,,He is not u, but he's like u.
Dialogue: 0,1:32:09.14,1:32:11.45,Default,,0000,0000,0000,,OK, not necessary.
Dialogue: 0,1:32:11.45,1:32:11.95,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:32:11.95,1:32:16.01,Default,,0000,0000,0000,,So you go 2 pi-- because\Nthere is no theta.
Dialogue: 0,1:32:16.01,1:32:20.84,Default,,0000,0000,0000,,So no theta means-- let me\Nwrite it one more time for you.
Dialogue: 0,1:32:20.84,1:32:23.98,Default,,0000,0000,0000,,The integral from\N0 to 2 pi 1d theta.
Dialogue: 0,1:32:23.98,1:32:26.02,Default,,0000,0000,0000,,And he goes out and has fun.
Dialogue: 0,1:32:26.02,1:32:27.85,Default,,0000,0000,0000,,This is 2 pi.
Dialogue: 0,1:32:27.85,1:32:33.53,Default,,0000,0000,0000,,But then all you have left\Ninside is the integral of u.
Dialogue: 0,1:32:33.53,1:32:43.76,Default,,0000,0000,0000,,Square root of u times\N1/8 du, close the bracket,
Dialogue: 0,1:32:43.76,1:32:52.89,Default,,0000,0000,0000,,where u is between 1 and 17.
Dialogue: 0,1:32:52.89,1:32:54.84,Default,,0000,0000,0000,,Isn't that beautiful?
Dialogue: 0,1:32:54.84,1:32:55.57,Default,,0000,0000,0000,,That's 17.
Dialogue: 0,1:32:55.57,1:33:00.40,Default,,0000,0000,0000,,So you have 2 squared\Ntimes 416 plus 117.
Dialogue: 0,1:33:00.40,1:33:03.61,Default,,0000,0000,0000,,But believe me that from this\Nviewpoint, from this point on,
Dialogue: 0,1:33:03.61,1:33:05.58,Default,,0000,0000,0000,,it's not really hard.
Dialogue: 0,1:33:05.58,1:33:07.79,Default,,0000,0000,0000,,It just looks like the\Nsurface of that egg
Dialogue: 0,1:33:07.79,1:33:12.80,Default,,0000,0000,0000,,is-- whenever it was produced,\Nin what factory, in whatever
Dialogue: 0,1:33:12.80,1:33:19.00,Default,,0000,0000,0000,,country is the toy factory, they\Nmust have done this area stage.
Dialogue: 0,1:33:19.00,1:33:21.68,Default,,0000,0000,0000,,So you have 2 pi.
Dialogue: 0,1:33:21.68,1:33:27.06,Default,,0000,0000,0000,,1/8 comes out, whether\Nhe wants out or not.
Dialogue: 0,1:33:27.06,1:33:28.32,Default,,0000,0000,0000,,Integral of square root of u.
Dialogue: 0,1:33:28.32,1:33:30.19,Default,,0000,0000,0000,,Do you like that?
Dialogue: 0,1:33:30.19,1:33:31.89,Default,,0000,0000,0000,,I don't.
Dialogue: 0,1:33:31.89,1:33:33.32,Default,,0000,0000,0000,,You have--
Dialogue: 0,1:33:33.32,1:33:34.11,Default,,0000,0000,0000,,STUDENT: 2/3.
Dialogue: 0,1:33:34.11,1:33:38.04,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\N2/3 u to the 3/2
Dialogue: 0,1:33:38.04,1:33:40.92,Default,,0000,0000,0000,,between-- down is u equals 1.
Dialogue: 0,1:33:40.92,1:33:42.58,Default,,0000,0000,0000,,Up is u equals 17.
Dialogue: 0,1:33:42.58,1:33:44.62,Default,,0000,0000,0000,,So I was asked,\Nbecause we've done
Dialogue: 0,1:33:44.62,1:33:47.80,Default,,0000,0000,0000,,this in the past\Nreviews for the finals--
Dialogue: 0,1:33:47.80,1:33:50.16,Default,,0000,0000,0000,,and several finals\Nare like that.
Dialogue: 0,1:33:50.16,1:33:53.64,Default,,0000,0000,0000,,My students asked me, what\Ndo I do in such a case?
Dialogue: 0,1:33:53.64,1:33:54.23,Default,,0000,0000,0000,,Nothing.
Dialogue: 0,1:33:54.23,1:33:55.84,Default,,0000,0000,0000,,I mean, you do nothing.
Dialogue: 0,1:33:55.84,1:33:57.87,Default,,0000,0000,0000,,You just plug it in\Nand leave it as is.
Dialogue: 0,1:33:57.87,1:34:02.24,Default,,0000,0000,0000,,So you have-- to simplify\Nyour life a little bit, what
Dialogue: 0,1:34:02.24,1:34:04.51,Default,,0000,0000,0000,,you can do is 2, 2, and 8.
Dialogue: 0,1:34:04.51,1:34:05.82,Default,,0000,0000,0000,,What is 2 times 2?
Dialogue: 0,1:34:05.82,1:34:06.32,Default,,0000,0000,0000,,4.
Dialogue: 0,1:34:06.32,1:34:13.01,Default,,0000,0000,0000,,Divided by 8-- so you\Nhave pi/6 overall.
Dialogue: 0,1:34:13.01,1:34:17.42,Default,,0000,0000,0000,,
Dialogue: 0,1:34:17.42,1:34:26.50,Default,,0000,0000,0000,,pi/6 times 17 to\Nthe 3/2 and minus 1.
Dialogue: 0,1:34:26.50,1:34:29.75,Default,,0000,0000,0000,,
Dialogue: 0,1:34:29.75,1:34:33.71,Default,,0000,0000,0000,,One of my students, after he\Ngot such an answer last time
Dialogue: 0,1:34:33.71,1:34:37.48,Default,,0000,0000,0000,,we did the review, he\Nsaid, I don't like it.
Dialogue: 0,1:34:37.48,1:34:41.35,Default,,0000,0000,0000,,I want to write this as\Nsquare root of 17 cubed.
Dialogue: 0,1:34:41.35,1:34:43.38,Default,,0000,0000,0000,,You can write it\Nwhatever you want.
Dialogue: 0,1:34:43.38,1:34:45.34,Default,,0000,0000,0000,,It can be-- it\Nhas to be correct.
Dialogue: 0,1:34:45.34,1:34:47.55,Default,,0000,0000,0000,,I don't care how you write it.
Dialogue: 0,1:34:47.55,1:34:49.22,Default,,0000,0000,0000,,What if you mess up?
Dialogue: 0,1:34:49.22,1:34:51.82,Default,,0000,0000,0000,,You say, well, this\Nwoman is killing me
Dialogue: 0,1:34:51.82,1:34:54.10,Default,,0000,0000,0000,,with her algebra over here.
Dialogue: 0,1:34:54.10,1:34:54.77,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:34:54.77,1:34:56.72,Default,,0000,0000,0000,,If you understood--\Nsuppose that you
Dialogue: 0,1:34:56.72,1:34:59.16,Default,,0000,0000,0000,,are taking the final right now.
Dialogue: 0,1:34:59.16,1:35:00.53,Default,,0000,0000,0000,,You drew the\Npicture beautifully.
Dialogue: 0,1:35:00.53,1:35:01.57,Default,,0000,0000,0000,,You remember the problem.
Dialogue: 0,1:35:01.57,1:35:02.77,Default,,0000,0000,0000,,You remember the formula.
Dialogue: 0,1:35:02.77,1:35:04.50,Default,,0000,0000,0000,,You write it down.
Dialogue: 0,1:35:04.50,1:35:05.45,Default,,0000,0000,0000,,You wrote it down.
Dialogue: 0,1:35:05.45,1:35:06.46,Default,,0000,0000,0000,,You got to this point.
Dialogue: 0,1:35:06.46,1:35:10.67,Default,,0000,0000,0000,,At this point, you already\Nhave 50% of the problem.
Dialogue: 0,1:35:10.67,1:35:11.37,Default,,0000,0000,0000,,Yup.
Dialogue: 0,1:35:11.37,1:35:16.09,Default,,0000,0000,0000,,And then from this point on,\Nyou do the polar coordinates,
Dialogue: 0,1:35:16.09,1:35:18.94,Default,,0000,0000,0000,,and you still get another 25%.
Dialogue: 0,1:35:18.94,1:35:20.15,Default,,0000,0000,0000,,You messed it up.
Dialogue: 0,1:35:20.15,1:35:21.91,Default,,0000,0000,0000,,You lose some partial credit.
Dialogue: 0,1:35:21.91,1:35:25.28,Default,,0000,0000,0000,,But everything you\Nwrite correctly
Dialogue: 0,1:35:25.28,1:35:28.95,Default,,0000,0000,0000,,earns and earns\Nand earns points.
Dialogue: 0,1:35:28.95,1:35:29.76,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:35:29.76,1:35:31.93,Default,,0000,0000,0000,,So don't freak out\Nthinking, I'm going
Dialogue: 0,1:35:31.93,1:35:34.24,Default,,0000,0000,0000,,to mess up my algebra for sure.
Dialogue: 0,1:35:34.24,1:35:38.51,Default,,0000,0000,0000,,If you do, it doesn't matter,\Nbecause even if this would
Dialogue: 0,1:35:38.51,1:35:40.63,Default,,0000,0000,0000,,be a multiple choice--\Nsome problems will
Dialogue: 0,1:35:40.63,1:35:43.22,Default,,0000,0000,0000,,be show work completely,\Nand some problems
Dialogue: 0,1:35:43.22,1:35:44.91,Default,,0000,0000,0000,,may be multiple\Nchoice questions.
Dialogue: 0,1:35:44.91,1:35:49.83,Default,,0000,0000,0000,,Even if this is going\Nto be a multiple choice,
Dialogue: 0,1:35:49.83,1:35:54.29,Default,,0000,0000,0000,,I will still go over the entire\Ncomputation for everybody
Dialogue: 0,1:35:54.29,1:35:55.35,Default,,0000,0000,0000,,and give partial credit.
Dialogue: 0,1:35:55.35,1:35:59.71,Default,,0000,0000,0000,,This is my policy.
Dialogue: 0,1:35:59.71,1:36:03.41,Default,,0000,0000,0000,,We are allowed to choose\Nour policies as instructors.
Dialogue: 0,1:36:03.41,1:36:07.88,Default,,0000,0000,0000,,So you earn partial credit\Nfor everything you write down.
Dialogue: 0,1:36:07.88,1:36:08.63,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:36:08.63,1:36:09.57,Default,,0000,0000,0000,,Was this hard?
Dialogue: 0,1:36:09.57,1:36:11.90,Default,,0000,0000,0000,,It's one of the harder\Nproblems in the book.
Dialogue: 0,1:36:11.90,1:36:17.34,Default,,0000,0000,0000,,It is he similar to\Nexample number-- well,
Dialogue: 0,1:36:17.34,1:36:21.73,Default,,0000,0000,0000,,this is exactly like\Nexample 2 in the section.
Dialogue: 0,1:36:21.73,1:36:24.36,Default,,0000,0000,0000,,
Dialogue: 0,1:36:24.36,1:36:27.78,Default,,0000,0000,0000,,So we did these two\Nexamples from the section.
Dialogue: 0,1:36:27.78,1:36:31.22,Default,,0000,0000,0000,,And I want to give you one\Nmore piece of information
Dialogue: 0,1:36:31.22,1:36:36.48,Default,,0000,0000,0000,,that I saw, that unfortunately\Nmy colleagues don't teach that.
Dialogue: 0,1:36:36.48,1:36:38.82,Default,,0000,0000,0000,,And it sort of bothers me.
Dialogue: 0,1:36:38.82,1:36:39.87,Default,,0000,0000,0000,,I wish they did.
Dialogue: 0,1:36:39.87,1:36:43.05,Default,,0000,0000,0000,,
Dialogue: 0,1:36:43.05,1:36:45.60,Default,,0000,0000,0000,,Once upon a time,\Na long time ago,
Dialogue: 0,1:36:45.60,1:36:53.76,Default,,0000,0000,0000,,I taught you a little bit\Nmore about the parametrization
Dialogue: 0,1:36:53.76,1:36:55.74,Default,,0000,0000,0000,,of a surface.
Dialogue: 0,1:36:55.74,1:36:58.70,Default,,0000,0000,0000,,And I want to give you yet\Nanother formula, not just
Dialogue: 0,1:36:58.70,1:37:02.23,Default,,0000,0000,0000,,this one but one more.
Dialogue: 0,1:37:02.23,1:37:04.38,Default,,0000,0000,0000,,So what if you have a\Ngeneralized surface that
Dialogue: 0,1:37:04.38,1:37:10.04,Default,,0000,0000,0000,,is parametrized, meaning\Nthat your surface is not
Dialogue: 0,1:37:10.04,1:37:13.39,Default,,0000,0000,0000,,given as explicitly\Nz equals f of x, y?
Dialogue: 0,1:37:13.39,1:37:15.68,Default,,0000,0000,0000,,That's the lucky case.
Dialogue: 0,1:37:15.68,1:37:16.79,Default,,0000,0000,0000,,That's a graph.
Dialogue: 0,1:37:16.79,1:37:20.11,Default,,0000,0000,0000,,We call that a graph,\Nz equals f of x and y.
Dialogue: 0,1:37:20.11,1:37:21.57,Default,,0000,0000,0000,,And we call ourselves lucky.
Dialogue: 0,1:37:21.57,1:37:25.29,Default,,0000,0000,0000,,But life is not always so easy.
Dialogue: 0,1:37:25.29,1:37:35.02,Default,,0000,0000,0000,,Sometimes all you can get\Nis a parametrization r
Dialogue: 0,1:37:35.02,1:37:38.25,Default,,0000,0000,0000,,of v, v for a surface.
Dialogue: 0,1:37:38.25,1:37:42.19,Default,,0000,0000,0000,,And from that, you\Nhave to deal with that.
Dialogue: 0,1:37:42.19,1:37:46.53,Default,,0000,0000,0000,,So suppose somebody says,\NI don't give you f of x, y,
Dialogue: 0,1:37:46.53,1:37:50.53,Default,,0000,0000,0000,,although locally every\Nsurface looks like the graph.
Dialogue: 0,1:37:50.53,1:37:52.86,Default,,0000,0000,0000,,But a surface doesn't have\Nto be a graph in general.
Dialogue: 0,1:37:52.86,1:37:57.07,Default,,0000,0000,0000,,Locally, it does look like\Na graph on a small length.
Dialogue: 0,1:37:57.07,1:38:02.33,Default,,0000,0000,0000,,But in general, it's\Ngiven by r, v, v equals--
Dialogue: 0,1:38:02.33,1:38:03.82,Default,,0000,0000,0000,,and that was what?
Dialogue: 0,1:38:03.82,1:38:12.08,Default,,0000,0000,0000,,I gave you something like x of\Nu, v I plus y of u, v J plus z
Dialogue: 0,1:38:12.08,1:38:23.16,Default,,0000,0000,0000,,of u, v-- let's not put\Nthings in alphabetical order.
Dialogue: 0,1:38:23.16,1:38:29.00,Default,,0000,0000,0000,,
Dialogue: 0,1:38:29.00,1:38:34.26,Default,,0000,0000,0000,,z of u, v J and K.
Dialogue: 0,1:38:34.26,1:38:38.38,Default,,0000,0000,0000,,And we said that\Nwe have a point.
Dialogue: 0,1:38:38.38,1:38:44.39,Default,,0000,0000,0000,,P is our coordinate u0, v0.
Dialogue: 0,1:38:44.39,1:38:46.22,Default,,0000,0000,0000,,And we said we\Nlook at that point,
Dialogue: 0,1:38:46.22,1:38:48.98,Default,,0000,0000,0000,,and we try to draw the partials.
Dialogue: 0,1:38:48.98,1:38:52.85,Default,,0000,0000,0000,,What are the partials from\Na geometric viewpoint?
Dialogue: 0,1:38:52.85,1:38:55.96,Default,,0000,0000,0000,,Well, if I want to\Nwrite the partials,
Dialogue: 0,1:38:55.96,1:38:57.26,Default,,0000,0000,0000,,they would be various.
Dialogue: 0,1:38:57.26,1:39:01.46,Default,,0000,0000,0000,,It's going to be the vector\Nx sub u, y sub u, z sub
Dialogue: 0,1:39:01.46,1:39:11.03,Default,,0000,0000,0000,,u, and the vector x sub v, y\Nsub v, z sub v, two vectors.
Dialogue: 0,1:39:11.03,1:39:13.81,Default,,0000,0000,0000,,Do you remember when I\Ndrew them, what they were?
Dialogue: 0,1:39:13.81,1:39:17.00,Default,,0000,0000,0000,,
Dialogue: 0,1:39:17.00,1:39:19.73,Default,,0000,0000,0000,,We said the following.
Dialogue: 0,1:39:19.73,1:39:23.03,Default,,0000,0000,0000,,We said, let's assume\Nv will be a constant.
Dialogue: 0,1:39:23.03,1:39:26.57,Default,,0000,0000,0000,,
Dialogue: 0,1:39:26.57,1:39:28.61,Default,,0000,0000,0000,,So we say, v is a constant.
Dialogue: 0,1:39:28.61,1:39:30.34,Default,,0000,0000,0000,,And then v equals v0.
Dialogue: 0,1:39:30.34,1:39:33.08,Default,,0000,0000,0000,,And then you have P of u0, v0.
Dialogue: 0,1:39:33.08,1:39:38.28,Default,,0000,0000,0000,,
Dialogue: 0,1:39:38.28,1:39:43.60,Default,,0000,0000,0000,,And then we have another,\Nand we have u equals u0.
Dialogue: 0,1:39:43.60,1:39:49.33,Default,,0000,0000,0000,,
Dialogue: 0,1:39:49.33,1:39:53.74,Default,,0000,0000,0000,,This guy is going to\Nbe who of the two guys?
Dialogue: 0,1:39:53.74,1:39:54.95,Default,,0000,0000,0000,,r sub u.
Dialogue: 0,1:39:54.95,1:39:58.81,Default,,0000,0000,0000,,When we measure out\Nthe point P, r sub u
Dialogue: 0,1:39:58.81,1:40:07.33,Default,,0000,0000,0000,,is this guy, who is tangent\Nto the line r of u, v zero.
Dialogue: 0,1:40:07.33,1:40:10.80,Default,,0000,0000,0000,,
Dialogue: 0,1:40:10.80,1:40:11.67,Default,,0000,0000,0000,,Does it look tangent?
Dialogue: 0,1:40:11.67,1:40:14.06,Default,,0000,0000,0000,,I hope it looks tangent.
Dialogue: 0,1:40:14.06,1:40:19.18,Default,,0000,0000,0000,,And this guy will be r of\Nu-- because u0 means what?
Dialogue: 0,1:40:19.18,1:40:23.51,Default,,0000,0000,0000,,u0 and v. So who\Nis free to move?
Dialogue: 0,1:40:23.51,1:40:32.87,Default,,0000,0000,0000,,v. So this guy, this r sub\Nv-- they are both tangents.
Dialogue: 0,1:40:32.87,1:40:36.23,Default,,0000,0000,0000,,So do you have a surface?
Dialogue: 0,1:40:36.23,1:40:38.16,Default,,0000,0000,0000,,This is the surface.
Dialogue: 0,1:40:38.16,1:40:39.33,Default,,0000,0000,0000,,This is the surface.
Dialogue: 0,1:40:39.33,1:40:42.61,Default,,0000,0000,0000,,And these two horns or\Nwhatever they are-- those
Dialogue: 0,1:40:42.61,1:40:46.59,Default,,0000,0000,0000,,are the tangents r sub u, r\Nsub v, the two tangent vectors,
Dialogue: 0,1:40:46.59,1:40:47.77,Default,,0000,0000,0000,,the partial velocities.
Dialogue: 0,1:40:47.77,1:40:51.18,Default,,0000,0000,0000,,And I told you before, they\Nform the tangent plane.
Dialogue: 0,1:40:51.18,1:40:52.46,Default,,0000,0000,0000,,They are partial velocities.
Dialogue: 0,1:40:52.46,1:40:56.10,Default,,0000,0000,0000,,They are both tangent to\Nthe surface at that point.
Dialogue: 0,1:40:56.10,1:40:57.17,Default,,0000,0000,0000,,They form a basis.
Dialogue: 0,1:40:57.17,1:40:58.52,Default,,0000,0000,0000,,They are linearly independent.
Dialogue: 0,1:40:58.52,1:40:59.48,Default,,0000,0000,0000,,Always?
Dialogue: 0,1:40:59.48,1:41:00.05,Default,,0000,0000,0000,,No.
Dialogue: 0,1:41:00.05,1:41:06.32,Default,,0000,0000,0000,,But we assume that r sub u\Nand r sub v are non-zero,
Dialogue: 0,1:41:06.32,1:41:07.93,Default,,0000,0000,0000,,and they are not co-linear.
Dialogue: 0,1:41:07.93,1:41:08.91,Default,,0000,0000,0000,,How do I write that?
Dialogue: 0,1:41:08.91,1:41:11.11,Default,,0000,0000,0000,,They are not parallel.
Dialogue: 0,1:41:11.11,1:41:12.55,Default,,0000,0000,0000,,So guys, what does it mean?
Dialogue: 0,1:41:12.55,1:41:14.69,Default,,0000,0000,0000,,It means-- we talked\Nabout this before.
Dialogue: 0,1:41:14.69,1:41:17.50,Default,,0000,0000,0000,,The velocities cannot be 0.
Dialogue: 0,1:41:17.50,1:41:20.35,Default,,0000,0000,0000,,And r sub u, r sub v\Ncannot be parallel,
Dialogue: 0,1:41:20.35,1:41:23.32,Default,,0000,0000,0000,,because if they are parallel,\Nthere is no area element.
Dialogue: 0,1:41:23.32,1:41:26.15,Default,,0000,0000,0000,,There is no tangent\Nplane between them.
Dialogue: 0,1:41:26.15,1:41:31.03,Default,,0000,0000,0000,,What they form is\Nthe area element.
Dialogue: 0,1:41:31.03,1:41:35.94,Default,,0000,0000,0000,,So what do you think the\Narea element will look like?
Dialogue: 0,1:41:35.94,1:41:39.20,Default,,0000,0000,0000,,It's a magic thing.
Dialogue: 0,1:41:39.20,1:41:44.19,Default,,0000,0000,0000,,The surface element\Nactually will
Dialogue: 0,1:41:44.19,1:41:59.44,Default,,0000,0000,0000,,be exactly the area between ru\Nand rv times the u derivative.
Dialogue: 0,1:41:59.44,1:42:00.83,Default,,0000,0000,0000,,Say it again, Magdalena.
Dialogue: 0,1:42:00.83,1:42:04.40,Default,,0000,0000,0000,,What the heck is the area\Nbetween the vectors r sub
Dialogue: 0,1:42:04.40,1:42:05.53,Default,,0000,0000,0000,,u, r sub v?
Dialogue: 0,1:42:05.53,1:42:09.89,Default,,0000,0000,0000,,You know it better than\Nme because you're younger,
Dialogue: 0,1:42:09.89,1:42:11.32,Default,,0000,0000,0000,,and your memory is better.
Dialogue: 0,1:42:11.32,1:42:16.22,Default,,0000,0000,0000,,And you just covered\Nthis in chapter nine.
Dialogue: 0,1:42:16.22,1:42:19.26,Default,,0000,0000,0000,,When you have a vector A\Nand a vector B that are not
Dialogue: 0,1:42:19.26,1:42:22.66,Default,,0000,0000,0000,,co-linear, what was the\Narea of the parallelogram
Dialogue: 0,1:42:22.66,1:42:24.02,Default,,0000,0000,0000,,that they form?
Dialogue: 0,1:42:24.02,1:42:25.48,Default,,0000,0000,0000,,STUDENT: The\Nmagnitude [INAUDIBLE].
Dialogue: 0,1:42:25.48,1:42:26.12,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NMagnitude of--
Dialogue: 0,1:42:26.12,1:42:26.57,Default,,0000,0000,0000,,STUDENT: The cross product.
Dialogue: 0,1:42:26.57,1:42:27.34,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NThe cross product.
Dialogue: 0,1:42:27.34,1:42:27.90,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,1:42:27.90,1:42:30.79,Default,,0000,0000,0000,,This is exactly what\NI was hoping for.
Dialogue: 0,1:42:30.79,1:42:35.46,Default,,0000,0000,0000,,The magnitude of the\Ncross product is the area.
Dialogue: 0,1:42:35.46,1:42:42.32,Default,,0000,0000,0000,,So you have ds, infinitesimal\Nof an answer plus area, surface
Dialogue: 0,1:42:42.32,1:42:46.72,Default,,0000,0000,0000,,element will be\Nexactly the magnitude
Dialogue: 0,1:42:46.72,1:42:51.65,Default,,0000,0000,0000,,of the cross product of the\Ntwo velocity vectors, dudv.
Dialogue: 0,1:42:51.65,1:42:55.80,Default,,0000,0000,0000,,dudv can also be\Nwritten dA in this case
Dialogue: 0,1:42:55.80,1:42:58.95,Default,,0000,0000,0000,,because it's a flat\Narea on the floor.
Dialogue: 0,1:42:58.95,1:43:02.57,Default,,0000,0000,0000,,It's the area of a tiny\Nsquare on the floor,
Dialogue: 0,1:43:02.57,1:43:04.58,Default,,0000,0000,0000,,infinitesimally small square.
Dialogue: 0,1:43:04.58,1:43:07.12,Default,,0000,0000,0000,,So remember that.
Dialogue: 0,1:43:07.12,1:43:10.37,Default,,0000,0000,0000,,And you say, well, Magdalena,\Nyou are just feeding us
Dialogue: 0,1:43:10.37,1:43:11.46,Default,,0000,0000,0000,,formula after formula.
Dialogue: 0,1:43:11.46,1:43:13.14,Default,,0000,0000,0000,,But we don't even know.
Dialogue: 0,1:43:13.14,1:43:14.27,Default,,0000,0000,0000,,OK, this makes sense.
Dialogue: 0,1:43:14.27,1:43:18.50,Default,,0000,0000,0000,,This looks like I have some\Nsort of tiny parallelogram,
Dialogue: 0,1:43:18.50,1:43:22.98,Default,,0000,0000,0000,,and I approximate the\Nactual curvilinear
Dialogue: 0,1:43:22.98,1:43:25.32,Default,,0000,0000,0000,,patch, curvilinear\Npatch on-- I'm
Dialogue: 0,1:43:25.32,1:43:26.98,Default,,0000,0000,0000,,going to draw it on my hand.
Dialogue: 0,1:43:26.98,1:43:29.39,Default,,0000,0000,0000,,So this is-- oh, my god.
Dialogue: 0,1:43:29.39,1:43:30.95,Default,,0000,0000,0000,,My son would make fun of me.
Dialogue: 0,1:43:30.95,1:43:34.96,Default,,0000,0000,0000,,So this curvilinear patch\Nbetween two curves on my hand
Dialogue: 0,1:43:34.96,1:43:38.88,Default,,0000,0000,0000,,will be actually\Napproximated by this.
Dialogue: 0,1:43:38.88,1:43:39.87,Default,,0000,0000,0000,,What is this rectangle?
Dialogue: 0,1:43:39.87,1:43:40.80,Default,,0000,0000,0000,,No, it's a--
Dialogue: 0,1:43:40.80,1:43:41.76,Default,,0000,0000,0000,,STUDENT: Parallelogram.
Dialogue: 0,1:43:41.76,1:43:42.36,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NParallelogram.
Dialogue: 0,1:43:42.36,1:43:43.57,Default,,0000,0000,0000,,Thank you so much.
Dialogue: 0,1:43:43.57,1:43:47.93,Default,,0000,0000,0000,,So this is an\Napproximation, again.
Dialogue: 0,1:43:47.93,1:43:50.48,Default,,0000,0000,0000,,So this is the area\Nof the parallelogram.
Dialogue: 0,1:43:50.48,1:43:55.05,Default,,0000,0000,0000,,
Dialogue: 0,1:43:55.05,1:44:00.16,Default,,0000,0000,0000,,And that's what we defined\Nas being the surface element.
Dialogue: 0,1:44:00.16,1:44:02.12,Default,,0000,0000,0000,,It has to do with\Nthe tangent plane.
Dialogue: 0,1:44:02.12,1:44:04.75,Default,,0000,0000,0000,,But now you're asking,\Nbut shouldn't this
Dialogue: 0,1:44:04.75,1:44:09.45,Default,,0000,0000,0000,,be the same as the formula\Nroot of 1 plus f sub x squared
Dialogue: 0,1:44:09.45,1:44:12.23,Default,,0000,0000,0000,,plus f sub y squared dxdy?
Dialogue: 0,1:44:12.23,1:44:12.73,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:44:12.73,1:44:13.90,Default,,0000,0000,0000,,Let's prove it.
Dialogue: 0,1:44:13.90,1:44:17.49,Default,,0000,0000,0000,,Let's finally prove that\Nthe meaning of this area
Dialogue: 0,1:44:17.49,1:44:21.94,Default,,0000,0000,0000,,will provide you\Nwith the surface
Dialogue: 0,1:44:21.94,1:44:26.47,Default,,0000,0000,0000,,element the terms of x\Nand y, just the way you--
Dialogue: 0,1:44:26.47,1:44:27.94,Default,,0000,0000,0000,,you did not prove it.
Dialogue: 0,1:44:27.94,1:44:29.07,Default,,0000,0000,0000,,You discovered it.
Dialogue: 0,1:44:29.07,1:44:30.34,Default,,0000,0000,0000,,Remember, guys?
Dialogue: 0,1:44:30.34,1:44:33.20,Default,,0000,0000,0000,,You came up with a\Nformula as a conjecture.
Dialogue: 0,1:44:33.20,1:44:36.94,Default,,0000,0000,0000,,You said, if we generalize\Nthe arch length,
Dialogue: 0,1:44:36.94,1:44:38.59,Default,,0000,0000,0000,,it should look like that.
Dialogue: 0,1:44:38.59,1:44:39.90,Default,,0000,0000,0000,,You sort of smelled it.
Dialogue: 0,1:44:39.90,1:44:41.00,Default,,0000,0000,0000,,You said, I think.
Dialogue: 0,1:44:41.00,1:44:42.27,Default,,0000,0000,0000,,I feel.
Dialogue: 0,1:44:42.27,1:44:43.19,Default,,0000,0000,0000,,I'm almost sure.
Dialogue: 0,1:44:43.19,1:44:45.41,Default,,0000,0000,0000,,But did you prove it?
Dialogue: 0,1:44:45.41,1:44:46.33,Default,,0000,0000,0000,,No.
Dialogue: 0,1:44:46.33,1:44:50.12,Default,,0000,0000,0000,,So starting from the\Nidea of the area element
Dialogue: 0,1:44:50.12,1:44:52.58,Default,,0000,0000,0000,,that I gave before,\Ndo you remember
Dialogue: 0,1:44:52.58,1:44:57.48,Default,,0000,0000,0000,,that we also had that signed\Narea between the dx and dy,
Dialogue: 0,1:44:57.48,1:45:00.38,Default,,0000,0000,0000,,and we used the area of\Nthe parallelogram before?
Dialogue: 0,1:45:00.38,1:45:05.63,Default,,0000,0000,0000,,We also allowed it to\Ngo oriented plus, minus.
Dialogue: 0,1:45:05.63,1:45:06.48,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:45:06.48,1:45:06.98,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:45:06.98,1:45:10.79,Default,,0000,0000,0000,,So this makes more sense\Nthan what you gave me.
Dialogue: 0,1:45:10.79,1:45:13.67,Default,,0000,0000,0000,,Can I prove what you\Ngave me based on this
Dialogue: 0,1:45:13.67,1:45:16.34,Default,,0000,0000,0000,,and show it's one\Nand the same thing?
Dialogue: 0,1:45:16.34,1:45:17.91,Default,,0000,0000,0000,,So hopefully, yes.
Dialogue: 0,1:45:17.91,1:45:23.82,Default,,0000,0000,0000,,If I have my explicit form\Nz equals f of x and y,
Dialogue: 0,1:45:23.82,1:45:27.03,Default,,0000,0000,0000,,I should be able to\Nparametrize this surface.
Dialogue: 0,1:45:27.03,1:45:28.99,Default,,0000,0000,0000,,How do I parametrize\Nthis surface
Dialogue: 0,1:45:28.99,1:45:30.61,Default,,0000,0000,0000,,in the simplest possible way?
Dialogue: 0,1:45:30.61,1:45:33.59,Default,,0000,0000,0000,,
Dialogue: 0,1:45:33.59,1:45:36.81,Default,,0000,0000,0000,,x is u.
Dialogue: 0,1:45:36.81,1:45:45.48,Default,,0000,0000,0000,,y is v. z is f of\Nu, v. And that's it.
Dialogue: 0,1:45:45.48,1:45:54.86,Default,,0000,0000,0000,,Then it's r of u, v as a vector\Nwill be angular bracket, u, v,
Dialogue: 0,1:45:54.86,1:46:01.53,Default,,0000,0000,0000,,f of u, v. Now you have to\Nhelp me compute r sub u and r
Dialogue: 0,1:46:01.53,1:46:04.46,Default,,0000,0000,0000,,sub v. They have to\Nbe these blue vectors,
Dialogue: 0,1:46:04.46,1:46:05.72,Default,,0000,0000,0000,,the partial velocities.
Dialogue: 0,1:46:05.72,1:46:08.18,Default,,0000,0000,0000,,r sub u, r sub v.
Dialogue: 0,1:46:08.18,1:46:09.19,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:46:09.19,1:46:10.26,Default,,0000,0000,0000,,Come on.
Dialogue: 0,1:46:10.26,1:46:11.22,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,1:46:11.22,1:46:12.73,Default,,0000,0000,0000,,I need to change colors.
Dialogue: 0,1:46:12.73,1:46:15.02,Default,,0000,0000,0000,,So can you tell\Nme what they are?
Dialogue: 0,1:46:15.02,1:46:19.64,Default,,0000,0000,0000,,
Dialogue: 0,1:46:19.64,1:46:21.50,Default,,0000,0000,0000,,What's the first-- 1?
Dialogue: 0,1:46:21.50,1:46:22.81,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:46:22.81,1:46:23.31,Default,,0000,0000,0000,,What's next?
Dialogue: 0,1:46:23.31,1:46:23.84,Default,,0000,0000,0000,,0.
Dialogue: 0,1:46:23.84,1:46:25.02,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,1:46:25.02,1:46:25.73,Default,,0000,0000,0000,,STUDENT: F sub u.
Dialogue: 0,1:46:25.73,1:46:26.90,Default,,0000,0000,0000,,DR. MAGDALENA TODA: f sub u.
Dialogue: 0,1:46:26.90,1:46:28.88,Default,,0000,0000,0000,,Very good.
Dialogue: 0,1:46:28.88,1:46:32.55,Default,,0000,0000,0000,,f sub u or f sub x is the same\Nbecause x and u are the same.
Dialogue: 0,1:46:32.55,1:46:37.42,Default,,0000,0000,0000,,So let me rewrite\Nit 1, 0, f sub x.
Dialogue: 0,1:46:37.42,1:46:41.08,Default,,0000,0000,0000,,Now the next batch.
Dialogue: 0,1:46:41.08,1:46:50.26,Default,,0000,0000,0000,,0, 1, f sub v, which\Nis 0, 1, f sub y.
Dialogue: 0,1:46:50.26,1:46:52.39,Default,,0000,0000,0000,,0, 1, f sub y.
Dialogue: 0,1:46:52.39,1:46:57.08,Default,,0000,0000,0000,,
Dialogue: 0,1:46:57.08,1:46:58.80,Default,,0000,0000,0000,,Now I need to cross them.
Dialogue: 0,1:46:58.80,1:47:02.44,Default,,0000,0000,0000,,And I need to cross them, and\NI'm too lazy because it's 2:20.
Dialogue: 0,1:47:02.44,1:47:03.19,Default,,0000,0000,0000,,But I'll do it.
Dialogue: 0,1:47:03.19,1:47:04.03,Default,,0000,0000,0000,,I'll do it.
Dialogue: 0,1:47:04.03,1:47:05.16,Default,,0000,0000,0000,,I'll cross.
Dialogue: 0,1:47:05.16,1:47:07.90,Default,,0000,0000,0000,,So with your help\Nand everything,
Dialogue: 0,1:47:07.90,1:47:09.99,Default,,0000,0000,0000,,I'm going to get to\Nwhere I need to get.
Dialogue: 0,1:47:09.99,1:47:22.94,Default,,0000,0000,0000,,
Dialogue: 0,1:47:22.94,1:47:23.79,Default,,0000,0000,0000,,You can start.
Dialogue: 0,1:47:23.79,1:47:27.31,Default,,0000,0000,0000,,I mean, don't wait for me.
Dialogue: 0,1:47:27.31,1:47:29.20,Default,,0000,0000,0000,,Try it yourselves\Nand see what you get.
Dialogue: 0,1:47:29.20,1:47:33.73,Default,,0000,0000,0000,,And how hard do you think\Nit is to compute the thing?
Dialogue: 0,1:47:33.73,1:47:34.56,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:47:34.56,1:47:38.38,Default,,0000,0000,0000,,
Dialogue: 0,1:47:38.38,1:47:40.17,Default,,0000,0000,0000,,DR. MAGDALENA TODA: I\Nwill do the normality
Dialogue: 0,1:47:40.17,1:47:41.94,Default,,0000,0000,0000,,at the magnitude later.
Dialogue: 0,1:47:41.94,1:47:57.69,Default,,0000,0000,0000,,r sub u, r sub v's cross product\Nwill be I, J, K. 1, 0, f sub x,
Dialogue: 0,1:47:57.69,1:47:59.27,Default,,0000,0000,0000,,0, 1, f sub y.
Dialogue: 0,1:47:59.27,1:48:00.19,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,1:48:00.19,1:48:01.58,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,1:48:01.58,1:48:10.23,Default,,0000,0000,0000,,So I have minus f sub x, what?
Dialogue: 0,1:48:10.23,1:48:11.13,Default,,0000,0000,0000,,I. I'm sorry.
Dialogue: 0,1:48:11.13,1:48:12.88,Default,,0000,0000,0000,,I for an I. OK.
Dialogue: 0,1:48:12.88,1:48:19.22,Default,,0000,0000,0000,,J, again minus because\NI need to change.
Dialogue: 0,1:48:19.22,1:48:23.27,Default,,0000,0000,0000,,When I expand along the row, I\Nhave plus, minus, plus, minus,
Dialogue: 0,1:48:23.27,1:48:24.75,Default,,0000,0000,0000,,plus, alternating.
Dialogue: 0,1:48:24.75,1:48:27.18,Default,,0000,0000,0000,,So I need to have minus.
Dialogue: 0,1:48:27.18,1:48:37.78,Default,,0000,0000,0000,,The determinant is f sub y times\NJ plus K times this fellow.
Dialogue: 0,1:48:37.78,1:48:41.92,Default,,0000,0000,0000,,But that fellow is 1, is the\Nminor 1, for god's sakes.
Dialogue: 0,1:48:41.92,1:48:44.59,Default,,0000,0000,0000,,So this is so easy.
Dialogue: 0,1:48:44.59,1:48:48.93,Default,,0000,0000,0000,,I got the vector,\Nbut I need the norm.
Dialogue: 0,1:48:48.93,1:48:50.19,Default,,0000,0000,0000,,But so what?
Dialogue: 0,1:48:50.19,1:48:50.90,Default,,0000,0000,0000,,Do you have it?
Dialogue: 0,1:48:50.90,1:48:51.97,Default,,0000,0000,0000,,I'm there, guys.
Dialogue: 0,1:48:51.97,1:48:54.22,Default,,0000,0000,0000,,I'm really there.
Dialogue: 0,1:48:54.22,1:48:55.65,Default,,0000,0000,0000,,It's a piece of cake.
Dialogue: 0,1:48:55.65,1:48:58.16,Default,,0000,0000,0000,,I take the components.
Dialogue: 0,1:48:58.16,1:48:59.46,Default,,0000,0000,0000,,I squeeze them a little bit.
Dialogue: 0,1:48:59.46,1:48:59.96,Default,,0000,0000,0000,,No.
Dialogue: 0,1:48:59.96,1:49:03.52,Default,,0000,0000,0000,,I square them,\Nand I sum them up.
Dialogue: 0,1:49:03.52,1:49:08.51,Default,,0000,0000,0000,,And I get the square\Nroot of 1 plus-- exactly.
Dialogue: 0,1:49:08.51,1:49:14.15,Default,,0000,0000,0000,,1 plus f sub x squared\Nplus f sub y squared d.
Dialogue: 0,1:49:14.15,1:49:19.71,Default,,0000,0000,0000,,This is u and v, and this\Nis dxdy, which is dA.
Dialogue: 0,1:49:19.71,1:49:25.87,Default,,0000,0000,0000,,This is the tiny floor square\Nof an infinitesimally square
Dialogue: 0,1:49:25.87,1:49:27.61,Default,,0000,0000,0000,,on the floor.
Dialogue: 0,1:49:27.61,1:49:28.71,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:49:28.71,1:49:30.86,Default,,0000,0000,0000,,And what is this again?
Dialogue: 0,1:49:30.86,1:49:36.14,Default,,0000,0000,0000,,This is the area of\Na tiny curvilinear
Dialogue: 0,1:49:36.14,1:49:40.75,Default,,0000,0000,0000,,patch on the surface that's\Nprojected on that tiny square
Dialogue: 0,1:49:40.75,1:49:42.72,Default,,0000,0000,0000,,on the floor.
Dialogue: 0,1:49:42.72,1:49:44.69,Default,,0000,0000,0000,,All right?
Dialogue: 0,1:49:44.69,1:49:46.65,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:49:46.65,1:49:54.02,Default,,0000,0000,0000,,So Now you know why\Nyou get what you get.
Dialogue: 0,1:49:54.02,1:49:56.76,Default,,0000,0000,0000,,One the last problem\Nbecause time is up.
Dialogue: 0,1:49:56.76,1:49:58.28,Default,,0000,0000,0000,,No, I'm just kidding.
Dialogue: 0,1:49:58.28,1:49:59.89,Default,,0000,0000,0000,,We still have plenty of time.
Dialogue: 0,1:49:59.89,1:50:09.34,Default,,0000,0000,0000,,
Dialogue: 0,1:50:09.34,1:50:11.37,Default,,0000,0000,0000,,This is a beautiful,\Nbeautiful problem.
Dialogue: 0,1:50:11.37,1:50:13.03,Default,,0000,0000,0000,,But I don't want to finish it.
Dialogue: 0,1:50:13.03,1:50:20.76,Default,,0000,0000,0000,,I want to give you\Nthe problem at home.
Dialogue: 0,1:50:20.76,1:50:22.71,Default,,0000,0000,0000,,It's like the one\Nin the book, but I
Dialogue: 0,1:50:22.71,1:50:25.37,Default,,0000,0000,0000,,don't want to give you\Nexactly the one in the book.
Dialogue: 0,1:50:25.37,1:50:28.52,Default,,0000,0000,0000,,
Dialogue: 0,1:50:28.52,1:50:30.72,Default,,0000,0000,0000,,I want to cover\Nsomething special today.
Dialogue: 0,1:50:30.72,1:50:36.59,Default,,0000,0000,0000,,
Dialogue: 0,1:50:36.59,1:50:43.65,Default,,0000,0000,0000,,We are all familiar with their\Nnotion of a spiral staircase.
Dialogue: 0,1:50:43.65,1:50:47.15,Default,,0000,0000,0000,,But spiral staircases\Nare everywhere,
Dialogue: 0,1:50:47.15,1:50:55.60,Default,,0000,0000,0000,,in elegant buildings, official\Nbuildings, palaces, theaters,
Dialogue: 0,1:50:55.60,1:51:00.13,Default,,0000,0000,0000,,houses of multimillionaires\Nin California.
Dialogue: 0,1:51:00.13,1:51:02.48,Default,,0000,0000,0000,,And even people who\Nare not millionaires
Dialogue: 0,1:51:02.48,1:51:05.37,Default,,0000,0000,0000,,have some spiral\Nstaircases in their houses,
Dialogue: 0,1:51:05.37,1:51:08.61,Default,,0000,0000,0000,,maybe made of wood\Nor even marble.
Dialogue: 0,1:51:08.61,1:51:14.36,Default,,0000,0000,0000,,Did you ever wonder why\Nthe spiral staircases
Dialogue: 0,1:51:14.36,1:51:15.52,Default,,0000,0000,0000,,were invented?
Dialogue: 0,1:51:15.52,1:51:18.69,Default,,0000,0000,0000,,If you go to most of the\Ncastles on the Loire Valley,
Dialogue: 0,1:51:18.69,1:51:22.21,Default,,0000,0000,0000,,or many European castles\Nhave spiral staircases.
Dialogue: 0,1:51:22.21,1:51:30.28,Default,,0000,0000,0000,,Many mosques, many churches\Nhave these spiral staircases.
Dialogue: 0,1:51:30.28,1:51:35.34,Default,,0000,0000,0000,,I think it was about a few\Nthousand years ago that it
Dialogue: 0,1:51:35.34,1:51:37.58,Default,,0000,0000,0000,,was documented\Nfor the first time
Dialogue: 0,1:51:37.58,1:51:42.99,Default,,0000,0000,0000,,that the spiral staircases\Nconsumed the least
Dialogue: 0,1:51:42.99,1:51:45.99,Default,,0000,0000,0000,,amount of materials to build.
Dialogue: 0,1:51:45.99,1:51:49.96,Default,,0000,0000,0000,,
Dialogue: 0,1:51:49.96,1:51:54.83,Default,,0000,0000,0000,,Also what's good about them\Nis that for confined spaces--
Dialogue: 0,1:51:54.83,1:52:00.71,Default,,0000,0000,0000,,you have something like a\Ncylinder tower like that--
Dialogue: 0,1:52:00.71,1:52:04.54,Default,,0000,0000,0000,,that's the only shape you\Ncan build that minimizes
Dialogue: 0,1:52:04.54,1:52:09.37,Default,,0000,0000,0000,,the area of the staircase\Nbecause if you start building
Dialogue: 0,1:52:09.37,1:52:11.40,Default,,0000,0000,0000,,a staircase like\Nours here, it's not
Dialogue: 0,1:52:11.40,1:52:14.46,Default,,0000,0000,0000,,efficient at all in\Nterms of construction,
Dialogue: 0,1:52:14.46,1:52:16.27,Default,,0000,0000,0000,,in terms of materials.
Dialogue: 0,1:52:16.27,1:52:21.59,Default,,0000,0000,0000,,So you get a struggle\Nat actually making
Dialogue: 0,1:52:21.59,1:52:24.29,Default,,0000,0000,0000,,these stairs that are not even.
Dialogue: 0,1:52:24.29,1:52:27.40,Default,,0000,0000,0000,,You know, they're not even even.
Dialogue: 0,1:52:27.40,1:52:32.57,Default,,0000,0000,0000,,Each of them will have a\Ntriangle, or what is this?
Dialogue: 0,1:52:32.57,1:52:37.24,Default,,0000,0000,0000,,Not a triangle, but\Nmore like a trapezoid.
Dialogue: 0,1:52:37.24,1:52:40.00,Default,,0000,0000,0000,,And it keeps going up.
Dialogue: 0,1:52:40.00,1:52:42.25,Default,,0000,0000,0000,,This comes from a\Nhelix, obviously.
Dialogue: 0,1:52:42.25,1:52:46.12,Default,,0000,0000,0000,,And we have to understand\Nwhy this happens.
Dialogue: 0,1:52:46.12,1:52:48.18,Default,,0000,0000,0000,,And I will introduce the\Nsurface called helicoid.
Dialogue: 0,1:52:48.18,1:52:50.95,Default,,0000,0000,0000,,
Dialogue: 0,1:52:50.95,1:52:56.78,Default,,0000,0000,0000,,And the helicoid will have\Nthe following parametrization
Dialogue: 0,1:52:56.78,1:52:58.31,Default,,0000,0000,0000,,by definition.
Dialogue: 0,1:52:58.31,1:53:09.56,Default,,0000,0000,0000,,u cosine v, u sine v, and v.
Dialogue: 0,1:53:09.56,1:53:21.03,Default,,0000,0000,0000,,Assume u is between 0 and 1 and\Nassume v is between 0 and 2 pi.
Dialogue: 0,1:53:21.03,1:53:22.00,Default,,0000,0000,0000,,Draw the surface.
Dialogue: 0,1:53:22.00,1:53:25.40,Default,,0000,0000,0000,,
Dialogue: 0,1:53:25.40,1:53:39.25,Default,,0000,0000,0000,,Also find the surface area of\Nthe patch u between 0 and 1,
Dialogue: 0,1:53:39.25,1:53:41.49,Default,,0000,0000,0000,,v between 0 and pi/2.
Dialogue: 0,1:53:41.49,1:53:44.57,Default,,0000,0000,0000,,
Dialogue: 0,1:53:44.57,1:53:46.17,Default,,0000,0000,0000,,So I go, uh oh.
Dialogue: 0,1:53:46.17,1:53:47.25,Default,,0000,0000,0000,,I'm in trouble.
Dialogue: 0,1:53:47.25,1:53:49.38,Default,,0000,0000,0000,,Now how in the world am I\Ngoing to do this problem?
Dialogue: 0,1:53:49.38,1:53:51.05,Default,,0000,0000,0000,,It looks horrible.
Dialogue: 0,1:53:51.05,1:53:53.06,Default,,0000,0000,0000,,And it looks hard.
Dialogue: 0,1:53:53.06,1:53:55.29,Default,,0000,0000,0000,,And it even looks hard to draw.
Dialogue: 0,1:53:55.29,1:53:57.84,Default,,0000,0000,0000,,It's not that hard.
Dialogue: 0,1:53:57.84,1:53:59.94,Default,,0000,0000,0000,,It's not hard at\Nall, because you
Dialogue: 0,1:53:59.94,1:54:04.22,Default,,0000,0000,0000,,have to think of these\Nextreme points, the limit
Dialogue: 0,1:54:04.22,1:54:11.04,Default,,0000,0000,0000,,points of u and v and see what\Nthey really represent for you.
Dialogue: 0,1:54:11.04,1:54:13.87,Default,,0000,0000,0000,,Put your imagination\Nto work and say,
Dialogue: 0,1:54:13.87,1:54:16.66,Default,,0000,0000,0000,,this is the frame\NI'm starting with.
Dialogue: 0,1:54:16.66,1:54:21.93,Default,,0000,0000,0000,,This is the x and y and\Nz frame with origin 0.
Dialogue: 0,1:54:21.93,1:54:26.72,Default,,0000,0000,0000,,And I better draw this helicoid\Nbecause it shouldn't be hard.
Dialogue: 0,1:54:26.72,1:54:29.60,Default,,0000,0000,0000,,So for u equals\N0, what do I have?
Dialogue: 0,1:54:29.60,1:54:33.31,Default,,0000,0000,0000,,
Dialogue: 0,1:54:33.31,1:54:34.13,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:54:34.13,1:54:35.43,Default,,0000,0000,0000,,It looks weird.
Dialogue: 0,1:54:35.43,1:54:36.76,Default,,0000,0000,0000,,But Alex said it.
Dialogue: 0,1:54:36.76,1:54:40.79,Default,,0000,0000,0000,,0, 0, v. v is my\Nparameter in real life.
Dialogue: 0,1:54:40.79,1:54:44.83,Default,,0000,0000,0000,,So I have the whole z-axis.
Dialogue: 0,1:54:44.83,1:54:49.68,Default,,0000,0000,0000,,So one edge is going to\Nbe the z-axis itself.
Dialogue: 0,1:54:49.68,1:54:51.61,Default,,0000,0000,0000,,Does it have to be\Nonly the positive one?
Dialogue: 0,1:54:51.61,1:54:52.11,Default,,0000,0000,0000,,No.
Dialogue: 0,1:54:52.11,1:54:54.78,Default,,0000,0000,0000,,Who said so?
Dialogue: 0,1:54:54.78,1:54:58.87,Default,,0000,0000,0000,,My problem said so,\Nthat v only takes
Dialogue: 0,1:54:58.87,1:55:00.13,Default,,0000,0000,0000,,values between 0 and 2 pi.
Dialogue: 0,1:55:00.13,1:55:03.58,Default,,0000,0000,0000,,Unfortunately, I'm limiting\Nv between 0 and 2 pi.
Dialogue: 0,1:55:03.58,1:55:06.67,Default,,0000,0000,0000,,But in general, v could\Nbe any real number.
Dialogue: 0,1:55:06.67,1:55:11.56,Default,,0000,0000,0000,,So I'll take it from\N0 to 2 pi, and this
Dialogue: 0,1:55:11.56,1:55:14.80,Default,,0000,0000,0000,,is going to be what I'm thinking\Nof, one edge of the staircase.
Dialogue: 0,1:55:14.80,1:55:17.77,Default,,0000,0000,0000,,
Dialogue: 0,1:55:17.77,1:55:19.67,Default,,0000,0000,0000,,It's the interior age, the axis.
Dialogue: 0,1:55:19.67,1:55:23.51,Default,,0000,0000,0000,,Let's see what happens\Nwhen u equals 1.
Dialogue: 0,1:55:23.51,1:55:27.13,Default,,0000,0000,0000,,That's another curve\Nof the surface.
Dialogue: 0,1:55:27.13,1:55:28.09,Default,,0000,0000,0000,,Let's see what I get.
Dialogue: 0,1:55:28.09,1:55:35.86,Default,,0000,0000,0000,,Cosine v, sine v, and v. And it\Nlooks like a friend of yours.
Dialogue: 0,1:55:35.86,1:55:37.59,Default,,0000,0000,0000,,And you have to\Ntell me who this is.
Dialogue: 0,1:55:37.59,1:55:43.27,Default,,0000,0000,0000,,If v were bt, what\Nis sine, cosine tt?
Dialogue: 0,1:55:43.27,1:55:44.37,Default,,0000,0000,0000,,STUDENT: A sphere.
Dialogue: 0,1:55:44.37,1:55:45.14,Default,,0000,0000,0000,,DR. MAGDALENA TODA: It's your--
Dialogue: 0,1:55:45.14,1:55:45.92,Default,,0000,0000,0000,,STUDENT: Helicoid.
Dialogue: 0,1:55:45.92,1:55:48.54,Default,,0000,0000,0000,,DR. MAGDALENA TODA: --helix\Nthat you loved in chapter 10
Dialogue: 0,1:55:48.54,1:55:50.37,Default,,0000,0000,0000,,and you made friends with.
Dialogue: 0,1:55:50.37,1:55:54.93,Default,,0000,0000,0000,,And it was a curve that had\Nconstant curvature and constant
Dialogue: 0,1:55:54.93,1:55:56.69,Default,,0000,0000,0000,,portion, and it\Nhad constant speed.
Dialogue: 0,1:55:56.69,1:55:58.24,Default,,0000,0000,0000,,And the speed of\Nthis, for example,
Dialogue: 0,1:55:58.24,1:56:00.41,Default,,0000,0000,0000,,would be square root of 2,\Nif you have the curiosity
Dialogue: 0,1:56:00.41,1:56:02.21,Default,,0000,0000,0000,,to compute it.
Dialogue: 0,1:56:02.21,1:56:04.46,Default,,0000,0000,0000,,It will have square root of 2.
Dialogue: 0,1:56:04.46,1:56:06.25,Default,,0000,0000,0000,,And can I draw it?
Dialogue: 0,1:56:06.25,1:56:08.49,Default,,0000,0000,0000,,I better draw it,\Nbut I don't know how.
Dialogue: 0,1:56:08.49,1:56:11.65,Default,,0000,0000,0000,,
Dialogue: 0,1:56:11.65,1:56:20.11,Default,,0000,0000,0000,,So I have to think of drawing\Nthis for v between 0 and 2 pi.
Dialogue: 0,1:56:20.11,1:56:22.88,Default,,0000,0000,0000,,
Dialogue: 0,1:56:22.88,1:56:27.74,Default,,0000,0000,0000,,When I'm at 0, when I\Nhave time v equals 0,
Dialogue: 0,1:56:27.74,1:56:30.86,Default,,0000,0000,0000,,I have the point 1, 0, and 0.
Dialogue: 0,1:56:30.86,1:56:31.70,Default,,0000,0000,0000,,And where am I?
Dialogue: 0,1:56:31.70,1:56:32.83,Default,,0000,0000,0000,,Here.
Dialogue: 0,1:56:32.83,1:56:35.70,Default,,0000,0000,0000,,1, 0, 0.
Dialogue: 0,1:56:35.70,1:56:39.37,Default,,0000,0000,0000,,And from here, I start moving\Non the helix and going up.
Dialogue: 0,1:56:39.37,1:56:47.53,Default,,0000,0000,0000,,And see, my hand should be\Non-- this is the stairs.
Dialogue: 0,1:56:47.53,1:56:50.18,Default,,0000,0000,0000,,It's obviously a smooth surface.
Dialogue: 0,1:56:50.18,1:56:52.63,Default,,0000,0000,0000,,This is a smooth\Nsurface, but the stairs
Dialogue: 0,1:56:52.63,1:56:55.24,Default,,0000,0000,0000,,that I was talking about\Nare a discretization
Dialogue: 0,1:56:55.24,1:56:57.55,Default,,0000,0000,0000,,of the smooth surface.
Dialogue: 0,1:56:57.55,1:57:00.73,Default,,0000,0000,0000,,I have a step, another step,\Nanother step, another step.
Dialogue: 0,1:57:00.73,1:57:06.62,Default,,0000,0000,0000,,So it's like a smooth helicoid\Nbut discretized step functions.
Dialogue: 0,1:57:06.62,1:57:08.24,Default,,0000,0000,0000,,Forget about the step functions.
Dialogue: 0,1:57:08.24,1:57:11.71,Default,,0000,0000,0000,,Assume that instead of the\Nstaircase in the church--
Dialogue: 0,1:57:11.71,1:57:13.00,Default,,0000,0000,0000,,you don't want to go to church.
Dialogue: 0,1:57:13.00,1:57:15.59,Default,,0000,0000,0000,,You want to go to\Nthe water park.
Dialogue: 0,1:57:15.59,1:57:17.43,Default,,0000,0000,0000,,You want to go to Six Flags.
Dialogue: 0,1:57:17.43,1:57:21.01,Default,,0000,0000,0000,,You want to go to whatever,\NDisney World, San Antonio,
Dialogue: 0,1:57:21.01,1:57:21.51,Default,,0000,0000,0000,,somewhere.
Dialogue: 0,1:57:21.51,1:57:23.70,Default,,0000,0000,0000,,This is a slide.
Dialogue: 0,1:57:23.70,1:57:25.15,Default,,0000,0000,0000,,You let yourself go.
Dialogue: 0,1:57:25.15,1:57:28.18,Default,,0000,0000,0000,,This is you going\Ndown, swimming--
Dialogue: 0,1:57:28.18,1:57:30.11,Default,,0000,0000,0000,,I don't know-- upside down.
Dialogue: 0,1:57:30.11,1:57:32.06,Default,,0000,0000,0000,,I don't know how.
Dialogue: 0,1:57:32.06,1:57:36.09,Default,,0000,0000,0000,,So this is a smooth\Nslide in a water park.
Dialogue: 0,1:57:36.09,1:57:38.39,Default,,0000,0000,0000,,That's how you should\Nbe imagining it.
Dialogue: 0,1:57:38.39,1:57:39.73,Default,,0000,0000,0000,,And it keeps going.
Dialogue: 0,1:57:39.73,1:57:43.82,Default,,0000,0000,0000,,If I start here-- if I\Nstart here, again, look,
Dialogue: 0,1:57:43.82,1:57:45.63,Default,,0000,0000,0000,,this is what I'm describing.
Dialogue: 0,1:57:45.63,1:57:48.35,Default,,0000,0000,0000,,
Dialogue: 0,1:57:48.35,1:57:49.33,Default,,0000,0000,0000,,A helicoid.
Dialogue: 0,1:57:49.33,1:57:53.24,Default,,0000,0000,0000,,My arm moved on this.
Dialogue: 0,1:57:53.24,1:57:56.48,Default,,0000,0000,0000,,Again, I draw the same motion.
Dialogue: 0,1:57:56.48,1:58:01.23,Default,,0000,0000,0000,,My elbow should not\Ndo something crazy.
Dialogue: 0,1:58:01.23,1:58:04.33,Default,,0000,0000,0000,,It should keep\Nmoving on the z-axis.
Dialogue: 0,1:58:04.33,1:58:13.67,Default,,0000,0000,0000,,And I perform the pi/2 motion\Nwhen the stair-- not the stair.
Dialogue: 0,1:58:13.67,1:58:15.38,Default,,0000,0000,0000,,I don't know what to call it.
Dialogue: 0,1:58:15.38,1:58:23.30,Default,,0000,0000,0000,,This line becomes horizontal\Nwhen v equals pi/2.
Dialogue: 0,1:58:23.30,1:58:29.56,Default,,0000,0000,0000,,So for v equals pi/2, I moved\Nfrom here straight to here.
Dialogue: 0,1:58:29.56,1:58:30.84,Default,,0000,0000,0000,,STUDENT: Doesn't it go around?
Dialogue: 0,1:58:30.84,1:58:32.30,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NIt goes around.
Dialogue: 0,1:58:32.30,1:58:35.74,Default,,0000,0000,0000,,But see, what I asked-- I\Nonly asked for the patch.
Dialogue: 0,1:58:35.74,1:58:37.70,Default,,0000,0000,0000,,First of all, I\Nsaid it goes around.
Dialogue: 0,1:58:37.70,1:58:39.46,Default,,0000,0000,0000,,So I'll try to go around.
Dialogue: 0,1:58:39.46,1:58:40.92,Default,,0000,0000,0000,,But it's hard.
Dialogue: 0,1:58:40.92,1:58:43.54,Default,,0000,0000,0000,,Oh, wish me luck.
Dialogue: 0,1:58:43.54,1:58:47.45,Default,,0000,0000,0000,,One, two, three.
Dialogue: 0,1:58:47.45,1:58:49.37,Default,,0000,0000,0000,,I cannot go higher.
Dialogue: 0,1:58:49.37,1:58:50.35,Default,,0000,0000,0000,,It goes to pi.
Dialogue: 0,1:58:50.35,1:58:52.34,Default,,0000,0000,0000,,STUDENT: If it went to\N2 pi, it would actually
Dialogue: 0,1:58:52.34,1:58:53.30,Default,,0000,0000,0000,,wrap completely around?
Dialogue: 0,1:58:53.30,1:58:54.06,Default,,0000,0000,0000,,DR. MAGDALENA TODA: It\Nwould wrap like that.
Dialogue: 0,1:58:54.06,1:58:56.14,Default,,0000,0000,0000,,STUDENT: Right above\Nwhere it started?
Dialogue: 0,1:58:56.14,1:58:57.31,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Exactly.
Dialogue: 0,1:58:57.31,1:59:02.95,Default,,0000,0000,0000,,So if I started here, I end\Nup parallel to that, up.
Dialogue: 0,1:59:02.95,1:59:05.17,Default,,0000,0000,0000,,But 2 pi is too high for me.
Dialogue: 0,1:59:05.17,1:59:07.05,Default,,0000,0000,0000,,So I should go slowly.
Dialogue: 0,1:59:07.05,1:59:13.25,Default,,0000,0000,0000,,
Dialogue: 0,1:59:13.25,1:59:17.36,Default,,0000,0000,0000,,I'm up after 2 pi in\Nthe same position.
Dialogue: 0,1:59:17.36,1:59:19.87,Default,,0000,0000,0000,,STUDENT: But since it's pi/2,\Nit's just kind of like--
Dialogue: 0,1:59:19.87,1:59:21.29,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NWhen I'm pi/2,
Dialogue: 0,1:59:21.29,1:59:24.56,Default,,0000,0000,0000,,I just performed\Nfrom here to here.
Dialogue: 0,1:59:24.56,1:59:26.60,Default,,0000,0000,0000,,STUDENT: So that's it's\Nasking you for the patch?
Dialogue: 0,1:59:26.60,1:59:28.61,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NAnd it's asking for,
Dialogue: 0,1:59:28.61,1:59:36.36,Default,,0000,0000,0000,,what area does my arm from here\Nto here sweep at this point?
Dialogue: 0,1:59:36.36,1:59:39.19,Default,,0000,0000,0000,,From this point to this point.
Dialogue: 0,1:59:39.19,1:59:40.86,Default,,0000,0000,0000,,It's a smooth surface.
Dialogue: 0,1:59:40.86,1:59:45.63,Default,,0000,0000,0000,,So it's generated by my motion.
Dialogue: 0,1:59:45.63,1:59:48.37,Default,,0000,0000,0000,,And stop.
Dialogue: 0,1:59:48.37,1:59:49.48,Default,,0000,0000,0000,,It's a root surface.
Dialogue: 0,1:59:49.48,1:59:51.19,Default,,0000,0000,0000,,It's a root patch of a surface.
Dialogue: 0,1:59:51.19,1:59:54.17,Default,,0000,0000,0000,,Somebody tell me how I'm going\Nto do this because this is not
Dialogue: 0,1:59:54.17,1:59:58.33,Default,,0000,0000,0000,,with square root of 1 plus f sub\Nx squared plus f sub y squared.
Dialogue: 0,1:59:58.33,1:59:59.42,Default,,0000,0000,0000,,That is for normal people.
Dialogue: 0,1:59:59.42,2:00:01.32,Default,,0000,0000,0000,,You are not normal people.
Dialogue: 0,2:00:01.32,2:00:05.10,Default,,0000,0000,0000,,They never teach this in honors.
Dialogue: 0,2:00:05.10,2:00:09.08,Default,,0000,0000,0000,,In honors, we don't\Ncover this formula.
Dialogue: 0,2:00:09.08,2:00:10.24,Default,,0000,0000,0000,,But you're honors.
Dialogue: 0,2:00:10.24,2:00:14.52,Default,,0000,0000,0000,,So do I want you to finish\Nit at home with a calculator?
Dialogue: 0,2:00:14.52,2:00:17.84,Default,,0000,0000,0000,,All I want is for you to be\Nable to set up the integral.
Dialogue: 0,2:00:17.84,2:00:21.61,Default,,0000,0000,0000,,And I think, knowing you\Nbetter and working with you--
Dialogue: 0,2:00:21.61,2:00:25.37,Default,,0000,0000,0000,,I think you have the potential\Nto do that without my help,
Dialogue: 0,2:00:25.37,2:00:29.35,Default,,0000,0000,0000,,with all the elements\NI gave you until now.
Dialogue: 0,2:00:29.35,2:00:38.60,Default,,0000,0000,0000,,So the area of s-- it\Nwill be the blue slide.
Dialogue: 0,2:00:38.60,2:00:40.74,Default,,0000,0000,0000,,These are all--\Nwhen you slide down,
Dialogue: 0,2:00:40.74,2:00:43.28,Default,,0000,0000,0000,,you slide down along helices.
Dialogue: 0,2:00:43.28,2:00:47.07,Default,,0000,0000,0000,,You and your friends-- you're\Ngoing down along helices.
Dialogue: 0,2:00:47.07,2:00:47.57,Default,,0000,0000,0000,,OK?
Dialogue: 0,2:00:47.57,2:00:49.73,Default,,0000,0000,0000,,So that's what you have.
Dialogue: 0,2:00:49.73,2:00:55.58,Default,,0000,0000,0000,,[INAUDIBLE] double integral\Nfor a certain domain D.
Dialogue: 0,2:00:55.58,2:00:56.54,Default,,0000,0000,0000,,Which is that domain D?
Dialogue: 0,2:00:56.54,2:01:03.14,Default,,0000,0000,0000,,That domain D is u between 0\Nand 1 and v between 0 and pi/2
Dialogue: 0,2:01:03.14,2:01:06.38,Default,,0000,0000,0000,,because that's what\NI said I want here.
Dialogue: 0,2:01:06.38,2:01:09.87,Default,,0000,0000,0000,,
Dialogue: 0,2:01:09.87,2:01:12.47,Default,,0000,0000,0000,,Of what?
Dialogue: 0,2:01:12.47,2:01:19.48,Default,,0000,0000,0000,,Of magnitude of r sub u times\Nr sub v cross product dudv.
Dialogue: 0,2:01:19.48,2:01:22.87,Default,,0000,0000,0000,,
Dialogue: 0,2:01:22.87,2:01:25.28,Default,,0000,0000,0000,,You need to help me\Nthough because I don't
Dialogue: 0,2:01:25.28,2:01:27.62,Default,,0000,0000,0000,,know what I'm going to do.
Dialogue: 0,2:01:27.62,2:01:32.99,Default,,0000,0000,0000,,So who starts?
Dialogue: 0,2:01:32.99,2:01:42.19,Default,,0000,0000,0000,,r sub u is-- I'm not doing\Nanything without you, I swear.
Dialogue: 0,2:01:42.19,2:01:43.01,Default,,0000,0000,0000,,OK?
Dialogue: 0,2:01:43.01,2:01:43.84,Default,,0000,0000,0000,,STUDENT: Cosine.
Dialogue: 0,2:01:43.84,2:01:45.76,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Cosine v.
Dialogue: 0,2:01:45.76,2:01:46.55,Default,,0000,0000,0000,,STUDENT: Sine v.
Dialogue: 0,2:01:46.55,2:01:47.67,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Sine v.
Dialogue: 0,2:01:47.67,2:01:48.38,Default,,0000,0000,0000,,STUDENT: And 0.
Dialogue: 0,2:01:48.38,2:01:49.30,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 0.
Dialogue: 0,2:01:49.30,2:01:52.14,Default,,0000,0000,0000,,r sub v equals--
Dialogue: 0,2:01:52.14,2:01:53.91,Default,,0000,0000,0000,,STUDENT: Negative u sine v.
Dialogue: 0,2:01:53.91,2:01:55.76,Default,,0000,0000,0000,,DR. MAGDALENA TODA: Very good.
Dialogue: 0,2:01:55.76,2:02:01.35,Default,,0000,0000,0000,,u cosine v. And 1, and\Nthis goes on my nerves.
Dialogue: 0,2:02:01.35,2:02:02.27,Default,,0000,0000,0000,,But what can I do?
Dialogue: 0,2:02:02.27,2:02:03.94,Default,,0000,0000,0000,,Nothing.
Dialogue: 0,2:02:03.94,2:02:05.15,Default,,0000,0000,0000,,OK?
Dialogue: 0,2:02:05.15,2:02:06.20,Default,,0000,0000,0000,,All right.
Dialogue: 0,2:02:06.20,2:02:11.81,Default,,0000,0000,0000,,So I have to compute the what?
Dialogue: 0,2:02:11.81,2:02:12.93,Default,,0000,0000,0000,,STUDENT: The cross product.
Dialogue: 0,2:02:12.93,2:02:14.52,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NThe cross product.
Dialogue: 0,2:02:14.52,2:02:21.85,Default,,0000,0000,0000,,
Dialogue: 0,2:02:21.85,2:02:24.20,Default,,0000,0000,0000,,I, J, K, of course, right?
Dialogue: 0,2:02:24.20,2:02:33.21,Default,,0000,0000,0000,,I, J, K, cosine v, sine v,\N0, minus u sine-- oh, my god.
Dialogue: 0,2:02:33.21,2:02:33.90,Default,,0000,0000,0000,,Where was it?
Dialogue: 0,2:02:33.90,2:02:34.40,Default,,0000,0000,0000,,It's there.
Dialogue: 0,2:02:34.40,2:02:35.69,Default,,0000,0000,0000,,You gave it to me.
Dialogue: 0,2:02:35.69,2:02:38.60,Default,,0000,0000,0000,,u cosine v and 1.
Dialogue: 0,2:02:38.60,2:02:44.33,Default,,0000,0000,0000,,Minus u sine v, u\Ncosine v, and 1.
Dialogue: 0,2:02:44.33,2:02:46.45,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:02:46.45,2:02:49.15,Default,,0000,0000,0000,,I times that.
Dialogue: 0,2:02:49.15,2:02:58.78,Default,,0000,0000,0000,,Sine v, I. sine v,\NI. J. J has a friend.
Dialogue: 0,2:02:58.78,2:03:02.02,Default,,0000,0000,0000,,Is this mine or--\Ncosine v times 1.
Dialogue: 0,2:03:02.02,2:03:03.55,Default,,0000,0000,0000,,But I have to change the sine.
Dialogue: 0,2:03:03.55,2:03:05.72,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,2:03:05.72,2:03:10.62,Default,,0000,0000,0000,,It's really serious that I have\Nto think of changing the sine.
Dialogue: 0,2:03:10.62,2:03:16.14,Default,,0000,0000,0000,,Minus cosine v times\NJ. Are you with me?
Dialogue: 0,2:03:16.14,2:03:18.80,Default,,0000,0000,0000,,
Dialogue: 0,2:03:18.80,2:03:22.42,Default,,0000,0000,0000,,This times that\Nwith a sign change.
Dialogue: 0,2:03:22.42,2:03:28.44,Default,,0000,0000,0000,,And then plus-- what is--\Nwell, this is not so obvious.
Dialogue: 0,2:03:28.44,2:03:30.62,Default,,0000,0000,0000,,But you have so much practice.
Dialogue: 0,2:03:30.62,2:03:31.70,Default,,0000,0000,0000,,Make me proud.
Dialogue: 0,2:03:31.70,2:03:34.48,Default,,0000,0000,0000,,What is the minor\Nthat multiplies K?
Dialogue: 0,2:03:34.48,2:03:36.19,Default,,0000,0000,0000,,This red fellow.
Dialogue: 0,2:03:36.19,2:03:39.17,Default,,0000,0000,0000,,You need to compute\Nit and simplify it.
Dialogue: 0,2:03:39.17,2:03:40.76,Default,,0000,0000,0000,,So I don't talk too much.
Dialogue: 0,2:03:40.76,2:03:43.34,Default,,0000,0000,0000,,
Dialogue: 0,2:03:43.34,2:03:44.24,Default,,0000,0000,0000,,u, excellent.
Dialogue: 0,2:03:44.24,2:03:47.19,Default,,0000,0000,0000,,How did you do it, [INAUDIBLE]?
Dialogue: 0,2:03:47.19,2:03:48.25,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,2:03:48.25,2:03:50.25,Default,,0000,0000,0000,,DR. MAGDALENA TODA: So\Nyou group together cosine
Dialogue: 0,2:03:50.25,2:03:51.95,Default,,0000,0000,0000,,squared plus sine squared.
Dialogue: 0,2:03:51.95,2:03:53.28,Default,,0000,0000,0000,,Minus, minus is a plus.
Dialogue: 0,2:03:53.28,2:03:55.21,Default,,0000,0000,0000,,And you said u, it's u.
Dialogue: 0,2:03:55.21,2:03:55.71,Default,,0000,0000,0000,,Good.
Dialogue: 0,2:03:55.71,2:03:58.58,Default,,0000,0000,0000,,Plus u times K. Good.
Dialogue: 0,2:03:58.58,2:04:00.55,Default,,0000,0000,0000,,It doesn't look so bad.
Dialogue: 0,2:04:00.55,2:04:02.99,Default,,0000,0000,0000,,Now that you look at it,\Nit doesn't look so bad.
Dialogue: 0,2:04:02.99,2:04:05.68,Default,,0000,0000,0000,,You have to set up the integral.
Dialogue: 0,2:04:05.68,2:04:08.66,Default,,0000,0000,0000,,And that's going to be what?
Dialogue: 0,2:04:08.66,2:04:13.12,Default,,0000,0000,0000,,The square root of\Nthis fellow squared
Dialogue: 0,2:04:13.12,2:04:15.65,Default,,0000,0000,0000,,plus that fellow squared\Nplus this fellow squared.
Dialogue: 0,2:04:15.65,2:04:17.85,Default,,0000,0000,0000,,Let's take our time.
Dialogue: 0,2:04:17.85,2:04:25.02,Default,,0000,0000,0000,,You take these three fellows,\Nsquare them, add them up,
Dialogue: 0,2:04:25.02,2:04:26.93,Default,,0000,0000,0000,,and put them under\Na square root.
Dialogue: 0,2:04:26.93,2:04:28.78,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,2:04:28.78,2:04:29.86,Default,,0000,0000,0000,,STUDENT: 1 plus u squared.
Dialogue: 0,2:04:29.86,2:04:33.18,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\N1 plus u squared.
Dialogue: 0,2:04:33.18,2:04:39.75,Default,,0000,0000,0000,,Now the thing is I\Ndon't have a Jacobian.
Dialogue: 0,2:04:39.75,2:04:41.80,Default,,0000,0000,0000,,This is dudv.
Dialogue: 0,2:04:41.80,2:04:44.11,Default,,0000,0000,0000,,The Jacobian is what?
Dialogue: 0,2:04:44.11,2:04:46.01,Default,,0000,0000,0000,,So this is what I have.
Dialogue: 0,2:04:46.01,2:04:49.99,Default,,0000,0000,0000,,Now between the end\Npoints, I have to think.
Dialogue: 0,2:04:49.99,2:04:52.43,Default,,0000,0000,0000,,v has to be between 0 and pi/2.
Dialogue: 0,2:04:52.43,2:04:55.56,Default,,0000,0000,0000,,
Dialogue: 0,2:04:55.56,2:04:58.47,Default,,0000,0000,0000,,And u has to be between 0 and 1.
Dialogue: 0,2:04:58.47,2:05:04.76,Default,,0000,0000,0000,,
Dialogue: 0,2:05:04.76,2:05:06.31,Default,,0000,0000,0000,,Do you notice anything?
Dialogue: 0,2:05:06.31,2:05:08.85,Default,,0000,0000,0000,,And that's exactly what\NI wanted to tell you.
Dialogue: 0,2:05:08.85,2:05:10.24,Default,,0000,0000,0000,,v is not inside.
Dialogue: 0,2:05:10.24,2:05:12.46,Default,,0000,0000,0000,,v says, I'm independent.
Dialogue: 0,2:05:12.46,2:05:13.76,Default,,0000,0000,0000,,Please leave me alone.
Dialogue: 0,2:05:13.76,2:05:17.31,Default,,0000,0000,0000,,I'll go off, take a break. pi/2.
Dialogue: 0,2:05:17.31,2:05:20.23,Default,,0000,0000,0000,,But then you have\Nintegral from 0
Dialogue: 0,2:05:20.23,2:05:26.87,Default,,0000,0000,0000,,to 1 square root of\N1 plus u squared du.
Dialogue: 0,2:05:26.87,2:05:28.24,Default,,0000,0000,0000,,I want to say a remark.
Dialogue: 0,2:05:28.24,2:05:32.09,Default,,0000,0000,0000,,
Dialogue: 0,2:05:32.09,2:05:35.48,Default,,0000,0000,0000,,Happy or not happy, shall I be?
Dialogue: 0,2:05:35.48,2:05:40.03,Default,,0000,0000,0000,,Here, you need either\Nthe calculator,
Dialogue: 0,2:05:40.03,2:05:42.20,Default,,0000,0000,0000,,which is the simplest\Nway to do it, just
Dialogue: 0,2:05:42.20,2:05:44.11,Default,,0000,0000,0000,,compute the simple integral.
Dialogue: 0,2:05:44.11,2:05:48.72,Default,,0000,0000,0000,,Integral from 0 to 1 square\Nroot of 1 plus u squared du.
Dialogue: 0,2:05:48.72,2:05:53.64,Default,,0000,0000,0000,,Or what do you have in this\Nbook that can still help you
Dialogue: 0,2:05:53.64,2:05:55.16,Default,,0000,0000,0000,,if you don't have a calculator?
Dialogue: 0,2:05:55.16,2:05:55.99,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:05:55.99,2:06:00.69,Default,,0000,0000,0000,,DR. MAGDALENA TODA: A table of\Nintegration, integration table.
Dialogue: 0,2:06:00.69,2:06:02.06,Default,,0000,0000,0000,,Please compute this.
Dialogue: 0,2:06:02.06,2:06:04.52,Default,,0000,0000,0000,,I mean, you cannot\Ngive me an exact value.
Dialogue: 0,2:06:04.52,2:06:11.96,Default,,0000,0000,0000,,But give me an approximate\Nvalue by Thursday.
Dialogue: 0,2:06:11.96,2:06:12.67,Default,,0000,0000,0000,,Is today Tuesday?
Dialogue: 0,2:06:12.67,2:06:13.82,Default,,0000,0000,0000,,Yes.
Dialogue: 0,2:06:13.82,2:06:15.25,Default,,0000,0000,0000,,By Thursday.
Dialogue: 0,2:06:15.25,2:06:18.39,Default,,0000,0000,0000,,So please let me know how much\Nyou got from the calculator
Dialogue: 0,2:06:18.39,2:06:20.93,Default,,0000,0000,0000,,or from integration tables.
Dialogue: 0,2:06:20.93,2:06:27.07,Default,,0000,0000,0000,,So we have this\Nresult. And then I
Dialogue: 0,2:06:27.07,2:06:30.70,Default,,0000,0000,0000,,would like to interpret\Nthis result geometrically.
Dialogue: 0,2:06:30.70,2:06:37.15,Default,,0000,0000,0000,,What we can say about the\Nhelicoid that I didn't tell you
Dialogue: 0,2:06:37.15,2:06:40.54,Default,,0000,0000,0000,,but I'm going to tell you\Njust to finish-- have you
Dialogue: 0,2:06:40.54,2:06:45.25,Default,,0000,0000,0000,,been to the OMNIMAX\NScience Spectrum, the one
Dialogue: 0,2:06:45.25,2:06:46.65,Default,,0000,0000,0000,,in Lubbock or any other?
Dialogue: 0,2:06:46.65,2:06:48.38,Default,,0000,0000,0000,,I think they are\Neverywhere, right?
Dialogue: 0,2:06:48.38,2:06:49.32,Default,,0000,0000,0000,,I mean-- everywhere.
Dialogue: 0,2:06:49.32,2:06:50.65,Default,,0000,0000,0000,,We are a large city.
Dialogue: 0,2:06:50.65,2:06:55.87,Default,,0000,0000,0000,,Only in the big cities can\Nyou come to a Science Spectrum
Dialogue: 0,2:06:55.87,2:06:57.63,Default,,0000,0000,0000,,museum kind of like that.
Dialogue: 0,2:06:57.63,2:07:00.89,Default,,0000,0000,0000,,Have you played\Nwith the soap films?
Dialogue: 0,2:07:00.89,2:07:01.52,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:07:01.52,2:07:05.42,Default,,0000,0000,0000,,Do you remember what kind\Nwire frames they had?
Dialogue: 0,2:07:05.42,2:07:10.61,Default,,0000,0000,0000,,They had the big tub with soapy\Nwater, with soap solution.
Dialogue: 0,2:07:10.61,2:07:14.02,Default,,0000,0000,0000,,And then there were all sorts\Nof [INAUDIBLE] in there.
Dialogue: 0,2:07:14.02,2:07:19.67,Default,,0000,0000,0000,,They had the wire with pig\Nrods that looked like a prism.
Dialogue: 0,2:07:19.67,2:07:24.66,Default,,0000,0000,0000,,They had a cube that they wanted\Nyou to dip into the soap tub.
Dialogue: 0,2:07:24.66,2:07:26.83,Default,,0000,0000,0000,,They had a heart.
Dialogue: 0,2:07:26.83,2:07:29.15,Default,,0000,0000,0000,,And there comes the\Nbeautiful thing.
Dialogue: 0,2:07:29.15,2:07:33.21,Default,,0000,0000,0000,,They had this, a\Nspring that they took
Dialogue: 0,2:07:33.21,2:07:35.88,Default,,0000,0000,0000,,from your grandfather's bed.
Dialogue: 0,2:07:35.88,2:07:36.42,Default,,0000,0000,0000,,No.
Dialogue: 0,2:07:36.42,2:07:38.75,Default,,0000,0000,0000,,I don't think it\Nwas flexible at all.
Dialogue: 0,2:07:38.75,2:07:43.80,Default,,0000,0000,0000,,It was a helix made with\Na rod inside, a metal rod
Dialogue: 0,2:07:43.80,2:07:49.16,Default,,0000,0000,0000,,inside and attached\Nto the frame of that.
Dialogue: 0,2:07:49.16,2:07:54.82,Default,,0000,0000,0000,,There was the metal rod and\Nthis helix made of hard iron,
Dialogue: 0,2:07:54.82,2:07:56.48,Default,,0000,0000,0000,,and they were sticking together.
Dialogue: 0,2:07:56.48,2:07:59.95,Default,,0000,0000,0000,,Have you dipped this\Ninto the soap solution?
Dialogue: 0,2:07:59.95,2:08:01.54,Default,,0000,0000,0000,,And what did you get?
Dialogue: 0,2:08:01.54,2:08:02.66,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:08:02.66,2:08:04.66,Default,,0000,0000,0000,,DR. MAGDALENA TODA: That's\Nexactly what you get.
Dialogue: 0,2:08:04.66,2:08:05.95,Default,,0000,0000,0000,,You can get several surfaces.
Dialogue: 0,2:08:05.95,2:08:09.58,Default,,0000,0000,0000,,You can even get the one on\Nthe outside that's unstable.
Dialogue: 0,2:08:09.58,2:08:11.14,Default,,0000,0000,0000,,It broke in my case.
Dialogue: 0,2:08:11.14,2:08:14.13,Default,,0000,0000,0000,,It's almost like a cylinder.
Dialogue: 0,2:08:14.13,2:08:18.95,Default,,0000,0000,0000,,The one that was pretty\Nstable was your helicoid.
Dialogue: 0,2:08:18.95,2:08:30.56,Default,,0000,0000,0000,,The helicoid is a so-called\Nsoap film, soap film
Dialogue: 0,2:08:30.56,2:08:33.00,Default,,0000,0000,0000,,or minimal surface.
Dialogue: 0,2:08:33.00,2:08:37.76,Default,,0000,0000,0000,,Minimal surface.
Dialogue: 0,2:08:37.76,2:08:40.46,Default,,0000,0000,0000,,So what is a minimal surface?
Dialogue: 0,2:08:40.46,2:08:42.50,Default,,0000,0000,0000,,A minimal surface\Nis a soap film.
Dialogue: 0,2:08:42.50,2:08:46.92,Default,,0000,0000,0000,,A minimal surface\Nis a surface that
Dialogue: 0,2:08:46.92,2:08:52.89,Default,,0000,0000,0000,,tends to minimize the area\Nenclosed in a certain frame.
Dialogue: 0,2:08:52.89,2:08:59.55,Default,,0000,0000,0000,,You take a wire that looks like\Na loop, its own skew curve.
Dialogue: 0,2:08:59.55,2:09:02.77,Default,,0000,0000,0000,,You dip that into\Nthe soap solution.
Dialogue: 0,2:09:02.77,2:09:03.63,Default,,0000,0000,0000,,You pull it out.
Dialogue: 0,2:09:03.63,2:09:05.13,Default,,0000,0000,0000,,You get a soap film.
Dialogue: 0,2:09:05.13,2:09:07.93,Default,,0000,0000,0000,,That a minimal surface.
Dialogue: 0,2:09:07.93,2:09:10.00,Default,,0000,0000,0000,,So all the things\Nthat you created
Dialogue: 0,2:09:10.00,2:09:14.58,Default,,0000,0000,0000,,by taking wires and dipping\Nthem into the soap tub
Dialogue: 0,2:09:14.58,2:09:18.24,Default,,0000,0000,0000,,and pulling them out-- they\Nare not just called soap films.
Dialogue: 0,2:09:18.24,2:09:19.97,Default,,0000,0000,0000,,They are called\Nminimal surfaces.
Dialogue: 0,2:09:19.97,2:09:22.64,Default,,0000,0000,0000,,Somewhere on the wall\Nof the Science Spectrum,
Dialogue: 0,2:09:22.64,2:09:24.53,Default,,0000,0000,0000,,they wrote that.
Dialogue: 0,2:09:24.53,2:09:27.47,Default,,0000,0000,0000,,They didn't write much about\Nthe theory of minimal surfaces.
Dialogue: 0,2:09:27.47,2:09:31.08,Default,,0000,0000,0000,,But there are people-- there\Nare famous mathematicians who
Dialogue: 0,2:09:31.08,2:09:34.30,Default,,0000,0000,0000,,all their life studied\Njust minimal surfaces, just
Dialogue: 0,2:09:34.30,2:09:35.48,Default,,0000,0000,0000,,soap films.
Dialogue: 0,2:09:35.48,2:09:37.13,Default,,0000,0000,0000,,They came up with the results.
Dialogue: 0,2:09:37.13,2:09:41.24,Default,,0000,0000,0000,,Some of them got very\Nprestigious awards
Dialogue: 0,2:09:41.24,2:09:44.35,Default,,0000,0000,0000,,for that kind of thing\Ntheory for minimal surfaces.
Dialogue: 0,2:09:44.35,2:09:50.85,Default,,0000,0000,0000,,And these have been known\Nsince approximately the middle
Dialogue: 0,2:09:50.85,2:09:54.17,Default,,0000,0000,0000,,of the 19th century.
Dialogue: 0,2:09:54.17,2:09:55.72,Default,,0000,0000,0000,,There were several\Nmathematicians
Dialogue: 0,2:09:55.72,2:10:00.03,Default,,0000,0000,0000,,who discovered the most\Nimportant minimal surfaces.
Dialogue: 0,2:10:00.03,2:10:03.92,Default,,0000,0000,0000,,There are several that\Nyou may be familiar with
Dialogue: 0,2:10:03.92,2:10:06.69,Default,,0000,0000,0000,,and several you may\Nnot be familiar with.
Dialogue: 0,2:10:06.69,2:10:12.46,Default,,0000,0000,0000,,But another one that you may\Nhave known is the catenoid.
Dialogue: 0,2:10:12.46,2:10:15.17,Default,,0000,0000,0000,,And that's the last thing I'm\Ngoing to talk about today.
Dialogue: 0,2:10:15.17,2:10:19.33,Default,,0000,0000,0000,,Have you heard of a catenary?
Dialogue: 0,2:10:19.33,2:10:21.63,Default,,0000,0000,0000,,Have you ever been to St. Louis?
Dialogue: 0,2:10:21.63,2:10:24.53,Default,,0000,0000,0000,,St. Louis, St. Louis.
Dialogue: 0,2:10:24.53,2:10:27.42,Default,,0000,0000,0000,,The city St. Louis\Nwith the arch.
Dialogue: 0,2:10:27.42,2:10:29.00,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:10:29.00,2:10:31.36,Default,,0000,0000,0000,,Did you go to the arch?
Dialogue: 0,2:10:31.36,2:10:32.61,Default,,0000,0000,0000,,No?
Dialogue: 0,2:10:32.61,2:10:33.96,Default,,0000,0000,0000,,You should go to the arch.
Dialogue: 0,2:10:33.96,2:10:34.76,Default,,0000,0000,0000,,It looks like that.
Dialogue: 0,2:10:34.76,2:10:37.01,Default,,0000,0000,0000,,It has a big base.
Dialogue: 0,2:10:37.01,2:10:39.48,Default,,0000,0000,0000,,So it looks so beautiful.
Dialogue: 0,2:10:39.48,2:10:41.74,Default,,0000,0000,0000,,It's thicker at the base.
Dialogue: 0,2:10:41.74,2:10:46.38,Default,,0000,0000,0000,,This was based on a\Nmathematical equation.
Dialogue: 0,2:10:46.38,2:10:54.76,Default,,0000,0000,0000,,The mathematical equation\Nit was based on was cosh x.
Dialogue: 0,2:10:54.76,2:10:56.44,Default,,0000,0000,0000,,What is cosh as a function?
Dialogue: 0,2:10:56.44,2:10:59.75,Default,,0000,0000,0000,,Now I'm testing you,\Nbut I'm not judging you.
Dialogue: 0,2:10:59.75,2:11:00.82,Default,,0000,0000,0000,,If you forgot--
Dialogue: 0,2:11:00.82,2:11:01.66,Default,,0000,0000,0000,,STUDENT: e to the x.
Dialogue: 0,2:11:01.66,2:11:04.87,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\Ne to the x plus e
Dialogue: 0,2:11:04.87,2:11:06.14,Default,,0000,0000,0000,,to the negative x over 3.
Dialogue: 0,2:11:06.14,2:11:08.04,Default,,0000,0000,0000,,If it's minus, it's called sinh.
Dialogue: 0,2:11:08.04,2:11:12.04,Default,,0000,0000,0000,,So the one with [? parts. ?]\NYou can have x/a,
Dialogue: 0,2:11:12.04,2:11:13.85,Default,,0000,0000,0000,,and you can have 1/a in front.
Dialogue: 0,2:11:13.85,2:11:17.29,Default,,0000,0000,0000,,It's still called a catenary.
Dialogue: 0,2:11:17.29,2:11:20.10,Default,,0000,0000,0000,,Now what is-- this\Nis a catenary.
Dialogue: 0,2:11:20.10,2:11:23.94,Default,,0000,0000,0000,,The shape of the arch of\NSt. Louis is a catenary.
Dialogue: 0,2:11:23.94,2:11:28.82,Default,,0000,0000,0000,,But you are more used to the\Ncatenary upside down, which
Dialogue: 0,2:11:28.82,2:11:29.93,Default,,0000,0000,0000,,is any necklace.
Dialogue: 0,2:11:29.93,2:11:30.89,Default,,0000,0000,0000,,Do you have a necklace?
Dialogue: 0,2:11:30.89,2:11:33.96,Default,,0000,0000,0000,,If you take any\Nnecklace-- that is,
Dialogue: 0,2:11:33.96,2:11:36.29,Default,,0000,0000,0000,,it has to be homogeneous,\Nnot one of those,
Dialogue: 0,2:11:36.29,2:11:40.12,Default,,0000,0000,0000,,like you have a pearl hanging,\Nor you have several beads.
Dialogue: 0,2:11:40.12,2:11:41.04,Default,,0000,0000,0000,,No.
Dialogue: 0,2:11:41.04,2:11:44.97,Default,,0000,0000,0000,,It has to be a\Nhomogeneous metal.
Dialogue: 0,2:11:44.97,2:11:48.36,Default,,0000,0000,0000,,Think gold, solid gold, but\Nthat kind of liquid gold.
Dialogue: 0,2:11:48.36,2:11:49.82,Default,,0000,0000,0000,,Do you know what\NI'm talking about?
Dialogue: 0,2:11:49.82,2:11:54.13,Default,,0000,0000,0000,,Those beautiful bracelets\Nor necklaces that are fluid,
Dialogue: 0,2:11:54.13,2:11:58.36,Default,,0000,0000,0000,,and you cannot even see\Nthe different links.
Dialogue: 0,2:11:58.36,2:12:03.97,Default,,0000,0000,0000,,So you hang it at\Nthe same height.
Dialogue: 0,2:12:03.97,2:12:07.99,Default,,0000,0000,0000,,What you get-- Galileo\Nproved it was not
Dialogue: 0,2:12:07.99,2:12:11.45,Default,,0000,0000,0000,,a parabola because people at\Nthat time were really stupid.
Dialogue: 0,2:12:11.45,2:12:16.20,Default,,0000,0000,0000,,So they thought, hang a\Nchain from a woman's neck
Dialogue: 0,2:12:16.20,2:12:18.66,Default,,0000,0000,0000,,or some sort of\Nbeautiful jewelry,
Dialogue: 0,2:12:18.66,2:12:21.61,Default,,0000,0000,0000,,it must be a parabola because\Nit looks like a parabola.
Dialogue: 0,2:12:21.61,2:12:24.53,Default,,0000,0000,0000,,And Galileo Galilei says,\Nthese guys are nuts.
Dialogue: 0,2:12:24.53,2:12:28.30,Default,,0000,0000,0000,,They don't know any mathematics,\Nany physics, any astronomy.
Dialogue: 0,2:12:28.30,2:12:32.69,Default,,0000,0000,0000,,So he proved in no time\Nthat thing, the chain,
Dialogue: 0,2:12:32.69,2:12:34.46,Default,,0000,0000,0000,,cannot be a parabola.
Dialogue: 0,2:12:34.46,2:12:37.94,Default,,0000,0000,0000,,And he actually\Ncame up with that.
Dialogue: 0,2:12:37.94,2:12:40.16,Default,,0000,0000,0000,,If a is 1, you just\Nhave the cosh x.
Dialogue: 0,2:12:40.16,2:12:41.82,Default,,0000,0000,0000,,So this is the chain.
Dialogue: 0,2:12:41.82,2:12:45.76,Default,,0000,0000,0000,,If you take the chain--\Nif you take the chain--
Dialogue: 0,2:12:45.76,2:12:47.61,Default,,0000,0000,0000,,that's a chain upside down.
Dialogue: 0,2:12:47.61,2:12:50.70,Default,,0000,0000,0000,,If you take a chain--\Nlet's say y equals cosh
Dialogue: 0,2:12:50.70,2:12:56.31,Default,,0000,0000,0000,,x to make it easier--\Nand revolve that chain,
Dialogue: 0,2:12:56.31,2:12:57.86,Default,,0000,0000,0000,,you get a surface of revolution.
Dialogue: 0,2:12:57.86,2:13:03.01,Default,,0000,0000,0000,,
Dialogue: 0,2:13:03.01,2:13:05.28,Default,,0000,0000,0000,,And this surface of\Nrevolution is called catenoid.
Dialogue: 0,2:13:05.28,2:13:08.75,Default,,0000,0000,0000,,
Dialogue: 0,2:13:08.75,2:13:11.60,Default,,0000,0000,0000,,How can you get the catenoid\Nas a minimal surface, expressed
Dialogue: 0,2:13:11.60,2:13:14.53,Default,,0000,0000,0000,,as the minimum surface?
Dialogue: 0,2:13:14.53,2:13:16.16,Default,,0000,0000,0000,,There are people\Nwho can do that.
Dialogue: 0,2:13:16.16,2:13:18.02,Default,,0000,0000,0000,,They have the\Nability to do that.
Dialogue: 0,2:13:18.02,2:13:21.23,Default,,0000,0000,0000,,And they tried to have\Nan experimental thing
Dialogue: 0,2:13:21.23,2:13:23.41,Default,,0000,0000,0000,,at the Science Spectrum as well.
Dialogue: 0,2:13:23.41,2:13:24.77,Default,,0000,0000,0000,,And they did a beautiful job.
Dialogue: 0,2:13:24.77,2:13:25.77,Default,,0000,0000,0000,,I was there.
Dialogue: 0,2:13:25.77,2:13:30.30,Default,,0000,0000,0000,,So they took two\Ncircles made of plastic.
Dialogue: 0,2:13:30.30,2:13:32.56,Default,,0000,0000,0000,,You can have them be\Ncircles made of wood,
Dialogue: 0,2:13:32.56,2:13:37.21,Default,,0000,0000,0000,,made of iron or\Nsteel or anything.
Dialogue: 0,2:13:37.21,2:13:41.19,Default,,0000,0000,0000,,But they have to be\Nequal, equal circles.
Dialogue: 0,2:13:41.19,2:13:42.69,Default,,0000,0000,0000,,And you touch them,\Nand you dip them
Dialogue: 0,2:13:42.69,2:13:45.57,Default,,0000,0000,0000,,both at the same time\Ninto the soap solution.
Dialogue: 0,2:13:45.57,2:13:47.78,Default,,0000,0000,0000,,And then you pull\Nit out very gently
Dialogue: 0,2:13:47.78,2:13:49.78,Default,,0000,0000,0000,,because you have\Nto be very smart
Dialogue: 0,2:13:49.78,2:13:51.95,Default,,0000,0000,0000,,and very-- be like a surgeon.
Dialogue: 0,2:13:51.95,2:13:54.99,Default,,0000,0000,0000,,If your hands start shaking,\Nit's goodbye minimal surfaces
Dialogue: 0,2:13:54.99,2:13:56.50,Default,,0000,0000,0000,,because they break.
Dialogue: 0,2:13:56.50,2:13:57.54,Default,,0000,0000,0000,,They collapse.
Dialogue: 0,2:13:57.54,2:14:01.97,Default,,0000,0000,0000,,So you have to pull those\Ncircles with the same force,
Dialogue: 0,2:14:01.97,2:14:04.85,Default,,0000,0000,0000,,gently, one away from the other.
Dialogue: 0,2:14:04.85,2:14:06.25,Default,,0000,0000,0000,,What you're going\Nto get is going
Dialogue: 0,2:14:06.25,2:14:09.49,Default,,0000,0000,0000,,to be a film that looks\Nexactly like that.
Dialogue: 0,2:14:09.49,2:14:11.61,Default,,0000,0000,0000,,These are the circles.
Dialogue: 0,2:14:11.61,2:14:15.87,Default,,0000,0000,0000,,After a certain distance\Nof moving them apart,
Dialogue: 0,2:14:15.87,2:14:18.68,Default,,0000,0000,0000,,the soap film will\Ncollapse and will burst.
Dialogue: 0,2:14:18.68,2:14:21.54,Default,,0000,0000,0000,,There's no more surface inside.
Dialogue: 0,2:14:21.54,2:14:25.40,Default,,0000,0000,0000,,But up to that moment,\Nyou have a catenoid.
Dialogue: 0,2:14:25.40,2:14:27.32,Default,,0000,0000,0000,,And this catenoid is\Na minimal surface.
Dialogue: 0,2:14:27.32,2:14:30.16,Default,,0000,0000,0000,,It's trying to-- of\Nthe frame you gave it,
Dialogue: 0,2:14:30.16,2:14:33.58,Default,,0000,0000,0000,,which is the wire frame--\Nminimize the area.
Dialogue: 0,2:14:33.58,2:14:34.97,Default,,0000,0000,0000,,It's not going to be a cylinder.
Dialogue: 0,2:14:34.97,2:14:38.18,Default,,0000,0000,0000,,It's way too much area.
Dialogue: 0,2:14:38.18,2:14:41.07,Default,,0000,0000,0000,,It's going to be something\Nsmaller than that,
Dialogue: 0,2:14:41.07,2:14:43.95,Default,,0000,0000,0000,,so something that says,\NI'm an elastic surface.
Dialogue: 0,2:14:43.95,2:14:45.86,Default,,0000,0000,0000,,I'm occupying as\Nlittle area as I
Dialogue: 0,2:14:45.86,2:14:51.02,Default,,0000,0000,0000,,can because I live in\Na world of scarcity,
Dialogue: 0,2:14:51.02,2:14:54.49,Default,,0000,0000,0000,,and I try to occupy\Nas little as I can.
Dialogue: 0,2:14:54.49,2:14:57.24,Default,,0000,0000,0000,,So it's based on a\Nprinciple of physics.
Dialogue: 0,2:14:57.24,2:15:00.90,Default,,0000,0000,0000,,The surface tension\Nof the soap films
Dialogue: 0,2:15:00.90,2:15:04.21,Default,,0000,0000,0000,,will create this\Nminimization of the area.
Dialogue: 0,2:15:04.21,2:15:08.32,Default,,0000,0000,0000,,So all you need to do is\Nremember you know the helicoid
Dialogue: 0,2:15:08.32,2:15:11.65,Default,,0000,0000,0000,,and catenoid only because\Nyou're honors students.
Dialogue: 0,2:15:11.65,2:15:15.01,Default,,0000,0000,0000,,So thank [INAUDIBLE] college\Nfor giving you this opportunity.
Dialogue: 0,2:15:15.01,2:15:15.94,Default,,0000,0000,0000,,All right?
Dialogue: 0,2:15:15.94,2:15:16.62,Default,,0000,0000,0000,,I'm not kidding.
Dialogue: 0,2:15:16.62,2:15:20.76,Default,,0000,0000,0000,,It may sound like a joke, but\Nit's half joke, half truth.
Dialogue: 0,2:15:20.76,2:15:23.83,Default,,0000,0000,0000,,We learn learn a little\Nbit more interesting stuff
Dialogue: 0,2:15:23.83,2:15:25.93,Default,,0000,0000,0000,,than other kids.
Dialogue: 0,2:15:25.93,2:15:27.03,Default,,0000,0000,0000,,Enjoy your week.
Dialogue: 0,2:15:27.03,2:15:29.12,Default,,0000,0000,0000,,Good luck with homework.
Dialogue: 0,2:15:29.12,2:15:32.02,Default,,0000,0000,0000,,Come bug me abut any kind\Nof homework [INAUDIBLE].
Dialogue: 0,2:15:32.02,2:15:42.57,Default,,0000,0000,0000,,
Dialogue: 0,2:15:42.57,2:15:43.99,Default,,0000,0000,0000,,STUDENT: Do you\Nknow if I can talk
Dialogue: 0,2:15:43.99,2:15:45.30,Default,,0000,0000,0000,,to people about [INAUDIBLE]?
Dialogue: 0,2:15:45.30,2:15:49.27,Default,,0000,0000,0000,,
Dialogue: 0,2:15:49.27,2:15:51.93,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NActually, my [INAUDIBLE].
Dialogue: 0,2:15:51.93,2:15:54.46,Default,,0000,0000,0000,,She is the one who\Ndoes [INAUDIBLE].
Dialogue: 0,2:15:54.46,2:16:01.83,Default,,0000,0000,0000,,But I can take you to her so\Nyou can start [INAUDIBLE].
Dialogue: 0,2:16:01.83,2:16:03.95,Default,,0000,0000,0000,,I think it would be a very\Ninteresting [INAUDIBLE].
Dialogue: 0,2:16:03.95,2:16:04.83,Default,,0000,0000,0000,,Maybe.
Dialogue: 0,2:16:04.83,2:16:05.33,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,2:16:05.33,2:16:15.30,Default,,0000,0000,0000,,
Dialogue: 0,2:16:15.30,2:16:16.15,Default,,0000,0000,0000,,STUDENT: OK.
Dialogue: 0,2:16:16.15,2:16:18.95,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NSecond floor, [INAUDIBLE].
Dialogue: 0,2:16:18.95,2:16:21.83,Default,,0000,0000,0000,,This is the [INAUDIBLE].
Dialogue: 0,2:16:21.83,2:16:24.32,Default,,0000,0000,0000,,You have to go all\Nthe way behind.
Dialogue: 0,2:16:24.32,2:16:24.93,Default,,0000,0000,0000,,STUDENT: OK.
Dialogue: 0,2:16:24.93,2:16:26.19,Default,,0000,0000,0000,,OK.
Dialogue: 0,2:16:26.19,2:16:28.40,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:16:28.40,2:16:32.47,Default,,0000,0000,0000,,
Dialogue: 0,2:16:32.47,2:16:35.42,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\N[INAUDIBLE] because I don't
Dialogue: 0,2:16:35.42,2:16:37.01,Default,,0000,0000,0000,,want to give you anything new.
Dialogue: 0,2:16:37.01,2:16:39.77,Default,,0000,0000,0000,,I don't want to get you\Nin any kind of trouble.
Dialogue: 0,2:16:39.77,2:16:44.08,Default,,0000,0000,0000,,The problems that I solved\Non the board are primarily,
Dialogue: 0,2:16:44.08,2:16:47.21,Default,,0000,0000,0000,,I would say, 60% of what\Nyou'll see on the midterm.
Dialogue: 0,2:16:47.21,2:16:49.72,Default,,0000,0000,0000,,It's something that\Nwe covered in class.
Dialogue: 0,2:16:49.72,2:16:53.07,Default,,0000,0000,0000,,And the other 40% will be\Nsomething not too hard,
Dialogue: 0,2:16:53.07,2:16:57.17,Default,,0000,0000,0000,,but something standard out of\Nyour WeBWork homework, the one
Dialogue: 0,2:16:57.17,2:16:58.37,Default,,0000,0000,0000,,that you studied.
Dialogue: 0,2:16:58.37,2:16:59.69,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,2:16:59.69,2:17:00.53,Default,,0000,0000,0000,,STUDENT: Thank you.
Dialogue: 0,2:17:00.53,2:17:02.17,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYou're welcome.
Dialogue: 0,2:17:02.17,2:17:03.68,Default,,0000,0000,0000,,STUDENT: We're trying to\Njoin the Honors Society.
Dialogue: 0,2:17:03.68,2:17:05.34,Default,,0000,0000,0000,,But we can't make\Nthat thing tomorrow.
Dialogue: 0,2:17:05.34,2:17:06.20,Default,,0000,0000,0000,,Can we still join?
Dialogue: 0,2:17:06.20,2:17:08.00,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYou can still join.
Dialogue: 0,2:17:08.00,2:17:11.29,Default,,0000,0000,0000,,Remind me to give you the\Ngolden pin, the brochures, all
Dialogue: 0,2:17:11.29,2:17:12.28,Default,,0000,0000,0000,,the information.
Dialogue: 0,2:17:12.28,2:17:15.54,Default,,0000,0000,0000,,And then when you get those,\Nyou'll give me the $35.
Dialogue: 0,2:17:15.54,2:17:17.07,Default,,0000,0000,0000,,It's a lifetime thing.
Dialogue: 0,2:17:17.07,2:17:17.78,Default,,0000,0000,0000,,STUDENT: Awesome.
Dialogue: 0,2:17:17.78,2:17:18.35,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,2:17:18.35,2:17:19.82,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYou're welcome.
Dialogue: 0,2:17:19.82,2:17:20.98,Default,,0000,0000,0000,,Both of you want to--
Dialogue: 0,2:17:20.98,2:17:21.56,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:17:21.56,2:17:22.69,Default,,0000,0000,0000,,DR. MAGDALENA TODA: And\Nyou cannot come tomorrow?
Dialogue: 0,2:17:22.69,2:17:23.27,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:17:23.27,2:17:23.87,Default,,0000,0000,0000,,I have my--
Dialogue: 0,2:17:23.87,2:17:27.60,Default,,0000,0000,0000,,DR. MAGDALENA TODA: I wanted to\Nbring something, some snacks.
Dialogue: 0,2:17:27.60,2:17:28.52,Default,,0000,0000,0000,,But I don't know.
Dialogue: 0,2:17:28.52,2:17:31.99,Default,,0000,0000,0000,,I need to count and see\Nhow many people can come.
Dialogue: 0,2:17:31.99,2:17:33.50,Default,,0000,0000,0000,,And it's going to\Nbe in my office.
Dialogue: 0,2:17:33.50,2:17:34.00,Default,,0000,0000,0000,,STUDENT: OK.
Dialogue: 0,2:17:34.00,2:17:35.04,Default,,0000,0000,0000,,DR. MAGDALENA TODA: All right.
Dialogue: 0,2:17:35.04,2:17:36.49,Default,,0000,0000,0000,,STUDENT: Were you in\Nthe tennis tournament?
Dialogue: 0,2:17:36.49,2:17:36.99,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:17:36.99,2:17:37.82,Default,,0000,0000,0000,,STUDENT: You're the guy who won?
Dialogue: 0,2:17:37.82,2:17:38.53,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:17:38.53,2:17:39.57,Default,,0000,0000,0000,,STUDENT: Congratulations.
Dialogue: 0,2:17:39.57,2:17:42.04,Default,,0000,0000,0000,,I was like, I know that name.
Dialogue: 0,2:17:42.04,2:17:43.43,Default,,0000,0000,0000,,He's in my [INAUDIBLE].
Dialogue: 0,2:17:43.43,2:17:44.72,Default,,0000,0000,0000,,DR. MAGDALENA TODA: You won it?
Dialogue: 0,2:17:44.72,2:17:45.20,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:17:45.20,2:17:45.96,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NThe tennis tournament?
Dialogue: 0,2:17:45.96,2:17:46.54,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,2:17:46.54,2:17:47.52,Default,,0000,0000,0000,,It was [INAUDIBLE].
Dialogue: 0,2:17:47.52,2:17:49.02,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NCongratulations.
Dialogue: 0,2:17:49.02,2:17:51.44,Default,,0000,0000,0000,,Why don't you blab a\Nlittle bit about yourself?
Dialogue: 0,2:17:51.44,2:17:52.81,Default,,0000,0000,0000,,You're so modest.
Dialogue: 0,2:17:52.81,2:17:54.18,Default,,0000,0000,0000,,You never say anything.
Dialogue: 0,2:17:54.18,2:17:55.55,Default,,0000,0000,0000,,STUDENT: It's all right.
Dialogue: 0,2:17:55.55,2:17:56.72,Default,,0000,0000,0000,,STUDENT: I'm sorry.
Dialogue: 0,2:17:56.72,2:17:57.22,Default,,0000,0000,0000,,STUDENT: No.
Dialogue: 0,2:17:57.22,2:17:57.84,Default,,0000,0000,0000,,STUDENT: It's not a big deal.
Dialogue: 0,2:17:57.84,2:17:59.29,Default,,0000,0000,0000,,STUDENT: Were people good?
Dialogue: 0,2:17:59.29,2:18:01.64,Default,,0000,0000,0000,,Yeah?
Dialogue: 0,2:18:01.64,2:18:02.89,Default,,0000,0000,0000,,DR. MAGDALENA TODA: All right.
Dialogue: 0,2:18:02.89,2:18:04.74,Default,,0000,0000,0000,,STUDENT: I have my extra credit.
Dialogue: 0,2:18:04.74,2:18:06.25,Default,,0000,0000,0000,,