[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.84,Default,,0000,0000,0000,,MAGDALENA TODA:\NAccording to my watch,
Dialogue: 0,0:00:01.84,0:00:04.22,Default,,0000,0000,0000,,we are right on time to start.
Dialogue: 0,0:00:04.22,0:00:09.68,Default,,0000,0000,0000,,I may be one minute\Nearly, or something.
Dialogue: 0,0:00:09.68,0:00:16.62,Default,,0000,0000,0000,,Do you have questions out of the\Nmaterial we covered last time?
Dialogue: 0,0:00:16.62,0:00:19.59,Default,,0000,0000,0000,,What I'm planning on\Ndoing-- let me tell you
Dialogue: 0,0:00:19.59,0:00:20.59,Default,,0000,0000,0000,,what I'm planning to do.
Dialogue: 0,0:00:20.59,0:00:24.06,Default,,0000,0000,0000,,I will cover triple\Nintegrals today.
Dialogue: 0,0:00:24.06,0:00:26.73,Default,,0000,0000,0000,,And this way, you\Nwould have accumulated
Dialogue: 0,0:00:26.73,0:00:31.95,Default,,0000,0000,0000,,enough to deal with most of\Nthe problems in homework four.
Dialogue: 0,0:00:31.95,0:00:36.31,Default,,0000,0000,0000,,You have mastered the\Ndouble integration by now,
Dialogue: 0,0:00:36.31,0:00:41.02,Default,,0000,0000,0000,,in all sorts of coordinates,\Nwhich is a good thing.
Dialogue: 0,0:00:41.02,0:00:46.05,Default,,0000,0000,0000,,Triple integrals\Nare your friend.
Dialogue: 0,0:00:46.05,0:00:51.01,Default,,0000,0000,0000,,If you have understood\Nthe double integration,
Dialogue: 0,0:00:51.01,0:00:52.100,Default,,0000,0000,0000,,you will have no\Nproblem understanding
Dialogue: 0,0:00:52.100,0:00:54.99,Default,,0000,0000,0000,,the triple integrals.
Dialogue: 0,0:00:54.99,0:00:56.79,Default,,0000,0000,0000,,The idea is the same.
Dialogue: 0,0:00:56.79,0:00:58.95,Default,,0000,0000,0000,,You look at different\Ndomains, and then you
Dialogue: 0,0:00:58.95,0:01:04.54,Default,,0000,0000,0000,,realize that there are\NFubini-Tunelli type of results.
Dialogue: 0,0:01:04.54,0:01:06.40,Default,,0000,0000,0000,,I'm going to present\None right now.
Dialogue: 0,0:01:06.40,0:01:12.55,Default,,0000,0000,0000,,And there are also regions\Nof a certain type, that
Dialogue: 0,0:01:12.55,0:01:15.60,Default,,0000,0000,0000,,can be treated\Ndifferentially, and then you
Dialogue: 0,0:01:15.60,0:01:20.33,Default,,0000,0000,0000,,have cases in which reversing\Nthe order of integration
Dialogue: 0,0:01:20.33,0:01:25.94,Default,,0000,0000,0000,,for those triple integrals\Nis going to help you a lot.
Dialogue: 0,0:01:25.94,0:01:30.87,Default,,0000,0000,0000,,OK, 12.5 is the name of the\Nsection, triple integrals.
Dialogue: 0,0:01:30.87,0:01:42.71,Default,,0000,0000,0000,,
Dialogue: 0,0:01:42.71,0:01:46.42,Default,,0000,0000,0000,,So what should you imagine?
Dialogue: 0,0:01:46.42,0:01:50.60,Default,,0000,0000,0000,,You should imagine\Nthat somebody gives you
Dialogue: 0,0:01:50.60,0:01:55.23,Default,,0000,0000,0000,,a function of three variables.
Dialogue: 0,0:01:55.23,0:02:00.31,Default,,0000,0000,0000,,Let's call that-- it doesn't\Noften have a name as a letter,
Dialogue: 0,0:02:00.31,0:02:03.64,Default,,0000,0000,0000,,but let's call it w.
Dialogue: 0,0:02:03.64,0:02:08.20,Default,,0000,0000,0000,,Being a function of three\Ncoordinates, x, y, and z,
Dialogue: 0,0:02:08.20,0:02:10.91,Default,,0000,0000,0000,,where x, y, z is in R3.
Dialogue: 0,0:02:10.91,0:02:14.89,Default,,0000,0000,0000,,
Dialogue: 0,0:02:14.89,0:02:22.48,Default,,0000,0000,0000,,And we have some assumptions\Nabout the domain D
Dialogue: 0,0:02:22.48,0:02:24.72,Default,,0000,0000,0000,,that you are working\Nover, and you
Dialogue: 0,0:02:24.72,0:02:36.64,Default,,0000,0000,0000,,have D as a closed-bounded\Ndomain in R3.
Dialogue: 0,0:02:36.64,0:02:39.39,Default,,0000,0000,0000,,
Dialogue: 0,0:02:39.39,0:02:44.01,Default,,0000,0000,0000,,Examples that you're going\Nto do use frequently.
Dialogue: 0,0:02:44.01,0:02:46.74,Default,,0000,0000,0000,,Frequently used.
Dialogue: 0,0:02:46.74,0:02:51.88,Default,,0000,0000,0000,,A sphere, a ball,\Nactually, because in here,
Dialogue: 0,0:02:51.88,0:02:56.07,Default,,0000,0000,0000,,if a sphere is together with\Na shell, it is the ball.
Dialogue: 0,0:02:56.07,0:03:07.17,Default,,0000,0000,0000,,Then you have some types of\Npolyhedra in r3 of all types.
Dialogue: 0,0:03:07.17,0:03:12.11,Default,,0000,0000,0000,,
Dialogue: 0,0:03:12.11,0:03:15.18,Default,,0000,0000,0000,,And by that, I mean\Nthe classical polyhedra
Dialogue: 0,0:03:15.18,0:03:23.50,Default,,0000,0000,0000,,whose sides are just polygons.
Dialogue: 0,0:03:23.50,0:03:27.11,Default,,0000,0000,0000,,But you will also have some\Ncurvilinear polyhedra, as well.
Dialogue: 0,0:03:27.11,0:03:32.95,Default,,0000,0000,0000,,
Dialogue: 0,0:03:32.95,0:03:34.96,Default,,0000,0000,0000,,What do I mean?
Dialogue: 0,0:03:34.96,0:03:36.62,Default,,0000,0000,0000,,I mean, we've seen that already.
Dialogue: 0,0:03:36.62,0:03:38.41,Default,,0000,0000,0000,,For example, somebody\Ngive you a graph
Dialogue: 0,0:03:38.41,0:03:49.56,Default,,0000,0000,0000,,of a function, g of x and y, a\Ncontinuous function, and says,
Dialogue: 0,0:03:49.56,0:03:53.59,Default,,0000,0000,0000,,OK, can you estimate the\Nvolume under the graph?
Dialogue: 0,0:03:53.59,0:03:55.85,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:03:55.85,0:03:59.05,Default,,0000,0000,0000,,And until now, we treated\Nthis volume under the graph
Dialogue: 0,0:03:59.05,0:04:05.61,Default,,0000,0000,0000,,as double integral of g of x, y,\Ncontinuous function over d, a,
Dialogue: 0,0:04:05.61,0:04:08.86,Default,,0000,0000,0000,,where d, a was dx dy.
Dialogue: 0,0:04:08.86,0:04:15.73,Default,,0000,0000,0000,,And we said double integral\Nover the projected domain
Dialogue: 0,0:04:15.73,0:04:16.36,Default,,0000,0000,0000,,in the plane.
Dialogue: 0,0:04:16.36,0:04:17.62,Default,,0000,0000,0000,,That's what I have.
Dialogue: 0,0:04:17.62,0:04:20.64,Default,,0000,0000,0000,,But can I treat it, this\Nvolume, can I treat it
Dialogue: 0,0:04:20.64,0:04:22.06,Default,,0000,0000,0000,,as a triple integral?
Dialogue: 0,0:04:22.06,0:04:25.38,Default,,0000,0000,0000,,This is the question,\Nand answer is--
Dialogue: 0,0:04:25.38,0:04:27.24,Default,,0000,0000,0000,,so can I make three snakes?
Dialogue: 0,0:04:27.24,0:04:29.69,Default,,0000,0000,0000,,The answer is yes.
Dialogue: 0,0:04:29.69,0:04:32.16,Default,,0000,0000,0000,,And the way I'm\Ngoing to define that
Dialogue: 0,0:04:32.16,0:04:39.29,Default,,0000,0000,0000,,would be a triple\Nintegral over a 3D domain.
Dialogue: 0,0:04:39.29,0:04:48.58,Default,,0000,0000,0000,,Let's call it curvilinear d in\Nr3, which is the volume-- which
Dialogue: 0,0:04:48.58,0:04:52.89,Default,,0000,0000,0000,,is the body under the graph\Nof this positive function,
Dialogue: 0,0:04:52.89,0:04:57.20,Default,,0000,0000,0000,,and above the projected\Narea in plane.
Dialogue: 0,0:04:57.20,0:05:00.24,Default,,0000,0000,0000,,So it's going to be a\Ncylinder in this case.
Dialogue: 0,0:05:00.24,0:05:02.83,Default,,0000,0000,0000,,And I'll put here 1 dv.
Dialogue: 0,0:05:02.83,0:05:07.84,Default,,0000,0000,0000,,And dv is a mysterious element.
Dialogue: 0,0:05:07.84,0:05:11.19,Default,,0000,0000,0000,,That's the volume element.
Dialogue: 0,0:05:11.19,0:05:15.22,Default,,0000,0000,0000,,And I will talk a little\Nbit about it right now.
Dialogue: 0,0:05:15.22,0:05:17.60,Default,,0000,0000,0000,,So what can you imagine?
Dialogue: 0,0:05:17.60,0:05:23.10,Default,,0000,0000,0000,,They give you a way to\Nlook at it in the book.
Dialogue: 0,0:05:23.10,0:05:28.25,Default,,0000,0000,0000,,I mean, we give you a way\Nto look at it in the book.
Dialogue: 0,0:05:28.25,0:05:31.33,Default,,0000,0000,0000,,It's not very thorough\Nin explanations,
Dialogue: 0,0:05:31.33,0:05:36.14,Default,,0000,0000,0000,,but it certainly gives you the\Ngeneral idea of what you want,
Dialogue: 0,0:05:36.14,0:05:38.24,Default,,0000,0000,0000,,what you need.
Dialogue: 0,0:05:38.24,0:05:42.63,Default,,0000,0000,0000,,So somebody gives you a potato.
Dialogue: 0,0:05:42.63,0:05:44.86,Default,,0000,0000,0000,,It doesn't have to\Nbe this cylinder.
Dialogue: 0,0:05:44.86,0:05:53.39,Default,,0000,0000,0000,,It's something beautiful, a\Nbody inside a compact surface.
Dialogue: 0,0:05:53.39,0:05:55.51,Default,,0000,0000,0000,,Let's say there are\Nno self-intersections.
Dialogue: 0,0:05:55.51,0:06:06.34,Default,,0000,0000,0000,,You have some compact surface,\Nlike a sphere or a polyhedron,
Dialogue: 0,0:06:06.34,0:06:09.93,Default,,0000,0000,0000,,assume it's simply\Nconnected, and it doesn't
Dialogue: 0,0:06:09.93,0:06:12.75,Default,,0000,0000,0000,,have any self-intersections.
Dialogue: 0,0:06:12.75,0:06:16.67,Default,,0000,0000,0000,,So a beautiful\Npotato that's smooth.
Dialogue: 0,0:06:16.67,0:06:19.74,Default,,0000,0000,0000,,If you imagine a potato\Nthat has singularities,
Dialogue: 0,0:06:19.74,0:06:23.50,Default,,0000,0000,0000,,like most potatoes have\Nsingularities, boo-boos,
Dialogue: 0,0:06:23.50,0:06:26.29,Default,,0000,0000,0000,,and cuts, so that's bad.
Dialogue: 0,0:06:26.29,0:06:31.76,Default,,0000,0000,0000,,So think about some\Nregular surface
Dialogue: 0,0:06:31.76,0:06:35.28,Default,,0000,0000,0000,,that's closed, no\Nself-intersections,
Dialogue: 0,0:06:35.28,0:06:39.23,Default,,0000,0000,0000,,and that is a potato\Nthat is [INAUDIBLE].
Dialogue: 0,0:06:39.23,0:06:42.55,Default,,0000,0000,0000,,Oh let's call it--\Np for potato, no,
Dialogue: 0,0:06:42.55,0:06:44.97,Default,,0000,0000,0000,,because I have got to\Nuse p for the partition.
Dialogue: 0,0:06:44.97,0:06:48.88,Default,,0000,0000,0000,,So let me call it\ND from 3D domain,
Dialogue: 0,0:06:48.88,0:06:54.07,Default,,0000,0000,0000,,because it's a three-dimensional\Ndomain, enclosed
Dialogue: 0,0:06:54.07,0:07:00.65,Default,,0000,0000,0000,,by a curve, enclosed\Nby a compact surface.
Dialogue: 0,0:07:00.65,0:07:04.16,Default,,0000,0000,0000,,
Dialogue: 0,0:07:04.16,0:07:05.17,Default,,0000,0000,0000,,So think potato.
Dialogue: 0,0:07:05.17,0:07:08.71,Default,,0000,0000,0000,,What do we do in\Nterms of partitions?
Dialogue: 0,0:07:08.71,0:07:12.60,Default,,0000,0000,0000,,Those pixels were pixels for\Nthe 2D world in flat line.
Dialogue: 0,0:07:12.60,0:07:15.70,Default,,0000,0000,0000,,But now, we don't\Nhave pixels anymore.
Dialogue: 0,0:07:15.70,0:07:16.96,Default,,0000,0000,0000,,Yes we do.
Dialogue: 0,0:07:16.96,0:07:20.46,Default,,0000,0000,0000,,I was watching lots of\Nsci-fi, and the holograms
Dialogue: 0,0:07:20.46,0:07:23.81,Default,,0000,0000,0000,,have the\Nthree-dimensional pixels.
Dialogue: 0,0:07:23.81,0:07:26.98,Default,,0000,0000,0000,,I'm going to try and\Nmake a partition.
Dialogue: 0,0:07:26.98,0:07:31.11,Default,,0000,0000,0000,,It's going to be a hard way\Nto partition this potato.
Dialogue: 0,0:07:31.11,0:07:39.14,Default,,0000,0000,0000,,But you have to imagine you\Nhave a rectangular partition,
Dialogue: 0,0:07:39.14,0:07:49.51,Default,,0000,0000,0000,,so every little pixel\Nwill be a-- is not cube.
Dialogue: 0,0:07:49.51,0:07:52.74,Default,,0000,0000,0000,,It has to be a little\Ntiny parallelepiped.
Dialogue: 0,0:07:52.74,0:07:59.85,Default,,0000,0000,0000,,So a 3D pixel, let me\Nput pixel in quotes,
Dialogue: 0,0:07:59.85,0:08:01.78,Default,,0000,0000,0000,,because this is\Nkind of the idea.
Dialogue: 0,0:08:01.78,0:08:04.27,Default,,0000,0000,0000,,Well have what\Nkind of dimensions?
Dialogue: 0,0:08:04.27,0:08:08.21,Default,,0000,0000,0000,,We'll have three\Ndimensions, right?
Dialogue: 0,0:08:08.21,0:08:16.73,Default,,0000,0000,0000,,Three dimensions, a delta xk,\Na delta yk, and a delta zk
Dialogue: 0,0:08:16.73,0:08:18.63,Default,,0000,0000,0000,,for the pixel number k.
Dialogue: 0,0:08:18.63,0:08:21.36,Default,,0000,0000,0000,,That's pixel number k.
Dialogue: 0,0:08:21.36,0:08:23.61,Default,,0000,0000,0000,,We have to number them,\Nsee how many they are.
Dialogue: 0,0:08:23.61,0:08:29.58,Default,,0000,0000,0000,,Where k is from 1 to n, and\Nis the total number of 3D
Dialogue: 0,0:08:29.58,0:08:31.93,Default,,0000,0000,0000,,pixels that I'm covering\Nthe whole thing.
Dialogue: 0,0:08:31.93,0:08:34.17,Default,,0000,0000,0000,,So don't think graphing\Npaper, anymore,
Dialogue: 0,0:08:34.17,0:08:36.45,Default,,0000,0000,0000,,because that's outdated.
Dialogue: 0,0:08:36.45,0:08:39.19,Default,,0000,0000,0000,,Don't even think of 2D image.
Dialogue: 0,0:08:39.19,0:08:46.08,Default,,0000,0000,0000,,Think of some hologram,\Nwhere you cover everything
Dialogue: 0,0:08:46.08,0:08:51.50,Default,,0000,0000,0000,,with tiny, tiny,\Ntiny 3D pixels so
Dialogue: 0,0:08:51.50,0:08:58.26,Default,,0000,0000,0000,,that in the limit, when you pass\Nto the limit, with respect to n
Dialogue: 0,0:08:58.26,0:09:04.47,Default,,0000,0000,0000,,and going to infinity,\Nthe discrete image,
Dialogue: 0,0:09:04.47,0:09:07.51,Default,,0000,0000,0000,,you're going to have something\Nlike a diamond shaped thingy,
Dialogue: 0,0:09:07.51,0:09:10.87,Default,,0000,0000,0000,,will convert to\Nthe smooth potato.
Dialogue: 0,0:09:10.87,0:09:16.71,Default,,0000,0000,0000,,So as n goes to\Ninfinity, that surface
Dialogue: 0,0:09:16.71,0:09:21.04,Default,,0000,0000,0000,,made of tiny, tiny squares\Nwill convert to the data.
Dialogue: 0,0:09:21.04,0:09:27.68,Default,,0000,0000,0000,,So how do you actually find\Na triple snake integral f
Dialogue: 0,0:09:27.68,0:09:33.46,Default,,0000,0000,0000,,of x, y, z over the\Ndomain D, dx dy dz.
Dialogue: 0,0:09:33.46,0:09:38.23,Default,,0000,0000,0000,,
Dialogue: 0,0:09:38.23,0:09:42.08,Default,,0000,0000,0000,,OK, this theoretically\Nshould be what?
Dialogue: 0,0:09:42.08,0:09:43.43,Default,,0000,0000,0000,,Think pixels.
Dialogue: 0,0:09:43.43,0:09:53.20,Default,,0000,0000,0000,,Limit as n goes to infinity\Nof the sum of the--
Dialogue: 0,0:09:53.20,0:09:54.62,Default,,0000,0000,0000,,what do I need to do?
Dialogue: 0,0:09:54.62,0:10:01.42,Default,,0000,0000,0000,,Think the whole\Npartition into pixels.
Dialogue: 0,0:10:01.42,0:10:09.43,Default,,0000,0000,0000,,How many pixels? n pixels\Ntotal is called a p.
Dialogue: 0,0:10:09.43,0:10:12.85,Default,,0000,0000,0000,,Script p.
Dialogue: 0,0:10:12.85,0:10:21.07,Default,,0000,0000,0000,,And the normal p will be the\Nhighest diameter of-- highest
Dialogue: 0,0:10:21.07,0:10:27.20,Default,,0000,0000,0000,,diameter among all pixels.
Dialogue: 0,0:10:27.20,0:10:32.03,Default,,0000,0000,0000,,
Dialogue: 0,0:10:32.03,0:10:34.62,Default,,0000,0000,0000,,And you're going to\Nsay, Oh, what the heck?
Dialogue: 0,0:10:34.62,0:10:36.03,Default,,0000,0000,0000,,I don't understand it.
Dialogue: 0,0:10:36.03,0:10:39.35,Default,,0000,0000,0000,,I have these three-dimensional\Ncubes, or three-dimensional
Dialogue: 0,0:10:39.35,0:10:42.88,Default,,0000,0000,0000,,barely by p that get tinier,\Nand tinier, and tinier.
Dialogue: 0,0:10:42.88,0:10:47.35,Default,,0000,0000,0000,,What in the world is going to\Nbe a diameter of such a pixel?
Dialogue: 0,0:10:47.35,0:10:49.04,Default,,0000,0000,0000,,Well, you have to\Ntake this pixel
Dialogue: 0,0:10:49.04,0:10:55.24,Default,,0000,0000,0000,,and magnify it so we can look\Nat it a little bit better.
Dialogue: 0,0:10:55.24,0:10:58.22,Default,,0000,0000,0000,,What do we mean by\Ndiameter of this pixel?
Dialogue: 0,0:10:58.22,0:11:00.90,Default,,0000,0000,0000,,
Dialogue: 0,0:11:00.90,0:11:02.91,Default,,0000,0000,0000,,Let's call this pixel k.
Dialogue: 0,0:11:02.91,0:11:08.18,Default,,0000,0000,0000,,
Dialogue: 0,0:11:08.18,0:11:13.09,Default,,0000,0000,0000,,The maximum of all the distances\Nyou can compute between two
Dialogue: 0,0:11:13.09,0:11:16.08,Default,,0000,0000,0000,,arbitrary points inside.
Dialogue: 0,0:11:16.08,0:11:17.71,Default,,0000,0000,0000,,Between two arbitrary\Npoints inside,
Dialogue: 0,0:11:17.71,0:11:19.13,Default,,0000,0000,0000,,you have many [INAUDIBLE].
Dialogue: 0,0:11:19.13,0:11:25.34,Default,,0000,0000,0000,,So the maximum of the\Ndistance between points,
Dialogue: 0,0:11:25.34,0:11:34.72,Default,,0000,0000,0000,,let's call them r and\Nq inside the pixel.
Dialogue: 0,0:11:34.72,0:11:37.93,Default,,0000,0000,0000,,So if it were for me\Nto ask you to find
Dialogue: 0,0:11:37.93,0:11:40.99,Default,,0000,0000,0000,,that diameter in this\Ncase, what would that be?
Dialogue: 0,0:11:40.99,0:11:42.12,Default,,0000,0000,0000,,STUDENT: It's the diagonal.
Dialogue: 0,0:11:42.12,0:11:46.24,Default,,0000,0000,0000,,MAGDALENA TODA: It's the\Ndiagonal between this corner
Dialogue: 0,0:11:46.24,0:11:49.29,Default,,0000,0000,0000,,and the opposite corner.
Dialogue: 0,0:11:49.29,0:11:55.71,Default,,0000,0000,0000,,So this would be the highest\Ndistance inside this pixel.
Dialogue: 0,0:11:55.71,0:11:58.75,Default,,0000,0000,0000,,If it's a cube-- you see\Nthat I wanted a cube.
Dialogue: 0,0:11:58.75,0:12:02.21,Default,,0000,0000,0000,,If it's a parallelepiped,\Nit's the same idea.
Dialogue: 0,0:12:02.21,0:12:08.63,Default,,0000,0000,0000,,So I have that opposite\Ncorner distance kind of thing.
Dialogue: 0,0:12:08.63,0:12:10.20,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:12:10.20,0:12:12.02,Default,,0000,0000,0000,,So I know what I want.
Dialogue: 0,0:12:12.02,0:12:15.01,Default,,0000,0000,0000,,I want n to go to infinity.
Dialogue: 0,0:12:15.01,0:12:19.49,Default,,0000,0000,0000,,That means I'm going to\Nhave the p going to 0.
Dialogue: 0,0:12:19.49,0:12:24.14,Default,,0000,0000,0000,,The length of the\Nhighest diameter
Dialogue: 0,0:12:24.14,0:12:26.27,Default,,0000,0000,0000,,will go shrinking to 0.
Dialogue: 0,0:12:26.27,0:12:29.67,Default,,0000,0000,0000,,And then I'm going to say\Nhere, what do I have inside?
Dialogue: 0,0:12:29.67,0:12:31.44,Default,,0000,0000,0000,,F of some intermediate point.
Dialogue: 0,0:12:31.44,0:12:34.30,Default,,0000,0000,0000,,In every pixel, I take a point.
Dialogue: 0,0:12:34.30,0:12:36.42,Default,,0000,0000,0000,,Another pixel, another\Npoint, and so on.
Dialogue: 0,0:12:36.42,0:12:39.01,Default,,0000,0000,0000,,So how many such\Npoints do I have?
Dialogue: 0,0:12:39.01,0:12:42.11,Default,,0000,0000,0000,,n, because it's the\Nnumber of pixels.
Dialogue: 0,0:12:42.11,0:12:45.86,Default,,0000,0000,0000,,So inside, let's call\Nthis as pixel p k.
Dialogue: 0,0:12:45.86,0:12:49.60,Default,,0000,0000,0000,,What is the little point that\NI took out of the [INAUDIBLE]
Dialogue: 0,0:12:49.60,0:12:51.63,Default,,0000,0000,0000,,inside the cube, or\Ninside the pixel?
Dialogue: 0,0:12:51.63,0:12:59.65,Default,,0000,0000,0000,,Let's call that mister x k\Nstar, y k star, x k star.
Dialogue: 0,0:12:59.65,0:13:01.16,Default,,0000,0000,0000,,Why do we put a star?
Dialogue: 0,0:13:01.16,0:13:02.59,Default,,0000,0000,0000,,Because he is a star.
Dialogue: 0,0:13:02.59,0:13:05.33,Default,,0000,0000,0000,,He wants to be number one\Nin these little domains,
Dialogue: 0,0:13:05.33,0:13:06.88,Default,,0000,0000,0000,,and says I'm a star.
Dialogue: 0,0:13:06.88,0:13:11.52,Default,,0000,0000,0000,,So we take that intermediate\Npoint, x k star,
Dialogue: 0,0:13:11.52,0:13:16.36,Default,,0000,0000,0000,,y k star, x k star.
Dialogue: 0,0:13:16.36,0:13:21.70,Default,,0000,0000,0000,,Then we have this function\Nmultiplied by the delta v k.
Dialogue: 0,0:13:21.70,0:13:25.43,Default,,0000,0000,0000,,somebody tell me what\Nthis delta v k will mean,
Dialogue: 0,0:13:25.43,0:13:27.36,Default,,0000,0000,0000,,because ir really looks weird.
Dialogue: 0,0:13:27.36,0:13:29.29,Default,,0000,0000,0000,,And then k will be from 1 to n.
Dialogue: 0,0:13:29.29,0:13:30.11,Default,,0000,0000,0000,,So what do you do?
Dialogue: 0,0:13:30.11,0:13:35.71,Default,,0000,0000,0000,,You sum up all these\Nweighted volumes.
Dialogue: 0,0:13:35.71,0:13:37.96,Default,,0000,0000,0000,,This is a weight.
Dialogue: 0,0:13:37.96,0:13:40.50,Default,,0000,0000,0000,,So this is a volume.
Dialogue: 0,0:13:40.50,0:13:41.88,Default,,0000,0000,0000,,All these weighted volumes.
Dialogue: 0,0:13:41.88,0:13:46.79,Default,,0000,0000,0000,,We sum them up for all\Nthe pixels k from 1 to n.
Dialogue: 0,0:13:46.79,0:13:51.14,Default,,0000,0000,0000,,We are going to get\Nsomething like this cover
Dialogue: 0,0:13:51.14,0:13:56.48,Default,,0000,0000,0000,,with tiny parallelepipeds\Nin the limit,
Dialogue: 0,0:13:56.48,0:13:59.31,Default,,0000,0000,0000,,as the partitions' [? norm ?]\Ngo to 0, or the number of pixels
Dialogue: 0,0:13:59.31,0:14:01.13,Default,,0000,0000,0000,,goes to infinity.
Dialogue: 0,0:14:01.13,0:14:06.73,Default,,0000,0000,0000,,This discrete\Nsurface will converge
Dialogue: 0,0:14:06.73,0:14:11.45,Default,,0000,0000,0000,,to the beautiful smooth\Npotato, and give you
Dialogue: 0,0:14:11.45,0:14:12.74,Default,,0000,0000,0000,,a perfect linear image.
Dialogue: 0,0:14:12.74,0:14:16.73,Default,,0000,0000,0000,,Actually, if we saw a\Nhologram, this is what it is.
Dialogue: 0,0:14:16.73,0:14:20.94,Default,,0000,0000,0000,,Our eyes actually\Nsee a bunch of I
Dialogue: 0,0:14:20.94,0:14:25.00,Default,,0000,0000,0000,,tiny-- many, many, many,\Nmany, millions of pixels
Dialogue: 0,0:14:25.00,0:14:27.90,Default,,0000,0000,0000,,that are cubes in 3D.
Dialogue: 0,0:14:27.90,0:14:29.50,Default,,0000,0000,0000,,But it's an optical illusion.
Dialogue: 0,0:14:29.50,0:14:33.01,Default,,0000,0000,0000,,We see, OK, it's a curvilinear,\Nit's a smooth body of a person.
Dialogue: 0,0:14:33.01,0:14:34.98,Default,,0000,0000,0000,,It's not smooth at all.
Dialogue: 0,0:14:34.98,0:14:37.34,Default,,0000,0000,0000,,If you get closer and\Ncloser to that diagram
Dialogue: 0,0:14:37.34,0:14:38.84,Default,,0000,0000,0000,,and put your eye\Nglasses on, you are
Dialogue: 0,0:14:38.84,0:14:41.43,Default,,0000,0000,0000,,going to see, oh, this\Nis not a real person.
Dialogue: 0,0:14:41.43,0:14:44.00,Default,,0000,0000,0000,,It's made of pixels\Nthat are all cubes.
Dialogue: 0,0:14:44.00,0:14:47.58,Default,,0000,0000,0000,,just the same, you see your\Ndigital image of your picture
Dialogue: 0,0:14:47.58,0:14:49.33,Default,,0000,0000,0000,,on Facebook, whatever it is.
Dialogue: 0,0:14:49.33,0:14:52.35,Default,,0000,0000,0000,,If you would be able\Nto be enlarge it,
Dialogue: 0,0:14:52.35,0:14:57.21,Default,,0000,0000,0000,,you would see the pixels, being\Nlittle tiny squares there.
Dialogue: 0,0:14:57.21,0:15:00.38,Default,,0000,0000,0000,,The graphical\Nimaging has improved.
Dialogue: 0,0:15:00.38,0:15:05.10,Default,,0000,0000,0000,,The quality of our digital\Nimaging has improved a lot.
Dialogue: 0,0:15:05.10,0:15:08.78,Default,,0000,0000,0000,,But of course, 20 years ago,\Nwhen you weren't even born,
Dialogue: 0,0:15:08.78,0:15:12.39,Default,,0000,0000,0000,,we could still see the pixels\Nin the photographic images
Dialogue: 0,0:15:12.39,0:15:14.07,Default,,0000,0000,0000,,in a digital camera.
Dialogue: 0,0:15:14.07,0:15:18.98,Default,,0000,0000,0000,,Those tiny first cameras,\Nwhat was that, '98?
Dialogue: 0,0:15:18.98,0:15:19.86,Default,,0000,0000,0000,,STUDENT: Kodak.
Dialogue: 0,0:15:19.86,0:15:23.72,Default,,0000,0000,0000,,MAGDALENA TODA: Like AOL\Ncameras that were so cheap.
Dialogue: 0,0:15:23.72,0:15:30.25,Default,,0000,0000,0000,,The cheaper the camera,\Nthe worse the resolution.
Dialogue: 0,0:15:30.25,0:15:33.03,Default,,0000,0000,0000,,I remember some resolutions\Nlike 400 by 600.
Dialogue: 0,0:15:33.03,0:15:34.07,Default,,0000,0000,0000,,STUDENT: Black and white.
Dialogue: 0,0:15:34.07,0:15:35.00,Default,,0000,0000,0000,,MAGDALENA TODA: Not\Nblack and white.
Dialogue: 0,0:15:35.00,0:15:36.92,Default,,0000,0000,0000,,Black and white\Nwould have been neat.
Dialogue: 0,0:15:36.92,0:15:40.61,Default,,0000,0000,0000,,But really nasty in the sense\Nthat you had the feeling
Dialogue: 0,0:15:40.61,0:15:43.29,Default,,0000,0000,0000,,that the colors\Nwere not even-- they
Dialogue: 0,0:15:43.29,0:15:45.94,Default,,0000,0000,0000,,were blending into each\Nother, because the resolution
Dialogue: 0,0:15:45.94,0:15:47.94,Default,,0000,0000,0000,,was so small.
Dialogue: 0,0:15:47.94,0:15:50.78,Default,,0000,0000,0000,,So it was not at all\Npleasing to the eye.
Dialogue: 0,0:15:50.78,0:15:53.22,Default,,0000,0000,0000,,What was good is that\Nany kind of defects you
Dialogue: 0,0:15:53.22,0:15:56.02,Default,,0000,0000,0000,,would have, something\Nlike a pimple
Dialogue: 0,0:15:56.02,0:15:58.82,Default,,0000,0000,0000,,could not be seen in that,\Nbecause the resolution was
Dialogue: 0,0:15:58.82,0:16:02.34,Default,,0000,0000,0000,,so slow that you couldn't\Nsee the boo-boos,
Dialogue: 0,0:16:02.34,0:16:06.10,Default,,0000,0000,0000,,the pimples, the defects\Nof a face or something.
Dialogue: 0,0:16:06.10,0:16:08.45,Default,,0000,0000,0000,,Now, you can see.
Dialogue: 0,0:16:08.45,0:16:11.62,Default,,0000,0000,0000,,With the digital cameras\Nwe have now, we can do,
Dialogue: 0,0:16:11.62,0:16:14.38,Default,,0000,0000,0000,,of course, Adobe\NPhotoshop, and all of us
Dialogue: 0,0:16:14.38,0:16:18.38,Default,,0000,0000,0000,,will look great if we\Nphotoshop our pictures.
Dialogue: 0,0:16:18.38,0:16:18.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:16:18.91,0:16:22.24,Default,,0000,0000,0000,,So this is what it\Nis in the limit.
Dialogue: 0,0:16:22.24,0:16:26.60,Default,,0000,0000,0000,,But in reality, in\Nthe everyday reality,
Dialogue: 0,0:16:26.60,0:16:30.84,Default,,0000,0000,0000,,you cannot take Riemann\Nsums like that--
Dialogue: 0,0:16:30.84,0:16:33.28,Default,,0000,0000,0000,,this is a Riemann\Napproximating sum--
Dialogue: 0,0:16:33.28,0:16:38.66,Default,,0000,0000,0000,,and then cast to the limit, and\Nget ideal curvilinear domains.
Dialogue: 0,0:16:38.66,0:16:39.23,Default,,0000,0000,0000,,No.
Dialogue: 0,0:16:39.23,0:16:40.78,Default,,0000,0000,0000,,You don't do that.
Dialogue: 0,0:16:40.78,0:16:44.19,Default,,0000,0000,0000,,You have to deal\Nwith the equivalent
Dialogue: 0,0:16:44.19,0:16:45.88,Default,,0000,0000,0000,,of the fundamental\Ntheorem of calculus
Dialogue: 0,0:16:45.88,0:16:51.38,Default,,0000,0000,0000,,from Calc I, which is called the\NFubini-Tonelli type of theorem
Dialogue: 0,0:16:51.38,0:16:53.27,Default,,0000,0000,0000,,in Calc III.
Dialogue: 0,0:16:53.27,0:16:54.33,Default,,0000,0000,0000,,So say it again.
Dialogue: 0,0:16:54.33,0:16:58.08,Default,,0000,0000,0000,,So the Fubini-Tunelli\Ntheorem that you
Dialogue: 0,0:16:58.08,0:17:04.13,Default,,0000,0000,0000,,learned for double\Nintegrals over a rectangle
Dialogue: 0,0:17:04.13,0:17:10.09,Default,,0000,0000,0000,,can be generalized to the\NFubini-Tonelli theorem
Dialogue: 0,0:17:10.09,0:17:12.85,Default,,0000,0000,0000,,over a parallelepiped.
Dialogue: 0,0:17:12.85,0:17:14.96,Default,,0000,0000,0000,,And it's the same\Nthing, practically,
Dialogue: 0,0:17:14.96,0:17:20.45,Default,,0000,0000,0000,,as applying the fundamental\Ntheorem of calculus in Calc I.
Dialogue: 0,0:17:20.45,0:17:25.21,Default,,0000,0000,0000,,So somebody says, well, let me\Nstart with a simple example.
Dialogue: 0,0:17:25.21,0:17:29.36,Default,,0000,0000,0000,,I give you a-- you\Nwill say, Magdalena,
Dialogue: 0,0:17:29.36,0:17:30.23,Default,,0000,0000,0000,,you are offending us.
Dialogue: 0,0:17:30.23,0:17:32.28,Default,,0000,0000,0000,,This is way too easy.
Dialogue: 0,0:17:32.28,0:17:35.35,Default,,0000,0000,0000,,What do you think, that we\Ncannot understand the concept?
Dialogue: 0,0:17:35.35,0:17:39.24,Default,,0000,0000,0000,,I'll just try to start with\Nthe simplest possible example
Dialogue: 0,0:17:39.24,0:17:40.64,Default,,0000,0000,0000,,that I think of.
Dialogue: 0,0:17:40.64,0:17:44.47,Default,,0000,0000,0000,,So x is between a and b.
Dialogue: 0,0:17:44.47,0:17:46.36,Default,,0000,0000,0000,,In my case, they will\Nbe positive numbers,
Dialogue: 0,0:17:46.36,0:17:50.17,Default,,0000,0000,0000,,because I want everything\Nto be in the first octant.
Dialogue: 0,0:17:50.17,0:17:55.71,Default,,0000,0000,0000,,First octant means x positive,\Ny positive, and z positive
Dialogue: 0,0:17:55.71,0:17:58.29,Default,,0000,0000,0000,,all together.
Dialogue: 0,0:17:58.29,0:18:01.87,Default,,0000,0000,0000,,To make my life easier,\NI take that example,
Dialogue: 0,0:18:01.87,0:18:06.03,Default,,0000,0000,0000,,and I say, I know the\Nnumbers for x, y, z.
Dialogue: 0,0:18:06.03,0:18:09.53,Default,,0000,0000,0000,,I would like you to\Ncompute two integrals.
Dialogue: 0,0:18:09.53,0:18:13.56,Default,,0000,0000,0000,,One would be the\Nvolume of this object.
Dialogue: 0,0:18:13.56,0:18:15.00,Default,,0000,0000,0000,,Let's call it body.
Dialogue: 0,0:18:15.00,0:18:19.03,Default,,0000,0000,0000,,It's not a dead body,\Nit's just body in 3D.
Dialogue: 0,0:18:19.03,0:18:22.69,Default,,0000,0000,0000,,The volume of the\Nbody, we say it
Dialogue: 0,0:18:22.69,0:18:27.12,Default,,0000,0000,0000,,like a mathematician, V of B.\NWhat is that by definition?
Dialogue: 0,0:18:27.12,0:18:28.50,Default,,0000,0000,0000,,Who's going to tell me?
Dialogue: 0,0:18:28.50,0:18:30.69,Default,,0000,0000,0000,,Triple snake.
Dialogue: 0,0:18:30.69,0:18:32.94,Default,,0000,0000,0000,,Don't say triple\Nsnake to other people,
Dialogue: 0,0:18:32.94,0:18:36.00,Default,,0000,0000,0000,,because other professors\Nare more orthodox than me.
Dialogue: 0,0:18:36.00,0:18:39.97,Default,,0000,0000,0000,,They will laugh-- they\Nwill not joke about it.
Dialogue: 0,0:18:39.97,0:18:50.89,Default,,0000,0000,0000,,So triple integral over the\Nbody of-- to get the volume,
Dialogue: 0,0:18:50.89,0:18:54.60,Default,,0000,0000,0000,,the weight must be 1.
Dialogue: 0,0:18:54.60,0:18:59.07,Default,,0000,0000,0000,,f integral must be 1, and\Nthen you have exactly dV.
Dialogue: 0,0:18:59.07,0:19:01.18,Default,,0000,0000,0000,,How can I convince\Nyou what we have here,
Dialogue: 0,0:19:01.18,0:19:02.52,Default,,0000,0000,0000,,in terms of Fubini-Tonelli?
Dialogue: 0,0:19:02.52,0:19:04.65,Default,,0000,0000,0000,,It's really beautiful.
Dialogue: 0,0:19:04.65,0:19:11.57,Default,,0000,0000,0000,,B is going to be a, b segment\Ncross product, c, d segment,
Dialogue: 0,0:19:11.57,0:19:12.90,Default,,0000,0000,0000,,cross product.
Dialogue: 0,0:19:12.90,0:19:15.61,Default,,0000,0000,0000,,What is the altitude?
Dialogue: 0,0:19:15.61,0:19:17.37,Default,,0000,0000,0000,,E, f, e to f.
Dialogue: 0,0:19:17.37,0:19:20.38,Default,,0000,0000,0000,,Interval e to f\Nmeans the height.
Dialogue: 0,0:19:20.38,0:19:23.61,Default,,0000,0000,0000,,So length, width, height.
Dialogue: 0,0:19:23.61,0:19:28.50,Default,,0000,0000,0000,,This is the box, or carry-on,\Nor USPS parcel, or whatever
Dialogue: 0,0:19:28.50,0:19:32.41,Default,,0000,0000,0000,,box you want to measure.
Dialogue: 0,0:19:32.41,0:19:37.73,Default,,0000,0000,0000,,So how am I going to set up\Nthe Fubini-Tonelli integral?
Dialogue: 0,0:19:37.73,0:19:44.04,Default,,0000,0000,0000,,a to b, c to d, e to f, 1.
Dialogue: 0,0:19:44.04,0:19:46.10,Default,,0000,0000,0000,,And now, who counts first?
Dialogue: 0,0:19:46.10,0:19:48.26,Default,,0000,0000,0000,,dz, dy, dx.
Dialogue: 0,0:19:48.26,0:19:54.51,Default,,0000,0000,0000,,So it is like the equivalent\Nof the vertical strip
Dialogue: 0,0:19:54.51,0:19:58.57,Default,,0000,0000,0000,,thingy in double corners,\Ndouble integrals.
Dialogue: 0,0:19:58.57,0:19:59.45,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:19:59.45,0:20:01.57,Default,,0000,0000,0000,,STUDENT: Professor,\Nwhy did you use 1 dV?
Dialogue: 0,0:20:01.57,0:20:02.93,Default,,0000,0000,0000,,Why 1?
Dialogue: 0,0:20:02.93,0:20:04.05,Default,,0000,0000,0000,,MAGDALENA TODA: OK.
Dialogue: 0,0:20:04.05,0:20:05.57,Default,,0000,0000,0000,,You'll see in a second.
Dialogue: 0,0:20:05.57,0:20:08.95,Default,,0000,0000,0000,,This is the same\Nthing we do for areas.
Dialogue: 0,0:20:08.95,0:20:14.40,Default,,0000,0000,0000,,So when you compute an\Narea-- very good question.
Dialogue: 0,0:20:14.40,0:20:18.29,Default,,0000,0000,0000,,If you use 1 here, and\Nyou put delta [? ak ?]
Dialogue: 0,0:20:18.29,0:20:21.36,Default,,0000,0000,0000,,that is the graphing paper area.
Dialogue: 0,0:20:21.36,0:20:24.66,Default,,0000,0000,0000,,It's going to be\Nall the tiny areas,
Dialogue: 0,0:20:24.66,0:20:27.70,Default,,0000,0000,0000,,summed up, sum of all\Nthe delta [? ak ?] which
Dialogue: 0,0:20:27.70,0:20:30.86,Default,,0000,0000,0000,,means this little pixel,\Nplus this little pixel,
Dialogue: 0,0:20:30.86,0:20:33.17,Default,,0000,0000,0000,,plus this little pixel,\Nplus this little pixel,
Dialogue: 0,0:20:33.17,0:20:38.36,Default,,0000,0000,0000,,plus 1,000 pixels all together\Nwill cover up the area.
Dialogue: 0,0:20:38.36,0:20:41.71,Default,,0000,0000,0000,,If you have the\Nvolume of a potato,
Dialogue: 0,0:20:41.71,0:20:44.47,Default,,0000,0000,0000,,a body that is alive,\Nbut shouldn't move.
Dialogue: 0,0:20:44.47,0:20:46.87,Default,,0000,0000,0000,,OK, it should stay in one place.
Dialogue: 0,0:20:46.87,0:20:50.14,Default,,0000,0000,0000,,Then, to compute the\Nvolume of the potato,
Dialogue: 0,0:20:50.14,0:20:53.60,Default,,0000,0000,0000,,you have to say, the\Npotato, the smooth potato,
Dialogue: 0,0:20:53.60,0:20:59.28,Default,,0000,0000,0000,,is the limit of the sum of\Nall the tiny cubes of potato,
Dialogue: 0,0:20:59.28,0:21:03.05,Default,,0000,0000,0000,,if you cut the potato in many\Ncubes, like you cut cheese.
Dialogue: 0,0:21:03.05,0:21:06.06,Default,,0000,0000,0000,,They got a bunch of cheddar\Ncheese into small cubes,
Dialogue: 0,0:21:06.06,0:21:10.80,Default,,0000,0000,0000,,and they feed us with crackers\Nand wine-- OK, no comments.
Dialogue: 0,0:21:10.80,0:21:15.35,Default,,0000,0000,0000,,So you have delta vk, you\Nhave 1,000 little cubes,
Dialogue: 0,0:21:15.35,0:21:17.80,Default,,0000,0000,0000,,tiny, tiny, tiny,\Nlike that Lego.
Dialogue: 0,0:21:17.80,0:21:20.16,Default,,0000,0000,0000,,OK, forget about the cheese.
Dialogue: 0,0:21:20.16,0:21:22.75,Default,,0000,0000,0000,,The cheese cubes\Nare way too big.
Dialogue: 0,0:21:22.75,0:21:26.00,Default,,0000,0000,0000,,So imagine Legos that\Nare really performing
Dialogue: 0,0:21:26.00,0:21:28.90,Default,,0000,0000,0000,,with millions of little pieces.
Dialogue: 0,0:21:28.90,0:21:35.98,Default,,0000,0000,0000,,Have you seen the exhibit, Lego\Nexhibit with almost invisible
Dialogue: 0,0:21:35.98,0:21:40.13,Default,,0000,0000,0000,,Legos at the Civic Center?
Dialogue: 0,0:21:40.13,0:21:42.85,Default,,0000,0000,0000,,They have that art festival.
Dialogue: 0,0:21:42.85,0:21:46.08,Default,,0000,0000,0000,,How many of you go\Nto the art festival?
Dialogue: 0,0:21:46.08,0:21:48.84,Default,,0000,0000,0000,,Is it every April?
Dialogue: 0,0:21:48.84,0:21:50.70,Default,,0000,0000,0000,,Something like that.
Dialogue: 0,0:21:50.70,0:21:53.52,Default,,0000,0000,0000,,So imagine those little\Ntiny Legos, but being cubes
Dialogue: 0,0:21:53.52,0:21:54.52,Default,,0000,0000,0000,,and put together.
Dialogue: 0,0:21:54.52,0:21:56.35,Default,,0000,0000,0000,,This is what it is.
Dialogue: 0,0:21:56.35,0:21:58.17,Default,,0000,0000,0000,,So f Vy.
Dialogue: 0,0:21:58.17,0:22:01.56,Default,,0000,0000,0000,,Now, can we verify\Nthe volume of a box?
Dialogue: 0,0:22:01.56,0:22:02.23,Default,,0000,0000,0000,,It's very easy.
Dialogue: 0,0:22:02.23,0:22:03.66,Default,,0000,0000,0000,,What do we do?
Dialogue: 0,0:22:03.66,0:22:08.07,Default,,0000,0000,0000,,Well, first of all, I\Nwould do it in a slow way,
Dialogue: 0,0:22:08.07,0:22:10.27,Default,,0000,0000,0000,,and you are going to\Nshout at me, I know.
Dialogue: 0,0:22:10.27,0:22:13.98,Default,,0000,0000,0000,,But I'll tell you why\Nyou need to bear with me.
Dialogue: 0,0:22:13.98,0:22:16.67,Default,,0000,0000,0000,,So integral of 1 dz goes first.
Dialogue: 0,0:22:16.67,0:22:18.74,Default,,0000,0000,0000,,That's z between f and e.
Dialogue: 0,0:22:18.74,0:22:21.03,Default,,0000,0000,0000,,So it's f minus e, am I right?
Dialogue: 0,0:22:21.03,0:22:22.82,Default,,0000,0000,0000,,You say, duh, that's\Nto easy for me.
Dialogue: 0,0:22:22.82,0:22:25.61,Default,,0000,0000,0000,,I'm know it's too\Neasy for me, but I'm
Dialogue: 0,0:22:25.61,0:22:26.80,Default,,0000,0000,0000,,going somewhere with it.
Dialogue: 0,0:22:26.80,0:22:29.56,Default,,0000,0000,0000,,dy dx.
Dialogue: 0,0:22:29.56,0:22:34.64,Default,,0000,0000,0000,,The one inside, f minus e\Nis a constant, pulls out,
Dialogue: 0,0:22:34.64,0:22:36.86,Default,,0000,0000,0000,,completely out of the product.
Dialogue: 0,0:22:36.86,0:22:42.89,Default,,0000,0000,0000,,And then I have integral from\Na to b of-- what is that left?
Dialogue: 0,0:22:42.89,0:22:46.70,Default,,0000,0000,0000,,1 dy, y between d and c.
Dialogue: 0,0:22:46.70,0:22:48.36,Default,,0000,0000,0000,,So d minus c, right?
Dialogue: 0,0:22:48.36,0:22:52.81,Default,,0000,0000,0000,,d minus c dx.
Dialogue: 0,0:22:52.81,0:22:56.51,Default,,0000,0000,0000,,And so on and so\Nforth, until I get it.
Dialogue: 0,0:22:56.51,0:23:02.45,Default,,0000,0000,0000,,If minus c times d\Nminus c times b minus a,
Dialogue: 0,0:23:02.45,0:23:06.23,Default,,0000,0000,0000,,and goodbye, because this\Nis the volume of the box.
Dialogue: 0,0:23:06.23,0:23:08.01,Default,,0000,0000,0000,,It's the height.
Dialogue: 0,0:23:08.01,0:23:10.31,Default,,0000,0000,0000,,This is the height.
Dialogue: 0,0:23:10.31,0:23:12.84,Default,,0000,0000,0000,,No, excuse me, guys.
Dialogue: 0,0:23:12.84,0:23:15.80,Default,,0000,0000,0000,,The height is-- this\None is the height.
Dialogue: 0,0:23:15.80,0:23:20.61,Default,,0000,0000,0000,,This is the width, and this is\Nthe length, whatever you want.
Dialogue: 0,0:23:20.61,0:23:21.52,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:23:21.52,0:23:25.43,Default,,0000,0000,0000,,How could I have done it if\NI were a little bit smarter?
Dialogue: 0,0:23:25.43,0:23:27.90,Default,,0000,0000,0000,,STUDENT: You could have just\Nput it in three integrals.
Dialogue: 0,0:23:27.90,0:23:29.90,Default,,0000,0000,0000,,MAGDALENA TODA:\NRight Hey, I have
Dialogue: 0,0:23:29.90,0:23:35.83,Default,,0000,0000,0000,,a theorem, just like\Nbefore, which says
Dialogue: 0,0:23:35.83,0:23:37.53,Default,,0000,0000,0000,,three integrals in a product.
Dialogue: 0,0:23:37.53,0:23:40.45,Default,,0000,0000,0000,,This is what Matt\Nimmediately remembered.
Dialogue: 0,0:23:40.45,0:23:43.79,Default,,0000,0000,0000,,We had two integrals\Nin a product last time.
Dialogue: 0,0:23:43.79,0:23:46.66,Default,,0000,0000,0000,,So what have we proved\Nin double integrals
Dialogue: 0,0:23:46.66,0:23:50.02,Default,,0000,0000,0000,,remains valid in\Ntriple integrals
Dialogue: 0,0:23:50.02,0:23:52.83,Default,,0000,0000,0000,,if we have something like that.
Dialogue: 0,0:23:52.83,0:23:54.79,Default,,0000,0000,0000,,So I'm going the same theorem.
Dialogue: 0,0:23:54.79,0:23:55.77,Default,,0000,0000,0000,,It's in the book.
Dialogue: 0,0:23:55.77,0:23:56.75,Default,,0000,0000,0000,,We have a proof.
Dialogue: 0,0:23:56.75,0:24:02.36,Default,,0000,0000,0000,,So you have integral from\Na to b, c to d, e to f.
Dialogue: 0,0:24:02.36,0:24:08.50,Default,,0000,0000,0000,,And then, some guys that you\Nlike, f of x, times g of y,
Dialogue: 0,0:24:08.50,0:24:10.95,Default,,0000,0000,0000,,times h of z.
Dialogue: 0,0:24:10.95,0:24:15.98,Default,,0000,0000,0000,,Functions of x, y, z,\Nseparated variables.
Dialogue: 0,0:24:15.98,0:24:18.65,Default,,0000,0000,0000,,So f, a function of\Nx only, g a function
Dialogue: 0,0:24:18.65,0:24:21.46,Default,,0000,0000,0000,,of y only, h a\Nfunction of z only.
Dialogue: 0,0:24:21.46,0:24:23.02,Default,,0000,0000,0000,,This is the complicated case.
Dialogue: 0,0:24:23.02,0:24:26.25,Default,,0000,0000,0000,,And then I have
Dialogue: 0,0:24:26.25,0:24:27.71,Default,,0000,0000,0000,,STUDENT: dz, dy, dx.
Dialogue: 0,0:24:27.71,0:24:29.04,Default,,0000,0000,0000,,MAGDALENA TODA: dz, dy, dx.
Dialogue: 0,0:24:29.04,0:24:29.69,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:24:29.69,0:24:32.62,Default,,0000,0000,0000,,Thanks for whispering,\Nbecause I was a little bit
Dialogue: 0,0:24:32.62,0:24:35.90,Default,,0000,0000,0000,,confused for a second.
Dialogue: 0,0:24:35.90,0:24:38.82,Default,,0000,0000,0000,,So, just as Matt said,\Ngo ahead and observe
Dialogue: 0,0:24:38.82,0:24:42.44,Default,,0000,0000,0000,,that you can treat them one\Nat a time like you did here,
Dialogue: 0,0:24:42.44,0:24:47.00,Default,,0000,0000,0000,,and integrate one at a time, and\Nintegrate again, and pull out
Dialogue: 0,0:24:47.00,0:24:49.71,Default,,0000,0000,0000,,a constant, integrate\Nagain, pull out a constant.
Dialogue: 0,0:24:49.71,0:24:53.28,Default,,0000,0000,0000,,But practically this is\Nexactly the same as integral
Dialogue: 0,0:24:53.28,0:25:02.19,Default,,0000,0000,0000,,from a to b of f of x\Nalone, dx, times integral
Dialogue: 0,0:25:02.19,0:25:11.61,Default,,0000,0000,0000,,from c to d, g of y alone,\Ndy, and times integral from e
Dialogue: 0,0:25:11.61,0:25:18.18,Default,,0000,0000,0000,,to f of h of z, dz, and close.
Dialogue: 0,0:25:18.18,0:25:22.22,Default,,0000,0000,0000,,So you've seen the version\Nof the double integral,
Dialogue: 0,0:25:22.22,0:25:26.52,Default,,0000,0000,0000,,and this is the same result\Nfor triple integrals.
Dialogue: 0,0:25:26.52,0:25:29.81,Default,,0000,0000,0000,,And it's practically--\Nwhat is the proof?
Dialogue: 0,0:25:29.81,0:25:33.60,Default,,0000,0000,0000,,You just pull out one\Nat a time, so the proof
Dialogue: 0,0:25:33.60,0:25:37.52,Default,,0000,0000,0000,,is that you start working and\Nsay, mister z counts here,
Dialogue: 0,0:25:37.52,0:25:39.80,Default,,0000,0000,0000,,and he's the only\None that counts.
Dialogue: 0,0:25:39.80,0:25:43.54,Default,,0000,0000,0000,,These guys get out for a\Nwalk one at a time outside
Dialogue: 0,0:25:43.54,0:25:46.21,Default,,0000,0000,0000,,of the first integral inside.
Dialogue: 0,0:25:46.21,0:25:48.85,Default,,0000,0000,0000,,And then, integral\Nof h of z, dz,
Dialogue: 0,0:25:48.85,0:25:53.44,Default,,0000,0000,0000,,over the corresponding domain,\Nwill be just a constant, c1,
Dialogue: 0,0:25:53.44,0:25:55.06,Default,,0000,0000,0000,,that pulls out.
Dialogue: 0,0:25:55.06,0:25:59.71,Default,,0000,0000,0000,,And that is that--\Nc1 that pulls out.
Dialogue: 0,0:25:59.71,0:26:02.84,Default,,0000,0000,0000,,Ans since you pull them out\Nin this product one at a time,
Dialogue: 0,0:26:02.84,0:26:04.63,Default,,0000,0000,0000,,that's what you get.
Dialogue: 0,0:26:04.63,0:26:08.72,Default,,0000,0000,0000,,I'm not going to give you this\Nas an exercise in the midterm
Dialogue: 0,0:26:08.72,0:26:12.21,Default,,0000,0000,0000,,with a proof, but this is\None of the first exercises
Dialogue: 0,0:26:12.21,0:26:16.31,Default,,0000,0000,0000,,I had as a freshman\Nin my multi--
Dialogue: 0,0:26:16.31,0:26:20.26,Default,,0000,0000,0000,,I took it as a freshman,\Nas multivariable calculus.
Dialogue: 0,0:26:20.26,0:26:23.22,Default,,0000,0000,0000,,And it was a pop quiz.
Dialogue: 0,0:26:23.22,0:26:26.66,Default,,0000,0000,0000,,My professor just came\None day, and said, guys,
Dialogue: 0,0:26:26.66,0:26:30.12,Default,,0000,0000,0000,,you have to try to do this\N[? before ?] by yourself.
Dialogue: 0,0:26:30.12,0:26:33.10,Default,,0000,0000,0000,,And some of us did,\Nsome of us didn't.
Dialogue: 0,0:26:33.10,0:26:35.09,Default,,0000,0000,0000,,To me, it really\Nlooked very easy.
Dialogue: 0,0:26:35.09,0:26:40.57,Default,,0000,0000,0000,,I was very happy to prove it,\Nin an elementary way, of course.
Dialogue: 0,0:26:40.57,0:26:41.12,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:26:41.12,0:26:45.89,Default,,0000,0000,0000,,So how hard is it\Nto generalize, to go
Dialogue: 0,0:26:45.89,0:26:47.87,Default,,0000,0000,0000,,to non-rectangular domains?
Dialogue: 0,0:26:47.87,0:26:49.35,Default,,0000,0000,0000,,Of course it's a pain.
Dialogue: 0,0:26:49.35,0:26:54.36,Default,,0000,0000,0000,,It's really a pain,\Nlike it was before.
Dialogue: 0,0:26:54.36,0:26:59.83,Default,,0000,0000,0000,,But you will be able to\Nfigure out what's going on.
Dialogue: 0,0:26:59.83,0:27:02.96,Default,,0000,0000,0000,,In most cases,\Nyou're going to have
Dialogue: 0,0:27:02.96,0:27:06.51,Default,,0000,0000,0000,,a domain that's really\Nnot bad, a domain that
Dialogue: 0,0:27:06.51,0:27:09.21,Default,,0000,0000,0000,,has x between fixed values.
Dialogue: 0,0:27:09.21,0:27:14.41,Default,,0000,0000,0000,,For example y between\Nyour favorite guys,
Dialogue: 0,0:27:14.41,0:27:20.02,Default,,0000,0000,0000,,something like f of x and\Ng of x, top and bottom.
Dialogue: 0,0:27:20.02,0:27:22.28,Default,,0000,0000,0000,,That's what you had\Nfor double integral.
Dialogue: 0,0:27:22.28,0:27:26.41,Default,,0000,0000,0000,,Well, in addition,\Nin this case, you
Dialogue: 0,0:27:26.41,0:27:31.26,Default,,0000,0000,0000,,will have z between--\Nlet's make this guy
Dialogue: 0,0:27:31.26,0:27:36.00,Default,,0000,0000,0000,,big F and big G,\Nother functions.
Dialogue: 0,0:27:36.00,0:27:37.92,Default,,0000,0000,0000,,This is going to be\Na function of x, y.
Dialogue: 0,0:27:37.92,0:27:40.70,Default,,0000,0000,0000,,This is going to be\Na function of x, y,
Dialogue: 0,0:27:40.70,0:27:44.03,Default,,0000,0000,0000,,and that's the\Nupper and the lower.
Dialogue: 0,0:27:44.03,0:27:49.87,Default,,0000,0000,0000,,And find the triple integral\Nof, let's say 1 over d dV
Dialogue: 0,0:27:49.87,0:27:52.86,Default,,0000,0000,0000,,will be a volume of the potato.
Dialogue: 0,0:27:52.86,0:27:55.89,Default,,0000,0000,0000,,Now, I'm sick of potatoes,\Nbecause they're not
Dialogue: 0,0:27:55.89,0:27:58.99,Default,,0000,0000,0000,,my favorite food.
Dialogue: 0,0:27:58.99,0:28:02.62,Default,,0000,0000,0000,,Let me imagine I'm making a\Ntetrahedron, a lot of cheese.
Dialogue: 0,0:28:02.62,0:28:08.95,Default,,0000,0000,0000,,
Dialogue: 0,0:28:08.95,0:28:13.78,Default,,0000,0000,0000,,I'm going to draw this same\Ntetrahedron from last time.
Dialogue: 0,0:28:13.78,0:28:15.46,Default,,0000,0000,0000,,So what did we do last time?
Dialogue: 0,0:28:15.46,0:28:17.94,Default,,0000,0000,0000,,We took a plane\Nthat was beautiful,
Dialogue: 0,0:28:17.94,0:28:21.98,Default,,0000,0000,0000,,and we said let's\Ncut with that plane.
Dialogue: 0,0:28:21.98,0:28:24.91,Default,,0000,0000,0000,,This is the plane we are\Ncutting the cheese with.
Dialogue: 0,0:28:24.91,0:28:25.85,Default,,0000,0000,0000,,It's a knife.
Dialogue: 0,0:28:25.85,0:28:28.04,Default,,0000,0000,0000,,x plus y plus z equals 1.
Dialogue: 0,0:28:28.04,0:28:29.83,Default,,0000,0000,0000,,Imagine that there's\Nan infinite knife that
Dialogue: 0,0:28:29.83,0:28:31.51,Default,,0000,0000,0000,,comes into the frame.
Dialogue: 0,0:28:31.51,0:28:33.15,Default,,0000,0000,0000,,Everything is cheese.
Dialogue: 0,0:28:33.15,0:28:37.02,Default,,0000,0000,0000,,The space, the universe is\Ncovered in solid cheese.
Dialogue: 0,0:28:37.02,0:28:42.29,Default,,0000,0000,0000,,So the whole thing,\Nthe Euclidean space
Dialogue: 0,0:28:42.29,0:28:43.80,Default,,0000,0000,0000,,is covered in cheddar cheese.
Dialogue: 0,0:28:43.80,0:28:44.51,Default,,0000,0000,0000,,That's all there.
Dialogue: 0,0:28:44.51,0:28:48.31,Default,,0000,0000,0000,,From everywhere, you\Ncome with this knife,
Dialogue: 0,0:28:48.31,0:28:52.71,Default,,0000,0000,0000,,and you cut along\Nthis plane-- hi
Dialogue: 0,0:28:52.71,0:28:54.96,Default,,0000,0000,0000,,let's call this\N[? high plane. ?]
Dialogue: 0,0:28:54.96,0:28:59.50,Default,,0000,0000,0000,,And then you cut the x\Nplane along the x, y plane,
Dialogue: 0,0:28:59.50,0:29:03.48,Default,,0000,0000,0000,,y, z plane and x, z plane.
Dialogue: 0,0:29:03.48,0:29:04.43,Default,,0000,0000,0000,,What are these called?
Dialogue: 0,0:29:04.43,0:29:06.52,Default,,0000,0000,0000,,Planes of coordinates.
Dialogue: 0,0:29:06.52,0:29:07.48,Default,,0000,0000,0000,,And what do you obtain?
Dialogue: 0,0:29:07.48,0:29:09.23,Default,,0000,0000,0000,,Then, you throw\Neverything away, and you
Dialogue: 0,0:29:09.23,0:29:15.25,Default,,0000,0000,0000,,maintain only the\Ntetrahedron made of cheese.
Dialogue: 0,0:29:15.25,0:29:17.97,Default,,0000,0000,0000,,Now, you remember\Nwhat the corners were.
Dialogue: 0,0:29:17.97,0:29:18.64,Default,,0000,0000,0000,,This is 0, 0, 0.
Dialogue: 0,0:29:18.64,0:29:19.70,Default,,0000,0000,0000,,It's a piece of cake.
Dialogue: 0,0:29:19.70,0:29:22.88,Default,,0000,0000,0000,,But I want to know the vertices.
Dialogue: 0,0:29:22.88,0:29:27.58,Default,,0000,0000,0000,,And you know them, and I\Ndon't want to spend time
Dialogue: 0,0:29:27.58,0:29:30.18,Default,,0000,0000,0000,,discussing why you know them.
Dialogue: 0,0:29:30.18,0:29:30.68,Default,,0000,0000,0000,,So
Dialogue: 0,0:29:30.68,0:29:30.99,Default,,0000,0000,0000,,STUDENT: 0--
Dialogue: 0,0:29:30.99,0:29:31.99,Default,,0000,0000,0000,,MAGDALENA TODA: 1, 0, 0.
Dialogue: 0,0:29:31.99,0:29:33.40,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:29:33.40,0:29:33.90,Default,,0000,0000,0000,,Huh?
Dialogue: 0,0:29:33.90,0:29:34.85,Default,,0000,0000,0000,,STUDENT: 0, 1, 0.
Dialogue: 0,0:29:34.85,0:29:35.68,Default,,0000,0000,0000,,MAGDALENA TODA: Yes.
Dialogue: 0,0:29:35.68,0:29:38.28,Default,,0000,0000,0000,,And 0, 0, 1.
Dialogue: 0,0:29:38.28,0:29:39.64,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:29:39.64,0:29:40.34,Default,,0000,0000,0000,,Great.
Dialogue: 0,0:29:40.34,0:29:44.02,Default,,0000,0000,0000,,The only thing is, if we see\Nthe cheese being a solid,
Dialogue: 0,0:29:44.02,0:29:49.87,Default,,0000,0000,0000,,we don't see this part, the\Nthree axes of corners behind.
Dialogue: 0,0:29:49.87,0:29:55.72,Default,,0000,0000,0000,,so I'm going to make them\Ndotted, and you see the slice,
Dialogue: 0,0:29:55.72,0:29:58.60,Default,,0000,0000,0000,,here, it has to\Nbe really planar.
Dialogue: 0,0:29:58.60,0:30:01.40,Default,,0000,0000,0000,,And you ask yourself,\Nhow do you set up
Dialogue: 0,0:30:01.40,0:30:03.66,Default,,0000,0000,0000,,the triple integral\Nthat represents
Dialogue: 0,0:30:03.66,0:30:05.81,Default,,0000,0000,0000,,the volume of this object?
Dialogue: 0,0:30:05.81,0:30:07.36,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,0:30:07.36,0:30:08.80,Default,,0000,0000,0000,,It shouldn't be hard.
Dialogue: 0,0:30:08.80,0:30:13.08,Default,,0000,0000,0000,,You just have to think what\Nthe domain will be like,
Dialogue: 0,0:30:13.08,0:30:16.74,Default,,0000,0000,0000,,and you say the domain is\Ninside the tetrahedron.
Dialogue: 0,0:30:16.74,0:30:18.23,Default,,0000,0000,0000,,Do you want d or t?
Dialogue: 0,0:30:18.23,0:30:20.94,Default,,0000,0000,0000,,T from tetrahedron.
Dialogue: 0,0:30:20.94,0:30:22.19,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:30:22.19,0:30:23.26,Default,,0000,0000,0000,,We have a new name.
Dialogue: 0,0:30:23.26,0:30:26.65,Default,,0000,0000,0000,,We get bored of all sorts\Nof names and notations.
Dialogue: 0,0:30:26.65,0:30:28.32,Default,,0000,0000,0000,,We change them.
Dialogue: 0,0:30:28.32,0:30:33.75,Default,,0000,0000,0000,,Mathematicians have imagination,\Nso we change our notations.
Dialogue: 0,0:30:33.75,0:30:35.29,Default,,0000,0000,0000,,Like we cannot change\Nour identities,
Dialogue: 0,0:30:35.29,0:30:37.84,Default,,0000,0000,0000,,and we suffer because of that.
Dialogue: 0,0:30:37.84,0:30:40.74,Default,,0000,0000,0000,,So you can be a nerd\Nmathematician imagining
Dialogue: 0,0:30:40.74,0:30:43.92,Default,,0000,0000,0000,,you're Spiderman,\Nand you can take,
Dialogue: 0,0:30:43.92,0:30:47.72,Default,,0000,0000,0000,,give any name you want, and\Nyou can adopt a new name,
Dialogue: 0,0:30:47.72,0:30:51.68,Default,,0000,0000,0000,,and this is behind\Nour motivation
Dialogue: 0,0:30:51.68,0:30:56.32,Default,,0000,0000,0000,,why we like to change names\Nand change notation so much.
Dialogue: 0,0:30:56.32,0:30:56.82,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:30:56.82,0:31:03.25,Default,,0000,0000,0000,,So we have triple integral of\Nthis T. All right, of what?
Dialogue: 0,0:31:03.25,0:31:05.94,Default,,0000,0000,0000,,1 dV.
Dialogue: 0,0:31:05.94,0:31:06.63,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:31:06.63,0:31:09.69,Default,,0000,0000,0000,,Now we understand\Nwhat we need to do,
Dialogue: 0,0:31:09.69,0:31:12.05,Default,,0000,0000,0000,,just like [? Miteish ?]\Nasked me why.
Dialogue: 0,0:31:12.05,0:31:15.60,Default,,0000,0000,0000,,OK, now we know this is going\Nto be a limit of little cubes.
Dialogue: 0,0:31:15.60,0:31:18.15,Default,,0000,0000,0000,,If were to cover\Nthis piece of cheese
Dialogue: 0,0:31:18.15,0:31:21.80,Default,,0000,0000,0000,,in tiny, tiny,\Ninfinitesimally small cubes.
Dialogue: 0,0:31:21.80,0:31:25.34,Default,,0000,0000,0000,,But now we know a\Nmethod to do it.
Dialogue: 0,0:31:25.34,0:31:29.51,Default,,0000,0000,0000,,So according to--\NFubini-Tonelli type of result.
Dialogue: 0,0:31:29.51,0:31:39.14,Default,,0000,0000,0000,,We would have a between--\Nno, x-- is first, dz.
Dialogue: 0,0:31:39.14,0:31:43.22,Default,,0000,0000,0000,,z is first, y is moving\Nnext, x is moving last.
Dialogue: 0,0:31:43.22,0:31:46.75,Default,,0000,0000,0000,,z is constrained to\Nmove between a and b.
Dialogue: 0,0:31:46.75,0:31:50.65,Default,,0000,0000,0000,,But in this case, a and b should\Nbe prescribed by you guys,
Dialogue: 0,0:31:50.65,0:31:55.67,Default,,0000,0000,0000,,because you should think\Nwhere everybody lives.
Dialogue: 0,0:31:55.67,0:31:59.78,Default,,0000,0000,0000,,Not you, I mean the coordinates\Nin their imaginary world.
Dialogue: 0,0:31:59.78,0:32:03.38,Default,,0000,0000,0000,,The coordinates\Nrepresent somebodies.
Dialogue: 0,0:32:03.38,0:32:03.88,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:32:03.88,0:32:07.52,Default,,0000,0000,0000,,MAGDALENA TODA: x, 0 to 1.
Dialogue: 0,0:32:07.52,0:32:10.55,Default,,0000,0000,0000,,How should I give you\Na feeling for that?
Dialogue: 0,0:32:10.55,0:32:11.93,Default,,0000,0000,0000,,Just draw this line.
Dialogue: 0,0:32:11.93,0:32:14.59,Default,,0000,0000,0000,,This red segment between 0 to 1.
Dialogue: 0,0:32:14.59,0:32:18.04,Default,,0000,0000,0000,,That expresses everything\Ninstead of words
Dialogue: 0,0:32:18.04,0:32:22.81,Default,,0000,0000,0000,,into pictures, because every\Npicture is worth 1,000 words.
Dialogue: 0,0:32:22.81,0:32:26.74,Default,,0000,0000,0000,,y is married to\Nx, unfortunately.
Dialogue: 0,0:32:26.74,0:32:30.52,Default,,0000,0000,0000,,y cannot say, oh, I am y,\NI'm going wherever I want.
Dialogue: 0,0:32:30.52,0:32:34.43,Default,,0000,0000,0000,,He hits his head against\Nthis purple line.
Dialogue: 0,0:32:34.43,0:32:36.82,Default,,0000,0000,0000,,He cannot go beyond\Nthat purple line.
Dialogue: 0,0:32:36.82,0:32:39.78,Default,,0000,0000,0000,,He's constrained, poor y.
Dialogue: 0,0:32:39.78,0:32:41.30,Default,,0000,0000,0000,,So he says, I'm moving.
Dialogue: 0,0:32:41.30,0:32:42.20,Default,,0000,0000,0000,,I'm mister y.
Dialogue: 0,0:32:42.20,0:32:46.27,Default,,0000,0000,0000,,I'm moving in this direction,\Nbut I cannot go past the purple
Dialogue: 0,0:32:46.27,0:32:49.17,Default,,0000,0000,0000,,line in plane here.
Dialogue: 0,0:32:49.17,0:32:52.13,Default,,0000,0000,0000,,
Dialogue: 0,0:32:52.13,0:32:56.64,Default,,0000,0000,0000,,I need you, because\Nif you go, I'm lost.
Dialogue: 0,0:32:56.64,0:32:58.74,Default,,0000,0000,0000,,y is between 0 and--
Dialogue: 0,0:32:58.74,0:32:59.65,Default,,0000,0000,0000,,STUDENT: 1 minus x.
Dialogue: 0,0:32:59.65,0:33:00.74,Default,,0000,0000,0000,,MAGDALENA TODA: 1 minus x.
Dialogue: 0,0:33:00.74,0:33:01.82,Default,,0000,0000,0000,,Excellent Roberto.
Dialogue: 0,0:33:01.82,0:33:04.86,Default,,0000,0000,0000,,How did we think about this?
Dialogue: 0,0:33:04.86,0:33:08.33,Default,,0000,0000,0000,,The purple line has\Nequation-- how do you
Dialogue: 0,0:33:08.33,0:33:10.99,Default,,0000,0000,0000,,get to the equation of the\Npurple line, first of all?
Dialogue: 0,0:33:10.99,0:33:15.25,Default,,0000,0000,0000,,In your imagination,\Nyour plug in z equals 0.
Dialogue: 0,0:33:15.25,0:33:19.40,Default,,0000,0000,0000,,So the purple line would\Nbe x plus y equals 1.
Dialogue: 0,0:33:19.40,0:33:24.76,Default,,0000,0000,0000,,And so mister y will\Nbe 1 minus x here.
Dialogue: 0,0:33:24.76,0:33:27.68,Default,,0000,0000,0000,,That's how you got it.
Dialogue: 0,0:33:27.68,0:33:31.96,Default,,0000,0000,0000,,And finally, z is that--\Nmister z foes from the floor
Dialogue: 0,0:33:31.96,0:33:34.19,Default,,0000,0000,0000,,all the way--\Nimagine somebody who
Dialogue: 0,0:33:34.19,0:33:39.76,Default,,0000,0000,0000,,is like-- z is a\Nhelium balloon, and he
Dialogue: 0,0:33:39.76,0:33:42.85,Default,,0000,0000,0000,,is left-- you let him\Ngo from the floor,
Dialogue: 0,0:33:42.85,0:33:44.78,Default,,0000,0000,0000,,and he goes all the\Nway to the ceiling.
Dialogue: 0,0:33:44.78,0:33:48.69,Default,,0000,0000,0000,,And the ceiling is not\Nflat like our ceiling.
Dialogue: 0,0:33:48.69,0:33:55.08,Default,,0000,0000,0000,,The ceiling is\Nthis oblique plane.
Dialogue: 0,0:33:55.08,0:33:59.38,Default,,0000,0000,0000,,So z is going to hit his head\Nagainst the roof at some point,
Dialogue: 0,0:33:59.38,0:34:01.44,Default,,0000,0000,0000,,and he doesn't know\Nwhere he is going
Dialogue: 0,0:34:01.44,0:34:05.87,Default,,0000,0000,0000,,to hit his head, unless you\Ntell him where that happens.
Dialogue: 0,0:34:05.87,0:34:11.25,Default,,0000,0000,0000,,So he knows he leaves at 0,\Nand he's going to end up where?
Dialogue: 0,0:34:11.25,0:34:12.66,Default,,0000,0000,0000,,STUDENT: 1 minus y minus x.
Dialogue: 0,0:34:12.66,0:34:13.74,Default,,0000,0000,0000,,MAGDALENA TODA: Excellent.
Dialogue: 0,0:34:13.74,0:34:15.62,Default,,0000,0000,0000,,1 minus x minus y.
Dialogue: 0,0:34:15.62,0:34:17.62,Default,,0000,0000,0000,,How do we do that?
Dialogue: 0,0:34:17.62,0:34:22.92,Default,,0000,0000,0000,,We pull z out of that, and\Nsay, 1 minus x minus y.
Dialogue: 0,0:34:22.92,0:34:27.31,Default,,0000,0000,0000,,So that is the equation of\Nthe shaded purple plane,
Dialogue: 0,0:34:27.31,0:34:29.52,Default,,0000,0000,0000,,and this is as\Nfar as you can go.
Dialogue: 0,0:34:29.52,0:34:33.55,Default,,0000,0000,0000,,You cannot go past the\Nroof of your house,
Dialogue: 0,0:34:33.55,0:34:38.13,Default,,0000,0000,0000,,which is the purple plane,\Nthe purple shaded plane.
Dialogue: 0,0:34:38.13,0:34:39.84,Default,,0000,0000,0000,,So here you are.
Dialogue: 0,0:34:39.84,0:34:40.48,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:34:40.48,0:34:40.98,Default,,0000,0000,0000,,No.
Dialogue: 0,0:34:40.98,0:34:44.06,Default,,0000,0000,0000,,In many problems on the\Nfinal and on the midterm,
Dialogue: 0,0:34:44.06,0:34:48.20,Default,,0000,0000,0000,,we tell you, don't even\Nthink about solving that,
Dialogue: 0,0:34:48.20,0:34:53.20,Default,,0000,0000,0000,,because we believe you.
Dialogue: 0,0:34:53.20,0:34:56.11,Default,,0000,0000,0000,,Just set up the integral.
Dialogue: 0,0:34:56.11,0:34:58.88,Default,,0000,0000,0000,,I might give you something\Nlike that again, just
Dialogue: 0,0:34:58.88,0:35:04.25,Default,,0000,0000,0000,,set up the integral and\Nyou have to do that.
Dialogue: 0,0:35:04.25,0:35:07.84,Default,,0000,0000,0000,,But now, I would like\Nto actually work it out,
Dialogue: 0,0:35:07.84,0:35:11.69,Default,,0000,0000,0000,,see how hard it is.
Dialogue: 0,0:35:11.69,0:35:14.65,Default,,0000,0000,0000,,So is this hard\Nto work this out?
Dialogue: 0,0:35:14.65,0:35:18.24,Default,,0000,0000,0000,,
Dialogue: 0,0:35:18.24,0:35:20.95,Default,,0000,0000,0000,,I have to do it one at\Na time, because you see,
Dialogue: 0,0:35:20.95,0:35:23.62,Default,,0000,0000,0000,,I don't have fixed endpoints.
Dialogue: 0,0:35:23.62,0:35:28.26,Default,,0000,0000,0000,,I cannot say, I'm applying the\Nproblem with the integral if f
Dialogue: 0,0:35:28.26,0:35:32.42,Default,,0000,0000,0000,,times the integral of g, times--\Nso I have to integrate one
Dialogue: 0,0:35:32.42,0:35:37.80,Default,,0000,0000,0000,,at a time, because I don't\Nhave fixed endpoints.
Dialogue: 0,0:35:37.80,0:35:42.41,Default,,0000,0000,0000,,And the integral of 1dz is\Nz between that and that.
Dialogue: 0,0:35:42.41,0:35:47.45,Default,,0000,0000,0000,,So z, 1 minus x minus y\Nwill be what's left over,
Dialogue: 0,0:35:47.45,0:35:49.71,Default,,0000,0000,0000,,and then I have dy,\Nand then I have dx.
Dialogue: 0,0:35:49.71,0:35:53.75,Default,,0000,0000,0000,,And at this point it\Nlooks horrible enough,
Dialogue: 0,0:35:53.75,0:35:56.22,Default,,0000,0000,0000,,but we have to pray\Nthat in the end
Dialogue: 0,0:35:56.22,0:36:01.60,Default,,0000,0000,0000,,it's not going to be so hard,\Nand I'm going to keep going.
Dialogue: 0,0:36:01.60,0:36:05.42,Default,,0000,0000,0000,,So we have integral from 0 to 1.
Dialogue: 0,0:36:05.42,0:36:12.03,Default,,0000,0000,0000,,We have integral\Nfrom 0 to 1 minus x..
Dialogue: 0,0:36:12.03,0:36:16.69,Default,,0000,0000,0000,,I'll just copy and paste it.
Dialogue: 0,0:36:16.69,0:36:19.64,Default,,0000,0000,0000,,Which is integral from 0 to 1.
Dialogue: 0,0:36:19.64,0:36:24.72,Default,,0000,0000,0000,,Now I have to think,\Nand that's dangerous.
Dialogue: 0,0:36:24.72,0:36:27.44,Default,,0000,0000,0000,,I have 1 minux x\Nwith respect to y.
Dialogue: 0,0:36:27.44,0:36:28.58,Default,,0000,0000,0000,,This is going to be ugly.
Dialogue: 0,0:36:28.58,0:36:34.48,Default,,0000,0000,0000,,That's a constant with\Nrespect to y, and times y,
Dialogue: 0,0:36:34.48,0:36:39.17,Default,,0000,0000,0000,,minus-- integrate with\Nrespect to y, y is [? what? ?]
Dialogue: 0,0:36:39.17,0:36:39.88,Default,,0000,0000,0000,,y squared over 2.
Dialogue: 0,0:36:39.88,0:36:45.62,Default,,0000,0000,0000,,
Dialogue: 0,0:36:45.62,0:36:49.14,Default,,0000,0000,0000,,Between y equals 0 down.
Dialogue: 0,0:36:49.14,0:36:51.93,Default,,0000,0000,0000,,That's going to save my\Nlife, because for y equals 0,
Dialogue: 0,0:36:51.93,0:36:55.44,Default,,0000,0000,0000,,0 is going to be a\Ngreat simplification.
Dialogue: 0,0:36:55.44,0:37:00.06,Default,,0000,0000,0000,,And for y equals\N1 minus x on top,
Dialogue: 0,0:37:00.06,0:37:02.23,Default,,0000,0000,0000,,hopefully it's not going\Nto be the end of the world.
Dialogue: 0,0:37:02.23,0:37:07.46,Default,,0000,0000,0000,,It looks ugly now, but\NI'm an optimistic person,
Dialogue: 0,0:37:07.46,0:37:11.16,Default,,0000,0000,0000,,so I hope that this is\Ngoing to get better.
Dialogue: 0,0:37:11.16,0:37:14.31,Default,,0000,0000,0000,,And I can see it's\Ngoing to get better.
Dialogue: 0,0:37:14.31,0:37:15.73,Default,,0000,0000,0000,,So I have integral\Nfrom here to 1.
Dialogue: 0,0:37:15.73,0:37:18.27,Default,,0000,0000,0000,,And now I say, OK, let me think.
Dialogue: 0,0:37:18.27,0:37:20.33,Default,,0000,0000,0000,,Life is not so bad.
Dialogue: 0,0:37:20.33,0:37:21.45,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:37:21.45,0:37:25.38,Default,,0000,0000,0000,,1 minus x, 1 minus x\Nis 1 minus x squared.
Dialogue: 0,0:37:25.38,0:37:28.30,Default,,0000,0000,0000,,I could think faster, you\Ncould think faster than me,
Dialogue: 0,0:37:28.30,0:37:30.01,Default,,0000,0000,0000,,but I don't want to rush.
Dialogue: 0,0:37:30.01,0:37:36.12,Default,,0000,0000,0000,,1 minus x squared over 2.
Dialogue: 0,0:37:36.12,0:37:37.36,Default,,0000,0000,0000,,So it's not bad at all.
Dialogue: 0,0:37:37.36,0:37:42.99,Default,,0000,0000,0000,,Look, I'm getting this guy who\Nis beautiful in the end, when
Dialogue: 0,0:37:42.99,0:37:45.43,Default,,0000,0000,0000,,I'm going to\Nintegrate, and you have
Dialogue: 0,0:37:45.43,0:37:48.35,Default,,0000,0000,0000,,to keep your fingers\Ncrossed for me,
Dialogue: 0,0:37:48.35,0:37:53.71,Default,,0000,0000,0000,,because I don't know\Nwhat I'm going to get.
Dialogue: 0,0:37:53.71,0:38:03.41,Default,,0000,0000,0000,,So I get integral from 0 to 1,\N1/2 out, 1 minus x squared dx.
Dialogue: 0,0:38:03.41,0:38:05.81,Default,,0000,0000,0000,,Is this bad?
Dialogue: 0,0:38:05.81,0:38:08.34,Default,,0000,0000,0000,,Can you do this by\Nyourself without my help?
Dialogue: 0,0:38:08.34,0:38:11.43,Default,,0000,0000,0000,,What are you going to do?
Dialogue: 0,0:38:11.43,0:38:16.43,Default,,0000,0000,0000,,x squared minus 2x plus 1.
Dialogue: 0,0:38:16.43,0:38:17.47,Default,,0000,0000,0000,,That's the square.
Dialogue: 0,0:38:17.47,0:38:19.37,Default,,0000,0000,0000,,STUDENT: Why not just change it?
Dialogue: 0,0:38:19.37,0:38:20.21,Default,,0000,0000,0000,,MAGDALENA TODA: Huh?
Dialogue: 0,0:38:20.21,0:38:22.36,Default,,0000,0000,0000,,STUDENT: Why not just change it?
Dialogue: 0,0:38:22.36,0:38:24.15,Default,,0000,0000,0000,,MAGDALENA TODA: You\Ncan do it in many ways.
Dialogue: 0,0:38:24.15,0:38:26.31,Default,,0000,0000,0000,,You can do whatever you want.
Dialogue: 0,0:38:26.31,0:38:27.69,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,0:38:27.69,0:38:31.56,Default,,0000,0000,0000,,I want you to the right\Nanswer one way or another.
Dialogue: 0,0:38:31.56,0:38:35.47,Default,,0000,0000,0000,,So I'm going to clean a\Nlittle bit around here.
Dialogue: 0,0:38:35.47,0:38:39.87,Default,,0000,0000,0000,,
Dialogue: 0,0:38:39.87,0:38:41.34,Default,,0000,0000,0000,,It's dirty.
Dialogue: 0,0:38:41.34,0:38:42.32,Default,,0000,0000,0000,,You do it.
Dialogue: 0,0:38:42.32,0:38:46.72,Default,,0000,0000,0000,,You have one minute\Nand a half to finish.
Dialogue: 0,0:38:46.72,0:38:49.65,Default,,0000,0000,0000,,And tell me what you get.
Dialogue: 0,0:38:49.65,0:38:52.39,Default,,0000,0000,0000,,STUDENT: 1 minus x\Ncubed over six negative.
Dialogue: 0,0:38:52.39,0:38:53.40,Default,,0000,0000,0000,,MAGDALENA TODA: No, no.
Dialogue: 0,0:38:53.40,0:38:54.59,Default,,0000,0000,0000,,In the end is the number.
Dialogue: 0,0:38:54.59,0:38:56.89,Default,,0000,0000,0000,,What number?
Dialogue: 0,0:38:56.89,0:38:58.74,Default,,0000,0000,0000,,But you have to go slow.
Dialogue: 0,0:38:58.74,0:39:01.50,Default,,0000,0000,0000,,I need three people to\Ngive me the same answer.
Dialogue: 0,0:39:01.50,0:39:04.16,Default,,0000,0000,0000,,Because then it's like in that\Nproverb, if two people tell
Dialogue: 0,0:39:04.16,0:39:05.77,Default,,0000,0000,0000,,you drunk, you go to bed.
Dialogue: 0,0:39:05.77,0:39:09.92,Default,,0000,0000,0000,,I need three people to tell\Nme what the answer is in order
Dialogue: 0,0:39:09.92,0:39:12.90,Default,,0000,0000,0000,,to believe them.
Dialogue: 0,0:39:12.90,0:39:14.39,Default,,0000,0000,0000,,Three witnesses.
Dialogue: 0,0:39:14.39,0:39:16.19,Default,,0000,0000,0000,,STUDENT: 1 [? by ?] 6.
Dialogue: 0,0:39:16.19,0:39:18.86,Default,,0000,0000,0000,,MAGDALENA TODA: Who got\N1 over 6, raise hand?
Dialogue: 0,0:39:18.86,0:39:20.41,Default,,0000,0000,0000,,Wow, guys, you're fast.
Dialogue: 0,0:39:20.41,0:39:22.78,Default,,0000,0000,0000,,Can you raise hands again?
Dialogue: 0,0:39:22.78,0:39:26.08,Default,,0000,0000,0000,,OK, being fast doesn't\Nmean you're the best,
Dialogue: 0,0:39:26.08,0:39:30.03,Default,,0000,0000,0000,,but I agree you do a very good\Njob, all of you in general.
Dialogue: 0,0:39:30.03,0:39:35.47,Default,,0000,0000,0000,,So I believe there were\Neight people or nine people.
Dialogue: 0,0:39:35.47,0:39:36.50,Default,,0000,0000,0000,,1 over 6.
Dialogue: 0,0:39:36.50,0:39:41.26,Default,,0000,0000,0000,,Now, how could I have cheated\Non this problem on the final?
Dialogue: 0,0:39:41.26,0:39:43.02,Default,,0000,0000,0000,,STUDENT: It's a\N[? junction ?] from this--
Dialogue: 0,0:39:43.02,0:39:43.94,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,0:39:43.94,0:39:48.35,Default,,0000,0000,0000,,In this case, being a volume,\NI would have been lucky enough,
Dialogue: 0,0:39:48.35,0:39:50.88,Default,,0000,0000,0000,,and say, it is the\Nvolume of a tetrahedron.
Dialogue: 0,0:39:50.88,0:39:55.94,Default,,0000,0000,0000,,I go, the tetrahedron\Nhas area of the base 1/2,
Dialogue: 0,0:39:55.94,0:39:57.41,Default,,0000,0000,0000,,the height is 1.
Dialogue: 0,0:39:57.41,0:40:01.11,Default,,0000,0000,0000,,1/2 times 1 divided by 3 is 1/6.
Dialogue: 0,0:40:01.11,0:40:06.15,Default,,0000,0000,0000,,And just pretend on the\Nfinal that I actually
Dialogue: 0,0:40:06.15,0:40:07.11,Default,,0000,0000,0000,,computed everything.
Dialogue: 0,0:40:07.11,0:40:11.53,Default,,0000,0000,0000,,I could have done that, from\Nhere jump to here, or from here
Dialogue: 0,0:40:11.53,0:40:13.17,Default,,0000,0000,0000,,jump straight to here.
Dialogue: 0,0:40:13.17,0:40:15.95,Default,,0000,0000,0000,,And ask you, how did you\Nget from here to here?
Dialogue: 0,0:40:15.95,0:40:18.78,Default,,0000,0000,0000,,And you say, I'm a genius.
Dialogue: 0,0:40:18.78,0:40:20.39,Default,,0000,0000,0000,,Could I not believe you?
Dialogue: 0,0:40:20.39,0:40:22.54,Default,,0000,0000,0000,,I have to give you full credit.
Dialogue: 0,0:40:22.54,0:40:28.31,Default,,0000,0000,0000,,However, what would you\Nhave done if I said compute,
Dialogue: 0,0:40:28.31,0:40:30.99,Default,,0000,0000,0000,,I don't know, something\Nworse, something
Dialogue: 0,0:40:30.99,0:40:37.33,Default,,0000,0000,0000,,like triple integral of x,\Ny, z over the tetrahedron 2.
Dialogue: 0,0:40:37.33,0:40:39.60,Default,,0000,0000,0000,,In that case, you cannot cheat.
Dialogue: 0,0:40:39.60,0:40:42.06,Default,,0000,0000,0000,,You're not lucky\Nenough to cheat.
Dialogue: 0,0:40:42.06,0:40:43.68,Default,,0000,0000,0000,,You're lucky enough\Nto cheat when
Dialogue: 0,0:40:43.68,0:40:46.63,Default,,0000,0000,0000,,you have a volume\Nof a prism, you
Dialogue: 0,0:40:46.63,0:40:49.66,Default,,0000,0000,0000,,have a volume of-- and volume\Nmeans this should be the number
Dialogue: 0,0:40:49.66,0:40:51.87,Default,,0000,0000,0000,,1 here, number 1.
Dialogue: 0,0:40:51.87,0:40:56.19,Default,,0000,0000,0000,,So if you have number 1, here,\Nor I ask you for the volume,
Dialogue: 0,0:40:56.19,0:40:58.61,Default,,0000,0000,0000,,and it's a prism, or\Ntetrahedron, or sphere,
Dialogue: 0,0:40:58.61,0:41:01.97,Default,,0000,0000,0000,,or something, go\Nahead and cheat,
Dialogue: 0,0:41:01.97,0:41:04.30,Default,,0000,0000,0000,,and pretend that you're\Nactually solving the integral.
Dialogue: 0,0:41:04.30,0:41:05.06,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:41:05.06,0:41:07.15,Default,,0000,0000,0000,,STUDENT: What would that\Nrepresent, geometrically,
Dialogue: 0,0:41:07.15,0:41:09.15,Default,,0000,0000,0000,,the triple integral of x, y, z?
Dialogue: 0,0:41:09.15,0:41:12.19,Default,,0000,0000,0000,,MAGDALENA TODA: It's a\Nweighted triple integral.
Dialogue: 0,0:41:12.19,0:41:16.39,Default,,0000,0000,0000,,I'm going to give\Nyou examples later.
Dialogue: 0,0:41:16.39,0:41:20.15,Default,,0000,0000,0000,,When you have mass and momentum,\Nwhen you compute the center
Dialogue: 0,0:41:20.15,0:41:25.14,Default,,0000,0000,0000,,map, or you compute the\Nmass, and somebody give you
Dialogue: 0,0:41:25.14,0:41:26.45,Default,,0000,0000,0000,,densities.
Dialogue: 0,0:41:26.45,0:41:32.84,Default,,0000,0000,0000,,Let me get -- If you have a\Ntriple integral over row at x,
Dialogue: 0,0:41:32.84,0:41:36.64,Default,,0000,0000,0000,,y, z, this could be it,\Nbut I [? recall ?] it row
Dialogue: 0,0:41:36.64,0:41:39.54,Default,,0000,0000,0000,,for a reason, not just for fun.
Dialogue: 0,0:41:39.54,0:41:42.60,Default,,0000,0000,0000,,And here, dx, dy, dz.
Dialogue: 0,0:41:42.60,0:41:45.02,Default,,0000,0000,0000,,Very good question, and\Nit's very insightful.
Dialogue: 0,0:41:45.02,0:41:48.14,Default,,0000,0000,0000,,For a physicist or\Nengineer, the guy
Dialogue: 0,0:41:48.14,0:41:52.03,Default,,0000,0000,0000,,needs to know why we take\Nthis weighted [? integral. ?]
Dialogue: 0,0:41:52.03,0:41:55.39,Default,,0000,0000,0000,,If row is the\Ndensity of an object,
Dialogue: 0,0:41:55.39,0:41:59.39,Default,,0000,0000,0000,,if it's everywhere the same, if\Nrow is a homogeneous density,
Dialogue: 0,0:41:59.39,0:42:02.09,Default,,0000,0000,0000,,for that piece of cheddar\Ncheese-- Oh my God
Dialogue: 0,0:42:02.09,0:42:05.11,Default,,0000,0000,0000,,I'm so hungry-- row\Nwould be constant.
Dialogue: 0,0:42:05.11,0:42:08.29,Default,,0000,0000,0000,,If it's a quality cheddar\Nmade in Vermont in the best
Dialogue: 0,0:42:08.29,0:42:12.62,Default,,0000,0000,0000,,factory, whatever, row would\Nbe considered to be a constant,
Dialogue: 0,0:42:12.62,0:42:13.72,Default,,0000,0000,0000,,right?
Dialogue: 0,0:42:13.72,0:42:15.02,Default,,0000,0000,0000,,And in that case, what happens?
Dialogue: 0,0:42:15.02,0:42:17.83,Default,,0000,0000,0000,,If it's a constant,\Nit's a gets out,
Dialogue: 0,0:42:17.83,0:42:21.00,Default,,0000,0000,0000,,and then you have row\Ntimes triple integral 1
Dialogue: 0,0:42:21.00,0:42:22.97,Default,,0000,0000,0000,,dV, which is what?
Dialogue: 0,0:42:22.97,0:42:25.26,Default,,0000,0000,0000,,The volume.
Dialogue: 0,0:42:25.26,0:42:28.08,Default,,0000,0000,0000,,And then the volume times the\Ndensity of the piece of cheese
Dialogue: 0,0:42:28.08,0:42:28.94,Default,,0000,0000,0000,,will be?
Dialogue: 0,0:42:28.94,0:42:29.93,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:42:29.93,0:42:32.92,Default,,0000,0000,0000,,MAGDALENA TODA: The mass\Nof the piece of cheese,
Dialogue: 0,0:42:32.92,0:42:36.68,Default,,0000,0000,0000,,in kilograms, because I\Nthink in kilograms because I
Dialogue: 0,0:42:36.68,0:42:38.42,Default,,0000,0000,0000,,can eat more.
Dialogue: 0,0:42:38.42,0:42:39.04,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:42:39.04,0:42:42.04,Default,,0000,0000,0000,,Actually, no, I'm just kidding.
Dialogue: 0,0:42:42.04,0:42:46.66,Default,,0000,0000,0000,,You guys have really-- I\Nmean, 2 pounds and 1 kilogram
Dialogue: 0,0:42:46.66,0:42:47.64,Default,,0000,0000,0000,,is not the same thin.
Dialogue: 0,0:42:47.64,0:42:50.08,Default,,0000,0000,0000,,Can somebody tell me why?
Dialogue: 0,0:42:50.08,0:42:52.12,Default,,0000,0000,0000,,I mean, you know it's not\Nthe same thing because,
Dialogue: 0,0:42:52.12,0:42:53.70,Default,,0000,0000,0000,,the approximation.
Dialogue: 0,0:42:53.70,0:42:57.87,Default,,0000,0000,0000,,But I'm claiming you cannot\Ncompare pounds with kilograms
Dialogue: 0,0:42:57.87,0:42:58.46,Default,,0000,0000,0000,,at all.
Dialogue: 0,0:42:58.46,0:43:00.08,Default,,0000,0000,0000,,STUDENT: Pounds is\Na measure of weight,
Dialogue: 0,0:43:00.08,0:43:01.72,Default,,0000,0000,0000,,whereas kilograms is\Na measure of mass.
Dialogue: 0,0:43:01.72,0:43:02.80,Default,,0000,0000,0000,,MAGDALENA TODA: Excellent.
Dialogue: 0,0:43:02.80,0:43:05.22,Default,,0000,0000,0000,,Kilogram is a measure\Nof mass, pound
Dialogue: 0,0:43:05.22,0:43:07.98,Default,,0000,0000,0000,,is a measure of the\Ngravitational force.
Dialogue: 0,0:43:07.98,0:43:11.30,Default,,0000,0000,0000,,It's a force measure.
Dialogue: 0,0:43:11.30,0:43:15.99,Default,,0000,0000,0000,,So OK.
Dialogue: 0,0:43:15.99,0:43:20.18,Default,,0000,0000,0000,,
Dialogue: 0,0:43:20.18,0:43:22.79,Default,,0000,0000,0000,,Which reminds me,\Nthere was-- I don't
Dialogue: 0,0:43:22.79,0:43:24.96,Default,,0000,0000,0000,,know if you saw this short\Nmovie for 15 minutes that
Dialogue: 0,0:43:24.96,0:43:29.21,Default,,0000,0000,0000,,got an award the\Nprevious Oscar last year,
Dialogue: 0,0:43:29.21,0:43:35.30,Default,,0000,0000,0000,,and there was an old lady\Ntelling another old lady
Dialogue: 0,0:43:35.30,0:43:41.97,Default,,0000,0000,0000,,in Great Britain, get\N2 pounds of sausage.
Dialogue: 0,0:43:41.97,0:43:45.13,Default,,0000,0000,0000,,And the other one says,\NI thought we got metric,
Dialogue: 0,0:43:45.13,0:43:47.32,Default,,0000,0000,0000,,because we are in\Nthe European Union.
Dialogue: 0,0:43:47.32,0:43:51.03,Default,,0000,0000,0000,,And she said, then get me\Njust the one meter of sausage,
Dialogue: 0,0:43:51.03,0:43:52.61,Default,,0000,0000,0000,,or something.
Dialogue: 0,0:43:52.61,0:43:55.09,Default,,0000,0000,0000,,So it was funny.
Dialogue: 0,0:43:55.09,0:43:56.45,Default,,0000,0000,0000,,So it can be mass.
Dialogue: 0,0:43:56.45,0:44:00.20,Default,,0000,0000,0000,,But what if this\Ndensity is not the same?
Dialogue: 0,0:44:00.20,0:44:04.43,Default,,0000,0000,0000,,This is exactly why we\Nneed to do the integral.
Dialogue: 0,0:44:04.43,0:44:07.91,Default,,0000,0000,0000,,Imagine that the\Ndensity is-- we have
Dialogue: 0,0:44:07.91,0:44:10.75,Default,,0000,0000,0000,,a piece of cake with layers.
Dialogue: 0,0:44:10.75,0:44:13.12,Default,,0000,0000,0000,,And again, you see\Nhow hungry I am.
Dialogue: 0,0:44:13.12,0:44:18.51,Default,,0000,0000,0000,,So you have a layer, and\Nthen cream, or whipped cream,
Dialogue: 0,0:44:18.51,0:44:22.07,Default,,0000,0000,0000,,or mousse, and another\Nlayer, and another mousse.
Dialogue: 0,0:44:22.07,0:44:25.18,Default,,0000,0000,0000,,The density will vary.
Dialogue: 0,0:44:25.18,0:44:29.35,Default,,0000,0000,0000,,But then there are bodies in\Nphysics where the density is
Dialogue: 0,0:44:29.35,0:44:30.82,Default,,0000,0000,0000,,even a smooth function.
Dialogue: 0,0:44:30.82,0:44:38.47,Default,,0000,0000,0000,,It doesn't matter that you have\Nsuch a discontinuous function.
Dialogue: 0,0:44:38.47,0:44:39.22,Default,,0000,0000,0000,,What would you do?
Dialogue: 0,0:44:39.22,0:44:40.09,Default,,0000,0000,0000,,You just split.
Dialogue: 0,0:44:40.09,0:44:47.38,Default,,0000,0000,0000,,You have triple row 1 for the\Nfirst layer, then triple row 2
Dialogue: 0,0:44:47.38,0:44:50.42,Default,,0000,0000,0000,,for the second later,\Nthe layer of mousse,
Dialogue: 0,0:44:50.42,0:44:52.70,Default,,0000,0000,0000,,and then let's\Nsay it's tiramisu,
Dialogue: 0,0:44:52.70,0:44:57.92,Default,,0000,0000,0000,,you have another layer, row\Nthree, dV3 for the top layer
Dialogue: 0,0:44:57.92,0:44:58.67,Default,,0000,0000,0000,,of the tiramisu.
Dialogue: 0,0:44:58.67,0:45:00.25,Default,,0000,0000,0000,,STUDENT: Can any row\Nbe kept constant?
Dialogue: 0,0:45:00.25,0:45:02.83,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nthese are discontinuous.
Dialogue: 0,0:45:02.83,0:45:04.65,Default,,0000,0000,0000,,They are all constant, though.
Dialogue: 0,0:45:04.65,0:45:06.60,Default,,0000,0000,0000,,That would be the\Ngreat advantage,
Dialogue: 0,0:45:06.60,0:45:11.29,Default,,0000,0000,0000,,because presumably mousse would\Nhave the constant density,
Dialogue: 0,0:45:11.29,0:45:14.85,Default,,0000,0000,0000,,the dough has a constant,\Nhomogeneous density, and so on.
Dialogue: 0,0:45:14.85,0:45:18.62,Default,,0000,0000,0000,,But what if the density\Nvaries in that body from point
Dialogue: 0,0:45:18.62,0:45:19.80,Default,,0000,0000,0000,,to point?
Dialogue: 0,0:45:19.80,0:45:23.06,Default,,0000,0000,0000,,Then nobody can do\Nit by approximation.
Dialogue: 0,0:45:23.06,0:45:26.98,Default,,0000,0000,0000,,You'd say volume, mass 1 plus\Nmass 2 plus mass 3 plus mass 1.
Dialogue: 0,0:45:26.98,0:45:31.25,Default,,0000,0000,0000,,You have to have a triple\Nintegral where this row varies,
Dialogue: 0,0:45:31.25,0:45:33.28,Default,,0000,0000,0000,,constantly varies.
Dialogue: 0,0:45:33.28,0:45:35.41,Default,,0000,0000,0000,,And for an engineer,\Nthat would be a puzzle.
Dialogue: 0,0:45:35.41,0:45:38.56,Default,,0000,0000,0000,,Poor engineers says,\Noh my God, the density
Dialogue: 0,0:45:38.56,0:45:40.19,Default,,0000,0000,0000,,is different from\None point to another.
Dialogue: 0,0:45:40.19,0:45:42.95,Default,,0000,0000,0000,,I have to find an\Napproximated function
Dialogue: 0,0:45:42.95,0:45:48.04,Default,,0000,0000,0000,,for that density moving from one\Npoint to another on that body.
Dialogue: 0,0:45:48.04,0:45:55.06,Default,,0000,0000,0000,,And then the only way to do it\Nwould be to solve an integral.
Dialogue: 0,0:45:55.06,0:45:57.44,Default,,0000,0000,0000,,Imagine that somebody--\Nnow it just occurred,
Dialogue: 0,0:45:57.44,0:46:01.71,Default,,0000,0000,0000,,I never thought\Nabout it-- we would
Dialogue: 0,0:46:01.71,0:46:05.34,Default,,0000,0000,0000,,be measured in terms of\Nthis type of integral.
Dialogue: 0,0:46:05.34,0:46:10.84,Default,,0000,0000,0000,,Of course, people would be able\Nto measure mass right away.
Dialogue: 0,0:46:10.84,0:46:13.39,Default,,0000,0000,0000,,But then, if you were\Nto know the density--
Dialogue: 0,0:46:13.39,0:46:17.67,Default,,0000,0000,0000,,you cannot even know the density\Nat every point of the body.
Dialogue: 0,0:46:17.67,0:46:21.47,Default,,0000,0000,0000,,It varies a lot, so\Nevery point of our bodies
Dialogue: 0,0:46:21.47,0:46:25.93,Default,,0000,0000,0000,,has a different\Nmaterial and a density.
Dialogue: 0,0:46:25.93,0:46:27.11,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:46:27.11,0:46:28.36,Default,,0000,0000,0000,,STUDENT: Tiramisu. [INAUDIBLE]
Dialogue: 0,0:46:28.36,0:46:31.34,Default,,0000,0000,0000,,
Dialogue: 0,0:46:31.34,0:46:32.17,Default,,0000,0000,0000,,MAGDALENA TODA: Huh?
Dialogue: 0,0:46:32.17,0:46:33.12,Default,,0000,0000,0000,,STUDENT: So you\Nuse the tiramasu,
Dialogue: 0,0:46:33.12,0:46:34.08,Default,,0000,0000,0000,,you're making me hungry.
Dialogue: 0,0:46:34.08,0:46:35.50,Default,,0000,0000,0000,,MAGDALENA TODA:\NYeah, because now,
Dialogue: 0,0:46:35.50,0:46:38.02,Default,,0000,0000,0000,,OK take your mind\Noff the tiramisu.
Dialogue: 0,0:46:38.02,0:46:40.25,Default,,0000,0000,0000,,Think about an exam.
Dialogue: 0,0:46:40.25,0:46:42.21,Default,,0000,0000,0000,,Then you don't--
Dialogue: 0,0:46:42.21,0:46:45.14,Default,,0000,0000,0000,,STUDENT: Now I'm sick.
Dialogue: 0,0:46:45.14,0:46:46.14,Default,,0000,0000,0000,,MAGDALENA TODA: Exactly.
Dialogue: 0,0:46:46.14,0:46:49.78,Default,,0000,0000,0000,,Now you need something\Nagainst nausea.
Dialogue: 0,0:46:49.78,0:46:53.53,Default,,0000,0000,0000,,Let's see what else\Nis interesting to do.
Dialogue: 0,0:46:53.53,0:46:57.76,Default,,0000,0000,0000,,
Dialogue: 0,0:46:57.76,0:47:00.43,Default,,0000,0000,0000,,I'll give you ten minutes.
Dialogue: 0,0:47:00.43,0:47:01.72,Default,,0000,0000,0000,,How much did I steal from you?
Dialogue: 0,0:47:01.72,0:47:07.22,Default,,0000,0000,0000,,I stole constantly about\Nfive minutes of your breaks
Dialogue: 0,0:47:07.22,0:47:09.96,Default,,0000,0000,0000,,for the last few Tuesdays.
Dialogue: 0,0:47:09.96,0:47:12.34,Default,,0000,0000,0000,,STUDENT: So the integral--
Dialogue: 0,0:47:12.34,0:47:14.02,Default,,0000,0000,0000,,MAGDALENA TODA: The\Nintegral of that.
Dialogue: 0,0:47:14.02,0:47:18.82,Default,,0000,0000,0000,,I think I would be fair\Nto give you 10 minutes
Dialogue: 0,0:47:18.82,0:47:22.30,Default,,0000,0000,0000,,as a gift today to compensate.
Dialogue: 0,0:47:22.30,0:47:28.27,Default,,0000,0000,0000,,OK, so remind me to let\Nyou go 10 minutes early.
Dialogue: 0,0:47:28.27,0:47:32.26,Default,,0000,0000,0000,,Especially since\Nspring break is coming.
Dialogue: 0,0:47:32.26,0:47:37.24,Default,,0000,0000,0000,,We have a 3D application.
Dialogue: 0,0:47:37.24,0:47:40.22,Default,,0000,0000,0000,,We have several 3D applications.
Dialogue: 0,0:47:40.22,0:47:44.12,Default,,0000,0000,0000,,Let me see which one\NI want to mimic first.
Dialogue: 0,0:47:44.12,0:47:49.29,Default,,0000,0000,0000,,
Dialogue: 0,0:47:49.29,0:47:51.37,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:47:51.37,0:47:56.48,Default,,0000,0000,0000,,I'm going to pick my favorite,\Nbecause I just want to.
Dialogue: 0,0:47:56.48,0:48:02.40,Default,,0000,0000,0000,,
Dialogue: 0,0:48:02.40,0:48:16.25,Default,,0000,0000,0000,,So imagine you\Nhave a disc that is
Dialogue: 0,0:48:16.25,0:48:19.20,Default,,0000,0000,0000,,x squared plus y squared\Nequals 1 would be the circle.
Dialogue: 0,0:48:19.20,0:48:21.65,Default,,0000,0000,0000,,That's the unit\Ndisc on the floor.
Dialogue: 0,0:48:21.65,0:48:27.39,Default,,0000,0000,0000,,
Dialogue: 0,0:48:27.39,0:48:38.08,Default,,0000,0000,0000,,And then I have the plane\Nx plus y plus z equals 8.
Dialogue: 0,0:48:38.08,0:48:40.06,Default,,0000,0000,0000,,Then I'm going to\Ndraw that plane.
Dialogue: 0,0:48:40.06,0:48:41.13,Default,,0000,0000,0000,,I'll try my best.
Dialogue: 0,0:48:41.13,0:48:48.96,Default,,0000,0000,0000,,
Dialogue: 0,0:48:48.96,0:48:50.95,Default,,0000,0000,0000,,It's similar to two\Nexamples from the book,
Dialogue: 0,0:48:50.95,0:48:53.79,Default,,0000,0000,0000,,but I did not want to\Nrepeat the ones in the book
Dialogue: 0,0:48:53.79,0:48:57.07,Default,,0000,0000,0000,,because I want you to\Nactually read them.
Dialogue: 0,0:48:57.07,0:48:59.00,Default,,0000,0000,0000,,That's kind of the idea.
Dialogue: 0,0:48:59.00,0:49:05.84,Default,,0000,0000,0000,,So you have this\Npicture, and you
Dialogue: 0,0:49:05.84,0:49:10.67,Default,,0000,0000,0000,,realize that we had that\Nin the first octant before.
Dialogue: 0,0:49:10.67,0:49:15.65,Default,,0000,0000,0000,,So I say, I don't\Nwant the volume
Dialogue: 0,0:49:15.65,0:49:21.00,Default,,0000,0000,0000,,of the body over the whole disc,\Nonly over the part of the disc
Dialogue: 0,0:49:21.00,0:49:23.83,Default,,0000,0000,0000,,which is in the first octant.
Dialogue: 0,0:49:23.83,0:49:33.06,Default,,0000,0000,0000,,So I say, I want this domain\ND, which is going to be what?
Dialogue: 0,0:49:33.06,0:49:35.59,Default,,0000,0000,0000,,x squared plus y squared\Nless than or equal to 1
Dialogue: 0,0:49:35.59,0:49:39.97,Default,,0000,0000,0000,,in plane, with x\Npositive, y positive.
Dialogue: 0,0:49:39.97,0:49:42.74,Default,,0000,0000,0000,,Do you know what we call\Nthat in trigonometry?
Dialogue: 0,0:49:42.74,0:49:46.84,Default,,0000,0000,0000,,
Dialogue: 0,0:49:46.84,0:49:50.35,Default,,0000,0000,0000,,Does anybody know what we\Ncall this in trigonometry?
Dialogue: 0,0:49:50.35,0:49:57.71,Default,,0000,0000,0000,,
Dialogue: 0,0:49:57.71,0:50:00.39,Default,,0000,0000,0000,,Let me put the points\Nwhile you think.
Dialogue: 0,0:50:00.39,0:50:03.14,Default,,0000,0000,0000,,Hopefully, you are\Nthinking about this.
Dialogue: 0,0:50:03.14,0:50:07.00,Default,,0000,0000,0000,,This is 1 in x-axis.
Dialogue: 0,0:50:07.00,0:50:13.91,Default,,0000,0000,0000,,1, 0, 0, and this is\N0, 1, 0, and this is y.
Dialogue: 0,0:50:13.91,0:50:18.08,Default,,0000,0000,0000,,If I were to go up\Nuntil I meet the plane,
Dialogue: 0,0:50:18.08,0:50:20.07,Default,,0000,0000,0000,,what point would this--
Dialogue: 0,0:50:20.07,0:50:22.26,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:50:22.26,0:50:27.39,Default,,0000,0000,0000,,MAGDALENA TODA: What point\Nwould this-- on the thing.
Dialogue: 0,0:50:27.39,0:50:30.24,Default,,0000,0000,0000,,
Dialogue: 0,0:50:30.24,0:50:34.92,Default,,0000,0000,0000,,STUDENT: 1, 0, 7\Nand then 0, 1, 7.
Dialogue: 0,0:50:34.92,0:50:36.09,Default,,0000,0000,0000,,MAGDALENA TODA: 1, 0, 7.
Dialogue: 0,0:50:36.09,0:50:41.03,Default,,0000,0000,0000,,
Dialogue: 0,0:50:41.03,0:50:46.96,Default,,0000,0000,0000,,This would be you said 0, 1, 6.
Dialogue: 0,0:50:46.96,0:50:49.08,Default,,0000,0000,0000,,And this would be 1, 0, 7.
Dialogue: 0,0:50:49.08,0:50:50.80,Default,,0000,0000,0000,,How did you think about this?
Dialogue: 0,0:50:50.80,0:50:52.29,Default,,0000,0000,0000,,How do you know?
Dialogue: 0,0:50:52.29,0:50:53.08,Default,,0000,0000,0000,,STUDENT: Y plus z--
Dialogue: 0,0:50:53.08,0:50:57.16,Default,,0000,0000,0000,,MAGDALENA TODA: Because z,\Nbecause it's on the y-axis,
Dialogue: 0,0:50:57.16,0:51:01.02,Default,,0000,0000,0000,,and since you are on the\Nx-axis here, y has to be 0.
Dialogue: 0,0:51:01.02,0:51:01.72,Default,,0000,0000,0000,,So you're right.
Dialogue: 0,0:51:01.72,0:51:02.22,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:51:02.22,0:51:03.27,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:51:03.27,0:51:11.04,Default,,0000,0000,0000,,Now I'm going to\Nsay, I'd like to know
Dialogue: 0,0:51:11.04,0:51:34.38,Default,,0000,0000,0000,,the-- compute the volume of\Nthe body that is bounded above
Dialogue: 0,0:51:34.38,0:51:52.37,Default,,0000,0000,0000,,from above by x plus y plus\Nz equals 8, who's projection
Dialogue: 0,0:51:52.37,0:52:00.87,Default,,0000,0000,0000,,on the floor is the\Ndomain D. And I'll say
Dialogue: 0,0:52:00.87,0:52:02.33,Default,,0000,0000,0000,,volume of the cylindrical body.
Dialogue: 0,0:52:02.33,0:52:10.35,Default,,0000,0000,0000,,
Dialogue: 0,0:52:10.35,0:52:15.02,Default,,0000,0000,0000,,So how could you obtain\Nsuch a, again-- No,
Dialogue: 0,0:52:15.02,0:52:17.46,Default,,0000,0000,0000,,this is Murphy's Law.
Dialogue: 0,0:52:17.46,0:52:25.01,Default,,0000,0000,0000,,OK, how could you obtain such\Nan object, such a cylinder?
Dialogue: 0,0:52:25.01,0:52:27.01,Default,,0000,0000,0000,,STUDENT: Take a pencil,\Nand cut it into fourths.
Dialogue: 0,0:52:27.01,0:52:27.84,Default,,0000,0000,0000,,MAGDALENA TODA: Huh?
Dialogue: 0,0:52:27.84,0:52:31.10,Default,,0000,0000,0000,,STUDENT: Take like a cylindrical\Npencil and cut it into fourths.
Dialogue: 0,0:52:31.10,0:52:33.91,Default,,0000,0000,0000,,MAGDALENA TODA: Take a\Nsalami, a piece of salami.
Dialogue: 0,0:52:33.91,0:52:39.70,Default,,0000,0000,0000,,Cut that piece of salami into\Nfour, into four quarters.
Dialogue: 0,0:52:39.70,0:52:43.87,Default,,0000,0000,0000,,
Dialogue: 0,0:52:43.87,0:52:50.44,Default,,0000,0000,0000,,And then we take, we slice,\Nand we slice like that.
Dialogue: 0,0:52:50.44,0:52:51.64,Default,,0000,0000,0000,,So we have something like--
Dialogue: 0,0:52:51.64,0:52:53.68,Default,,0000,0000,0000,,STUDENT: I tried to think\Nof a non- food example.
Dialogue: 0,0:52:53.68,0:52:55.72,Default,,0000,0000,0000,,MAGDALENA TODA: --a quarter.
Dialogue: 0,0:52:55.72,0:52:59.48,Default,,0000,0000,0000,,How can I draw this?
Dialogue: 0,0:52:59.48,0:53:01.36,Default,,0000,0000,0000,,OK, this is what it means.
Dialogue: 0,0:53:01.36,0:53:03.46,Default,,0000,0000,0000,,You don't see this one.
Dialogue: 0,0:53:03.46,0:53:04.86,Default,,0000,0000,0000,,You don't see this part.
Dialogue: 0,0:53:04.86,0:53:06.13,Default,,0000,0000,0000,,You don't see this part.
Dialogue: 0,0:53:06.13,0:53:07.09,Default,,0000,0000,0000,,This is curved.
Dialogue: 0,0:53:07.09,0:53:10.61,Default,,0000,0000,0000,,And here, instead of cutting\Nwith another perpendicular
Dialogue: 0,0:53:10.61,0:53:13.14,Default,,0000,0000,0000,,plane, along the\Nsalami-- so this
Dialogue: 0,0:53:13.14,0:53:17.50,Default,,0000,0000,0000,,is the axis of the salami--\Ninstead of taking the knife
Dialogue: 0,0:53:17.50,0:53:20.82,Default,,0000,0000,0000,,and cutting like that, I'm\Ncutting an oblique plane,
Dialogue: 0,0:53:20.82,0:53:26.48,Default,,0000,0000,0000,,and this is what this\Noblique plane will do.
Dialogue: 0,0:53:26.48,0:53:28.14,Default,,0000,0000,0000,,STUDENT: If you cut\Nthat way, then you
Dialogue: 0,0:53:28.14,0:53:30.77,Default,,0000,0000,0000,,would have only squares.
Dialogue: 0,0:53:30.77,0:53:31.73,Default,,0000,0000,0000,,MAGDALENA TODA: Hmm?
Dialogue: 0,0:53:31.73,0:53:42.54,Default,,0000,0000,0000,,So I'm going to have some\Noblique-- I cannot draw better.
Dialogue: 0,0:53:42.54,0:53:45.44,Default,,0000,0000,0000,,I don't know how to draw better.
Dialogue: 0,0:53:45.44,0:53:47.48,Default,,0000,0000,0000,,So it's going to be an\Noblique cut in the salami.
Dialogue: 0,0:53:47.48,0:53:50.25,Default,,0000,0000,0000,,
Dialogue: 0,0:53:50.25,0:53:53.15,Default,,0000,0000,0000,,Let's think how we\Ndo this problem.
Dialogue: 0,0:53:53.15,0:53:55.14,Default,,0000,0000,0000,,Elementary, it will\Nbe a piece of cake--
Dialogue: 0,0:53:55.14,0:53:57.72,Default,,0000,0000,0000,,it would be a piece of--
Dialogue: 0,0:53:57.72,0:53:58.84,Default,,0000,0000,0000,,STUDENT: A piece of salami.
Dialogue: 0,0:53:58.84,0:53:59.63,Default,,0000,0000,0000,,MAGDALENA TODA: No.
Dialogue: 0,0:53:59.63,0:54:01.20,Default,,0000,0000,0000,,It wouldn't be apiece of salami.
Dialogue: 0,0:54:01.20,0:54:02.86,Default,,0000,0000,0000,,STUDENT: It could be done.
Dialogue: 0,0:54:02.86,0:54:05.68,Default,,0000,0000,0000,,MAGDALENA TODA: How could we do\Nthat quickly with the Calculus
Dialogue: 0,0:54:05.68,0:54:07.29,Default,,0000,0000,0000,,III we know?
Dialogue: 0,0:54:07.29,0:54:09.20,Default,,0000,0000,0000,,STUDENT: Find the\Ntriple integral.
Dialogue: 0,0:54:09.20,0:54:11.60,Default,,0000,0000,0000,,Oh, you want us to do\Nthe double integral?
Dialogue: 0,0:54:11.60,0:54:14.45,Default,,0000,0000,0000,,MAGDALENA TODA: Double, triple,\NI don't know what to do.
Dialogue: 0,0:54:14.45,0:54:16.00,Default,,0000,0000,0000,,What do you think is best?
Dialogue: 0,0:54:16.00,0:54:17.70,Default,,0000,0000,0000,,Let's do that triple\Nintegral first,
Dialogue: 0,0:54:17.70,0:54:20.72,Default,,0000,0000,0000,,and you'll see that it's the\Nsame thing as double integral.
Dialogue: 0,0:54:20.72,0:54:31.29,Default,,0000,0000,0000,,Triple integral over B, the\Nbody of the salami, 1 dV.
Dialogue: 0,0:54:31.29,0:54:34.28,Default,,0000,0000,0000,,How can we set it up?
Dialogue: 0,0:54:34.28,0:54:37.09,Default,,0000,0000,0000,,Well, this is a\Nlittle bit tricky.
Dialogue: 0,0:54:37.09,0:54:38.90,Default,,0000,0000,0000,,It's going to be like that.
Dialogue: 0,0:54:38.90,0:54:42.15,Default,,0000,0000,0000,,
Dialogue: 0,0:54:42.15,0:54:45.42,Default,,0000,0000,0000,,We can say, I have a double\Nintegral over my domain,
Dialogue: 0,0:54:45.42,0:54:51.57,Default,,0000,0000,0000,,D. When it comes to the z,\Nmister z has to be first.
Dialogue: 0,0:54:51.57,0:54:54.50,Default,,0000,0000,0000,,So mister z says, I'm first.
Dialogue: 0,0:54:54.50,0:54:56.50,Default,,0000,0000,0000,,I know where I'm going.
Dialogue: 0,0:54:56.50,0:54:59.63,Default,,0000,0000,0000,,You guys, x and y\Nare bound together,
Dialogue: 0,0:54:59.63,0:55:02.73,Default,,0000,0000,0000,,mired in the element\Nof area of the circles.
Dialogue: 0,0:55:02.73,0:55:06.35,Default,,0000,0000,0000,,This is like dx dy.
Dialogue: 0,0:55:06.35,0:55:07.65,Default,,0000,0000,0000,,But I am independent from you.
Dialogue: 0,0:55:07.65,0:55:08.85,Default,,0000,0000,0000,,I am z.
Dialogue: 0,0:55:08.85,0:55:12.96,Default,,0000,0000,0000,,So I'm going all the way\Nfrom the floor to what?
Dialogue: 0,0:55:12.96,0:55:14.66,Default,,0000,0000,0000,,You taught me that.
Dialogue: 0,0:55:14.66,0:55:20.25,Default,,0000,0000,0000,,8 minus x minus y, and 1.
Dialogue: 0,0:55:20.25,0:55:22.40,Default,,0000,0000,0000,,This is the way to do\Nit as a triple integral,
Dialogue: 0,0:55:22.40,0:55:24.56,Default,,0000,0000,0000,,but then Alex will\Nsay, I could have
Dialogue: 0,0:55:24.56,0:55:26.35,Default,,0000,0000,0000,,done this as a double integral.
Dialogue: 0,0:55:26.35,0:55:28.51,Default,,0000,0000,0000,,Let me show you how.
Dialogue: 0,0:55:28.51,0:55:32.71,Default,,0000,0000,0000,,I could have done it over\Nthe domain D in plane.
Dialogue: 0,0:55:32.71,0:55:35.84,Default,,0000,0000,0000,,Put the function,\N8 minus x minus y
Dialogue: 0,0:55:35.84,0:55:38.25,Default,,0000,0000,0000,,is [? B and ?] z from\Nthe very beginning,
Dialogue: 0,0:55:38.25,0:55:42.64,Default,,0000,0000,0000,,because that's my altitude\Nfunction, f of x and y.
Dialogue: 0,0:55:42.64,0:55:47.08,Default,,0000,0000,0000,,So then I say dx dy, dx\Ndy, it doesn't matter.
Dialogue: 0,0:55:47.08,0:55:48.50,Default,,0000,0000,0000,,That's the only theory element.
Dialogue: 0,0:55:48.50,0:55:48.100,Default,,0000,0000,0000,,Fine.
Dialogue: 0,0:55:48.100,0:55:51.24,Default,,0000,0000,0000,,It's the same thing.
Dialogue: 0,0:55:51.24,0:55:53.58,Default,,0000,0000,0000,,This is what I wanted\Nyou to observe.
Dialogue: 0,0:55:53.58,0:55:55.87,Default,,0000,0000,0000,,Whether you view it like the\Ntriple integral like that,
Dialogue: 0,0:55:55.87,0:55:58.41,Default,,0000,0000,0000,,or you view it as the\Ndouble integral like that,
Dialogue: 0,0:55:58.41,0:56:01.81,Default,,0000,0000,0000,,it's the same thing.
Dialogue: 0,0:56:01.81,0:56:03.26,Default,,0000,0000,0000,,This is not a headache.
Dialogue: 0,0:56:03.26,0:56:06.44,Default,,0000,0000,0000,,The headache is coming next.
Dialogue: 0,0:56:06.44,0:56:07.84,Default,,0000,0000,0000,,This is not a headache.
Dialogue: 0,0:56:07.84,0:56:11.08,Default,,0000,0000,0000,,So you can do it in two ways.
Dialogue: 0,0:56:11.08,0:56:13.79,Default,,0000,0000,0000,,And I'd like to\Nlook at the-- check
Dialogue: 0,0:56:13.79,0:56:19.67,Default,,0000,0000,0000,,the two methods of doing this.
Dialogue: 0,0:56:19.67,0:56:23.59,Default,,0000,0000,0000,,
Dialogue: 0,0:56:23.59,0:56:27.41,Default,,0000,0000,0000,,And set up the integrals\Nwithout solving them.
Dialogue: 0,0:56:27.41,0:56:37.52,Default,,0000,0000,0000,,
Dialogue: 0,0:56:37.52,0:56:38.64,Default,,0000,0000,0000,,Can you read my mind?
Dialogue: 0,0:56:38.64,0:56:41.96,Default,,0000,0000,0000,,Do you realize what I'm asking?
Dialogue: 0,0:56:41.96,0:56:43.50,Default,,0000,0000,0000,,Imagine that would\Nbe on the midterm.
Dialogue: 0,0:56:43.50,0:56:46.52,Default,,0000,0000,0000,,What do you think I'm\Nasking, the two methods?
Dialogue: 0,0:56:46.52,0:56:49.25,Default,,0000,0000,0000,,This can be interpreted\Nin many ways.
Dialogue: 0,0:56:49.25,0:56:50.25,Default,,0000,0000,0000,,There are two methods.
Dialogue: 0,0:56:50.25,0:56:53.67,Default,,0000,0000,0000,,I mean, one method by\Ndoing it with Cartesian
Dialogue: 0,0:56:53.67,0:56:56.30,Default,,0000,0000,0000,,coordinates x and y.
Dialogue: 0,0:56:56.30,0:56:58.42,Default,,0000,0000,0000,,The other method is switching\Nto polar coordinates
Dialogue: 0,0:56:58.42,0:57:01.42,Default,,0000,0000,0000,,and set up the integral\Nwithout solving.
Dialogue: 0,0:57:01.42,0:57:03.79,Default,,0000,0000,0000,,And you say, why not solving?
Dialogue: 0,0:57:03.79,0:57:05.45,Default,,0000,0000,0000,,Because I'm going to cheat.
Dialogue: 0,0:57:05.45,0:57:07.63,Default,,0000,0000,0000,,I'm going to use a\NTI-92 to solve it,
Dialogue: 0,0:57:07.63,0:57:11.24,Default,,0000,0000,0000,,or I'm going to use\Na Matlab or Maple.
Dialogue: 0,0:57:11.24,0:57:12.82,Default,,0000,0000,0000,,If it looks a little\Nbit complicated,
Dialogue: 0,0:57:12.82,0:57:15.26,Default,,0000,0000,0000,,then I don't want\Nto spend my time.
Dialogue: 0,0:57:15.26,0:57:18.74,Default,,0000,0000,0000,,Actually, engineers,\Nafter taking Calc III,
Dialogue: 0,0:57:18.74,0:57:19.69,Default,,0000,0000,0000,,they know a lot.
Dialogue: 0,0:57:19.69,0:57:22.57,Default,,0000,0000,0000,,They understand a lot\Nabout volumes, areas.
Dialogue: 0,0:57:22.57,0:57:27.17,Default,,0000,0000,0000,,But do you think if you work on\Na real-life problem like that,
Dialogue: 0,0:57:27.17,0:57:29.17,Default,,0000,0000,0000,,that your boss will\Nlet you waste your time
Dialogue: 0,0:57:29.17,0:57:30.72,Default,,0000,0000,0000,,and do the integral by hand?
Dialogue: 0,0:57:30.72,0:57:31.22,Default,,0000,0000,0000,,STUDENT: No.
Dialogue: 0,0:57:31.22,0:57:33.54,Default,,0000,0000,0000,,MAGDALENA TODA: Most integrals\Nare really complicated
Dialogue: 0,0:57:33.54,0:57:34.70,Default,,0000,0000,0000,,in everyday life.
Dialogue: 0,0:57:34.70,0:57:36.39,Default,,0000,0000,0000,,So what you're\Ngoing to do is going
Dialogue: 0,0:57:36.39,0:57:40.18,Default,,0000,0000,0000,,to be a scientific software,\Nlike Matlab, which is primarily
Dialogue: 0,0:57:40.18,0:57:44.04,Default,,0000,0000,0000,,for engineers, Mathematica,\Nwhich is similar to Matlab,
Dialogue: 0,0:57:44.04,0:57:46.18,Default,,0000,0000,0000,,but is mainly for\Nmathematicians.
Dialogue: 0,0:57:46.18,0:57:48.16,Default,,0000,0000,0000,,It was invented\Nat the University
Dialogue: 0,0:57:48.16,0:57:50.66,Default,,0000,0000,0000,,of Illinois Urbana-Champaign,\Nand they're still
Dialogue: 0,0:57:50.66,0:57:51.79,Default,,0000,0000,0000,,very proud of it.
Dialogue: 0,0:57:51.79,0:57:54.48,Default,,0000,0000,0000,,I prefer Matlab\Nbecause I feel Matlab
Dialogue: 0,0:57:54.48,0:57:58.00,Default,,0000,0000,0000,,is stronger, has higher\Ncapabilities than Mathematica.
Dialogue: 0,0:57:58.00,0:57:59.19,Default,,0000,0000,0000,,You can use Maple.
Dialogue: 0,0:57:59.19,0:58:05.18,Default,,0000,0000,0000,,Maple lets you set up the\Nendpoints even as functions.
Dialogue: 0,0:58:05.18,0:58:08.50,Default,,0000,0000,0000,,And then it's user\Nfriendly, you type in this,
Dialogue: 0,0:58:08.50,0:58:10.40,Default,,0000,0000,0000,,you type in the endpoints.
Dialogue: 0,0:58:10.40,0:58:12.32,Default,,0000,0000,0000,,It has little windows, here.
Dialogue: 0,0:58:12.32,0:58:14.30,Default,,0000,0000,0000,,You don't need to\Nknow any programming.
Dialogue: 0,0:58:14.30,0:58:17.100,Default,,0000,0000,0000,,It's made for people who\Nhave no programming skills.
Dialogue: 0,0:58:17.100,0:58:20.38,Default,,0000,0000,0000,,So it's going to show\Na little window on top,
Dialogue: 0,0:58:20.38,0:58:22.15,Default,,0000,0000,0000,,here, here, here and here.
Dialogue: 0,0:58:22.15,0:58:24.26,Default,,0000,0000,0000,,You [? have ?] those,\Nand you press Enter,
Dialogue: 0,0:58:24.26,0:58:26.75,Default,,0000,0000,0000,,and it's going to spit\Nthe answer back at you.
Dialogue: 0,0:58:26.75,0:58:29.00,Default,,0000,0000,0000,,So this is how\Nengineers actually
Dialogue: 0,0:58:29.00,0:58:30.83,Default,,0000,0000,0000,,solve the everyday integrals.
Dialogue: 0,0:58:30.83,0:58:32.68,Default,,0000,0000,0000,,Not by hand.
Dialogue: 0,0:58:32.68,0:58:36.21,Default,,0000,0000,0000,,I want to be able to\Nset it up in both ways
Dialogue: 0,0:58:36.21,0:58:39.20,Default,,0000,0000,0000,,before I go home or\Neat something, right?
Dialogue: 0,0:58:39.20,0:58:43.92,Default,,0000,0000,0000,,So we don't have to spend\Na lot of time on it.
Dialogue: 0,0:58:43.92,0:58:47.66,Default,,0000,0000,0000,,But if you want to tell me\Nhow I am going to set it up,
Dialogue: 0,0:58:47.66,0:58:49.73,Default,,0000,0000,0000,,I would be very grateful.
Dialogue: 0,0:58:49.73,0:58:53.72,Default,,0000,0000,0000,,So this is Cartesian,\Nand this is polar.
Dialogue: 0,0:58:53.72,0:59:05.71,Default,,0000,0000,0000,,
Dialogue: 0,0:59:05.71,0:59:07.00,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:59:07.00,0:59:07.84,Default,,0000,0000,0000,,Who helps me?
Dialogue: 0,0:59:07.84,0:59:10.34,Default,,0000,0000,0000,,In Cartesian-- which\None do you prefer?
Dialogue: 0,0:59:10.34,0:59:11.42,Default,,0000,0000,0000,,I mean, it doesn't matter.
Dialogue: 0,0:59:11.42,0:59:13.84,Default,,0000,0000,0000,,You guys are good\Nand smart, and you'll
Dialogue: 0,0:59:13.84,0:59:15.90,Default,,0000,0000,0000,,figure out what I need to do.
Dialogue: 0,0:59:15.90,0:59:19.38,Default,,0000,0000,0000,,If I want to do it in terms\Nof vertical strip-- so
Dialogue: 0,0:59:19.38,0:59:21.27,Default,,0000,0000,0000,,for vertical strip\Nmethod-- first
Dialogue: 0,0:59:21.27,0:59:25.01,Default,,0000,0000,0000,,I integrate with respect to\Ny, and then with respect to x.
Dialogue: 0,0:59:25.01,0:59:27.35,Default,,0000,0000,0000,,And maybe, to test\Nyour understanding,
Dialogue: 0,0:59:27.35,0:59:29.64,Default,,0000,0000,0000,,let me change the\Norder of integrals
Dialogue: 0,0:59:29.64,0:59:33.28,Default,,0000,0000,0000,,and see how much you\Nunderstood from that last time.
Dialogue: 0,0:59:33.28,0:59:35.28,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:59:35.28,0:59:39.07,Default,,0000,0000,0000,,MAGDALENA TODA: So x is\Nbetween what and what?
Dialogue: 0,0:59:39.07,0:59:39.78,Default,,0000,0000,0000,,STUDENT: 0 and 1.
Dialogue: 0,0:59:39.78,0:59:41.32,Default,,0000,0000,0000,,MAGDALENA TODA: Look\Nat this picture.
Dialogue: 0,0:59:41.32,0:59:43.49,Default,,0000,0000,0000,,I have to reproduce\Nthis picture like that.
Dialogue: 0,0:59:43.49,0:59:46.93,Default,,0000,0000,0000,,0 to 1, says Alex,\Nand he's right.
Dialogue: 0,0:59:46.93,0:59:49.24,Default,,0000,0000,0000,,And why will he decide against--
Dialogue: 0,0:59:49.24,0:59:50.37,Default,,0000,0000,0000,,STUDENT: 1 minus x squared.
Dialogue: 0,0:59:50.37,0:59:52.82,Default,,0000,0000,0000,,MAGDALENA TODA: --square\Nroot 1 minus x squared.
Dialogue: 0,0:59:52.82,0:59:55.28,Default,,0000,0000,0000,,So we know very well\Nwhat we are going to do,
Dialogue: 0,0:59:55.28,0:59:57.26,Default,,0000,0000,0000,,what Maple is\Ngoing to do for us.
Dialogue: 0,0:59:57.26,0:59:59.99,Default,,0000,0000,0000,,1 square root 1 minus x squared.
Dialogue: 0,0:59:59.99,1:00:01.61,Default,,0000,0000,0000,,And then what do I put here?
Dialogue: 0,1:00:01.61,1:00:03.45,Default,,0000,0000,0000,,8 minus x minus y.
Dialogue: 0,1:00:03.45,1:00:06.04,Default,,0000,0000,0000,,Can I do it by hand?
Dialogue: 0,1:00:06.04,1:00:08.20,Default,,0000,0000,0000,,Yes, I guarantee to you\NI can do it by hand.
Dialogue: 0,1:00:08.20,1:00:10.29,Default,,0000,0000,0000,,Let me tell you why.
Dialogue: 0,1:00:10.29,1:00:13.88,Default,,0000,0000,0000,,Because when we integrate\Nwith respect to y, I get xy.
Dialogue: 0,1:00:13.88,1:00:17.60,Default,,0000,0000,0000,,So I get xy, and y will be\Nplugged in 1 minus x squared.
Dialogue: 0,1:00:17.60,1:00:23.60,Default,,0000,0000,0000,,How am I going to solve\Nan integral like this?
Dialogue: 0,1:00:23.60,1:00:28.08,Default,,0000,0000,0000,,I can the first one with a\Ntable, the second one with a u
Dialogue: 0,1:00:28.08,1:00:30.50,Default,,0000,0000,0000,,substitution.
Dialogue: 0,1:00:30.50,1:00:33.47,Default,,0000,0000,0000,,On the last one is a\Nlittle bit painful.
Dialogue: 0,1:00:33.47,1:00:34.97,Default,,0000,0000,0000,,I'm going to have\Ny squared over 2--
Dialogue: 0,1:00:34.97,1:00:36.34,Default,,0000,0000,0000,,STUDENT: That's\Nthe easiest part.
Dialogue: 0,1:00:36.34,1:00:38.92,Default,,0000,0000,0000,,MAGDALENA TODA: According\Nto Alex, yes, you're right.
Dialogue: 0,1:00:38.92,1:00:40.57,Default,,0000,0000,0000,,Maybe that is the easiest.
Dialogue: 0,1:00:40.57,1:00:42.90,Default,,0000,0000,0000,,STUDENT: That's the [INAUDIBLE]\Npart you can integrate--
Dialogue: 0,1:00:42.90,1:00:45.00,Default,,0000,0000,0000,,MAGDALENA TODA: And I can\Nintegrate one at a time,
Dialogue: 0,1:00:45.00,1:00:46.99,Default,,0000,0000,0000,,and I'm going to\Nwaste all my time.
Dialogue: 0,1:00:46.99,1:00:49.25,Default,,0000,0000,0000,,So if I want to be an\Nefficient engineer,
Dialogue: 0,1:00:49.25,1:00:53.54,Default,,0000,0000,0000,,and my boss is waiting for\Nthe end-of-the-day project,
Dialogue: 0,1:00:53.54,1:00:56.16,Default,,0000,0000,0000,,of course I'm not going\Nto do this by hand.
Dialogue: 0,1:00:56.16,1:00:59.01,Default,,0000,0000,0000,,How about the other integral?
Dialogue: 0,1:00:59.01,1:01:01.64,Default,,0000,0000,0000,,Same integral.
Dialogue: 0,1:01:01.64,1:01:04.79,Default,,0000,0000,0000,,Same idea, y between 0 and 1.
Dialogue: 0,1:01:04.79,1:01:06.69,Default,,0000,0000,0000,,And x between 0 and
Dialogue: 0,1:01:06.69,1:01:08.44,Default,,0000,0000,0000,,STUDENT: Square root\Nof 1 minus y squared.
Dialogue: 0,1:01:08.44,1:01:10.96,Default,,0000,0000,0000,,MAGDALENA TODA: Square\Nroot of 1 minus y squared.
Dialogue: 0,1:01:10.96,1:01:16.34,Default,,0000,0000,0000,,Because I'll do this guy\Nwith horizontal strips,
Dialogue: 0,1:01:16.34,1:01:19.45,Default,,0000,0000,0000,,and forget about\Nthe vertical strips.
Dialogue: 0,1:01:19.45,1:01:23.33,Default,,0000,0000,0000,,And here's the y-- I rotate\Nmy head and it cracks,
Dialogue: 0,1:01:23.33,1:01:27.71,Default,,0000,0000,0000,,so that means that\NI need some yoga.
Dialogue: 0,1:01:27.71,1:01:30.53,Default,,0000,0000,0000,,y is between 0 and 1.
Dialogue: 0,1:01:30.53,1:01:31.41,Default,,0000,0000,0000,,Or gymnastics.
Dialogue: 0,1:01:31.41,1:01:35.77,Default,,0000,0000,0000,,So x is between\N0 and square root
Dialogue: 0,1:01:35.77,1:01:40.47,Default,,0000,0000,0000,,1 minus y squared. [INAUDIBLE].
Dialogue: 0,1:01:40.47,1:01:41.97,Default,,0000,0000,0000,,And I'll leave it\Nhere on the meter.
Dialogue: 0,1:01:41.97,1:01:46.16,Default,,0000,0000,0000,,And I'm going to make a\Nsample like I promised.
Dialogue: 0,1:01:46.16,1:01:48.53,Default,,0000,0000,0000,,OK, good.
Dialogue: 0,1:01:48.53,1:01:53.23,Default,,0000,0000,0000,,How would you do this to set up\Nthe polar coordinate integral?
Dialogue: 0,1:01:53.23,1:01:57.78,Default,,0000,0000,0000,,And that is why Alex said\Nmaybe that's a pain because
Dialogue: 0,1:01:57.78,1:02:00.04,Default,,0000,0000,0000,,of a reason.
Dialogue: 0,1:02:00.04,1:02:03.86,Default,,0000,0000,0000,,And he's right, it's a little\Nbit painful to solve by hand.
Dialogue: 0,1:02:03.86,1:02:06.73,Default,,0000,0000,0000,,But again, once you\Nswitch to polar,
Dialogue: 0,1:02:06.73,1:02:10.56,Default,,0000,0000,0000,,you can solve it with a\Ncalculator or a computer
Dialogue: 0,1:02:10.56,1:02:14.27,Default,,0000,0000,0000,,software, scientific\Nsoftware in no time.
Dialogue: 0,1:02:14.27,1:02:19.18,Default,,0000,0000,0000,,In Maple, you just have\Nto plug in the numbers.
Dialogue: 0,1:02:19.18,1:02:22.06,Default,,0000,0000,0000,,You cannot plug in theta,\NI think, as a symbol.
Dialogue: 0,1:02:22.06,1:02:22.60,Default,,0000,0000,0000,,I'm not sure.
Dialogue: 0,1:02:22.60,1:02:26.63,Default,,0000,0000,0000,,But you can put theta\Nas t and r will be r,
Dialogue: 0,1:02:26.63,1:02:28.70,Default,,0000,0000,0000,,or you can use\Nwhatever letters you
Dialogue: 0,1:02:28.70,1:02:31.28,Default,,0000,0000,0000,,want that are roman letters.
Dialogue: 0,1:02:31.28,1:02:35.43,Default,,0000,0000,0000,,So you have to\Nintegrate smartly, here,
Dialogue: 0,1:02:35.43,1:02:38.22,Default,,0000,0000,0000,,switching to r and\Ntheta, and think
Dialogue: 0,1:02:38.22,1:02:41.01,Default,,0000,0000,0000,,about the meaning of that.
Dialogue: 0,1:02:41.01,1:02:45.49,Default,,0000,0000,0000,,So first of all, if\NI put dr d theta,
Dialogue: 0,1:02:45.49,1:02:48.65,Default,,0000,0000,0000,,I'm not worried that you won't\Nbe able to get r and theta,
Dialogue: 0,1:02:48.65,1:02:51.12,Default,,0000,0000,0000,,because I know you can do it.
Dialogue: 0,1:02:51.12,1:02:56.38,Default,,0000,0000,0000,,You can prove it to me\Nright now. r between 0 and
Dialogue: 0,1:02:56.38,1:02:56.88,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,1:02:56.88,1:02:57.96,Default,,0000,0000,0000,,MAGDALENA TODA: Excellent.
Dialogue: 0,1:02:57.96,1:03:01.50,Default,,0000,0000,0000,,And theta, pay\Nattention, between 0 and
Dialogue: 0,1:03:01.50,1:03:02.24,Default,,0000,0000,0000,,STUDENT: pi over 2
Dialogue: 0,1:03:02.24,1:03:03.33,Default,,0000,0000,0000,,MAGDALENA TODA: Excellent.
Dialogue: 0,1:03:03.33,1:03:03.97,Default,,0000,0000,0000,,I'm proud.
Dialogue: 0,1:03:03.97,1:03:04.50,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:03:04.50,1:03:06.57,Default,,0000,0000,0000,,STUDENT: Is it supposed\Nto be r dr 2 theta,
Dialogue: 0,1:03:06.57,1:03:08.04,Default,,0000,0000,0000,,or are you going\Nto add that later?
Dialogue: 0,1:03:08.04,1:03:10.47,Default,,0000,0000,0000,,MAGDALENA TODA: I\Nwill add it here.
Dialogue: 0,1:03:10.47,1:03:13.40,Default,,0000,0000,0000,,So the integrand\Nwill contain the r.
Dialogue: 0,1:03:13.40,1:03:16.74,Default,,0000,0000,0000,,Now what do I put\Nin terms of this?
Dialogue: 0,1:03:16.74,1:03:19.08,Default,,0000,0000,0000,,I left enough room.
Dialogue: 0,1:03:19.08,1:03:23.08,Default,,0000,0000,0000,,STUDENT: Is it pi over 2,\Nor is it negative pi over 2?
Dialogue: 0,1:03:23.08,1:03:24.60,Default,,0000,0000,0000,,MAGDALENA TODA:\NIt doesn't matter,
Dialogue: 0,1:03:24.60,1:03:30.96,Default,,0000,0000,0000,,because I'll have to take that--\Nwe assume always theta to go
Dialogue: 0,1:03:30.96,1:03:35.28,Default,,0000,0000,0000,,counterclockwise, and go\Nbetween 0 and pi over 2,
Dialogue: 0,1:03:35.28,1:03:38.76,Default,,0000,0000,0000,,so that when you start--\Nlet me make this motion.
Dialogue: 0,1:03:38.76,1:03:41.34,Default,,0000,0000,0000,,You are here at theta equals 0.
Dialogue: 0,1:03:41.34,1:03:42.04,Default,,0000,0000,0000,,STUDENT: Oh, OK.
Dialogue: 0,1:03:42.04,1:03:42.32,Default,,0000,0000,0000,,Sorry.
Dialogue: 0,1:03:42.32,1:03:43.80,Default,,0000,0000,0000,,I got my coordinates\Nmixed around--
Dialogue: 0,1:03:43.80,1:03:46.42,Default,,0000,0000,0000,,MAGDALENA TODA: --and\Ncounterclockwise to pi over 2.
Dialogue: 0,1:03:46.42,1:03:47.25,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:03:47.25,1:03:49.42,Default,,0000,0000,0000,,
Dialogue: 0,1:03:49.42,1:03:50.29,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:03:50.29,1:03:54.55,Default,,0000,0000,0000,,So you go in the trigonometric--\NHere, you have 8 minus,
Dialogue: 0,1:03:54.55,1:03:57.75,Default,,0000,0000,0000,,and who tells me what\NI'm supposed to type?
Dialogue: 0,1:03:57.75,1:03:58.98,Default,,0000,0000,0000,,STUDENT: r over x.
Dialogue: 0,1:03:58.98,1:04:07.92,Default,,0000,0000,0000,,MAGDALENA TODA: r\Ncosine theta minus
Dialogue: 0,1:04:07.92,1:04:08.50,Default,,0000,0000,0000,,STUDENT: Sine.
Dialogue: 0,1:04:08.50,1:04:11.27,Default,,0000,0000,0000,,MAGDALENA TODA: r sine theta.
Dialogue: 0,1:04:11.27,1:04:13.83,Default,,0000,0000,0000,,And let mister\Nwhatever his name is,
Dialogue: 0,1:04:13.83,1:04:17.35,Default,,0000,0000,0000,,the computer, find the answer.
Dialogue: 0,1:04:17.35,1:04:18.88,Default,,0000,0000,0000,,Can I do it by hand?
Dialogue: 0,1:04:18.88,1:04:20.71,Default,,0000,0000,0000,,Actually, I can.
Dialogue: 0,1:04:20.71,1:04:27.20,Default,,0000,0000,0000,,I can, but again, it's not worth\Nit, because it drives me crazy.
Dialogue: 0,1:04:27.20,1:04:29.53,Default,,0000,0000,0000,,How would I do it by hand?
Dialogue: 0,1:04:29.53,1:04:32.18,Default,,0000,0000,0000,,I would split the\Nintegral into three,
Dialogue: 0,1:04:32.18,1:04:36.09,Default,,0000,0000,0000,,and I would easily\Ncompute 8 times r,
Dialogue: 0,1:04:36.09,1:04:37.34,Default,,0000,0000,0000,,integrand is going to be easy.
Dialogue: 0,1:04:37.34,1:04:37.84,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:04:37.84,1:04:39.02,Default,,0000,0000,0000,,Agree with me?
Dialogue: 0,1:04:39.02,1:04:40.62,Default,,0000,0000,0000,,Then what am I going to do?
Dialogue: 0,1:04:40.62,1:04:46.94,Default,,0000,0000,0000,,I'm going to say, an r out\Ntimes an r, out comes r squared.
Dialogue: 0,1:04:46.94,1:04:50.85,Default,,0000,0000,0000,,And I have integral of r\Nsquared times a function
Dialogue: 0,1:04:50.85,1:04:53.50,Default,,0000,0000,0000,,of theta only,\Nwhich is going to be
Dialogue: 0,1:04:53.50,1:04:56.42,Default,,0000,0000,0000,,sine theta plus cosine theta.
Dialogue: 0,1:04:56.42,1:04:58.81,Default,,0000,0000,0000,,We are going to say, yes,\Nwith a minus, with a minus.
Dialogue: 0,1:04:58.81,1:05:01.66,Default,,0000,0000,0000,,
Dialogue: 0,1:05:01.66,1:05:06.59,Default,,0000,0000,0000,,Now, when I compute\Nr and theta thingy,
Dialogue: 0,1:05:06.59,1:05:09.80,Default,,0000,0000,0000,,theta will be between\N0 and pi over 2.
Dialogue: 0,1:05:09.80,1:05:12.26,Default,,0000,0000,0000,,r will be between 0 and 1.
Dialogue: 0,1:05:12.26,1:05:16.35,Default,,0000,0000,0000,,But I don't care, because\NMatthew reminded me,
Dialogue: 0,1:05:16.35,1:05:19.16,Default,,0000,0000,0000,,if you have a product\Nof separate variables,
Dialogue: 0,1:05:19.16,1:05:22.32,Default,,0000,0000,0000,,life becomes all of the\Nsudden easier for you.
Dialogue: 0,1:05:22.32,1:05:24.93,Default,,0000,0000,0000,,STUDENT: You've also got to\Nadd your integral of [? 8r ?]
Dialogue: 0,1:05:24.93,1:05:25.43,Default,,0000,0000,0000,,[? dr. ?]
Dialogue: 0,1:05:25.43,1:05:25.90,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:05:25.90,1:05:28.18,Default,,0000,0000,0000,,At the end, I'm going to\Nadd the integral of 8r.
Dialogue: 0,1:05:28.18,1:05:30.12,Default,,0000,0000,0000,,So I take them separately.
Dialogue: 0,1:05:30.12,1:05:32.97,Default,,0000,0000,0000,,I just look at one chunk.
Dialogue: 0,1:05:32.97,1:05:35.08,Default,,0000,0000,0000,,And this chunk will be what?
Dialogue: 0,1:05:35.08,1:05:39.28,Default,,0000,0000,0000,,Can you even see how easy it's\Ngoing to be with the naked eye?
Dialogue: 0,1:05:39.28,1:05:41.47,Default,,0000,0000,0000,,Firs of all,\Nintegral from 0 to 1,
Dialogue: 0,1:05:41.47,1:05:43.91,Default,,0000,0000,0000,,r squared dr is a piece of cake.
Dialogue: 0,1:05:43.91,1:05:45.98,Default,,0000,0000,0000,,How much is that--\Npiece of salami.
Dialogue: 0,1:05:45.98,1:05:46.52,Default,,0000,0000,0000,,STUDENT: 1/3.
Dialogue: 0,1:05:46.52,1:05:49.10,Default,,0000,0000,0000,,MAGDALENA TODA: 1/3.
Dialogue: 0,1:05:49.10,1:05:49.60,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:05:49.60,1:05:52.08,Default,,0000,0000,0000,,Because it's r cubed over 3.
Dialogue: 0,1:05:52.08,1:05:53.07,Default,,0000,0000,0000,,Then you have 1/3.
Dialogue: 0,1:05:53.07,1:05:54.70,Default,,0000,0000,0000,,That's easy.
Dialogue: 0,1:05:54.70,1:05:58.47,Default,,0000,0000,0000,,With a minus in front, but I\Ndon't care about it in the end.
Dialogue: 0,1:05:58.47,1:06:03.79,Default,,0000,0000,0000,,What is the integral of\Nsine theta cosine theta?
Dialogue: 0,1:06:03.79,1:06:06.24,Default,,0000,0000,0000,,STUDENT: Negative [INAUDIBLE].
Dialogue: 0,1:06:06.24,1:06:11.45,Default,,0000,0000,0000,,MAGDALENA TODA: Minus\Ncosine theta plus sine theta
Dialogue: 0,1:06:11.45,1:06:14.28,Default,,0000,0000,0000,,taken between 0 and pi over 2.
Dialogue: 0,1:06:14.28,1:06:15.94,Default,,0000,0000,0000,,Will this be hard?
Dialogue: 0,1:06:15.94,1:06:19.59,Default,,0000,0000,0000,,Who's going to tell me what,\Nor how I'm going to get what--
Dialogue: 0,1:06:19.59,1:06:22.69,Default,,0000,0000,0000,,we don't compute it now,\Nbut I just give you.
Dialogue: 0,1:06:22.69,1:06:24.16,Default,,0000,0000,0000,,Cosine of pi over 3 is?
Dialogue: 0,1:06:24.16,1:06:24.66,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:06:24.66,1:06:25.41,Default,,0000,0000,0000,,MAGDALENA TODA: 0.
Dialogue: 0,1:06:25.41,1:06:26.87,Default,,0000,0000,0000,,Sine of pi over 2 is?
Dialogue: 0,1:06:26.87,1:06:27.57,Default,,0000,0000,0000,,STUDENT: Oh yeah.
Dialogue: 0,1:06:27.57,1:06:27.70,Default,,0000,0000,0000,,1.
Dialogue: 0,1:06:27.70,1:06:28.45,Default,,0000,0000,0000,,MAGDALENA TODA: 1.
Dialogue: 0,1:06:28.45,1:06:30.97,Default,,0000,0000,0000,,So this is going to\Nbe 1 minus, what's
Dialogue: 0,1:06:30.97,1:06:34.56,Default,,0000,0000,0000,,the whole thingy computed at 0?
Dialogue: 0,1:06:34.56,1:06:35.43,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:06:35.43,1:06:38.71,Default,,0000,0000,0000,,MAGDALENA TODA: It's going to\Nbe minus 1, but minus minus 1
Dialogue: 0,1:06:38.71,1:06:40.50,Default,,0000,0000,0000,,is plus 1.
Dialogue: 0,1:06:40.50,1:06:42.93,Default,,0000,0000,0000,,So I have 2.
Dialogue: 0,1:06:42.93,1:06:46.31,Default,,0000,0000,0000,,So only this chunk of the\Nintegral would be easy.
Dialogue: 0,1:06:46.31,1:06:47.21,Default,,0000,0000,0000,,Minus 2/3.
Dialogue: 0,1:06:47.21,1:06:48.32,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:06:48.32,1:06:51.09,Default,,0000,0000,0000,,So it can be done by hand,\Nbut why waste the time when
Dialogue: 0,1:06:51.09,1:06:52.52,Default,,0000,0000,0000,,you can do it with Maple?
Dialogue: 0,1:06:52.52,1:06:53.11,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:06:53.11,1:06:56.33,Default,,0000,0000,0000,,STUDENT: Where did\Nyou get rid of 8?
Dialogue: 0,1:06:56.33,1:06:58.23,Default,,0000,0000,0000,,On the second, after the 8--
Dialogue: 0,1:06:58.23,1:06:59.73,Default,,0000,0000,0000,,MAGDALENA TODA: No, I didn't.
Dialogue: 0,1:06:59.73,1:07:01.87,Default,,0000,0000,0000,,That's exactly what\Nwe were talking.
Dialogue: 0,1:07:01.87,1:07:05.65,Default,,0000,0000,0000,,Alex says, but you just\Ntalked about integral of 8r,
Dialogue: 0,1:07:05.65,1:07:06.97,Default,,0000,0000,0000,,but you didn't want to do it.
Dialogue: 0,1:07:06.97,1:07:09.10,Default,,0000,0000,0000,,I said, I didn't want to do it.
Dialogue: 0,1:07:09.10,1:07:12.76,Default,,0000,0000,0000,,This is just the second\Nchunk of this integral.
Dialogue: 0,1:07:12.76,1:07:17.13,Default,,0000,0000,0000,,So I know that I can do integral\Nof integral of 8r in no time.
Dialogue: 0,1:07:17.13,1:07:20.14,Default,,0000,0000,0000,,Then I would need to\Ntake this and add that,
Dialogue: 0,1:07:20.14,1:07:21.11,Default,,0000,0000,0000,,and get the number.
Dialogue: 0,1:07:21.11,1:07:23.06,Default,,0000,0000,0000,,I don't care about the number.
Dialogue: 0,1:07:23.06,1:07:24.51,Default,,0000,0000,0000,,I just care about the method.
Dialogue: 0,1:07:24.51,1:07:25.01,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:07:25.01,1:07:27.61,Default,,0000,0000,0000,,STUDENT: Why are the limits\Nfrom 0 to 1 instead of like 0
Dialogue: 0,1:07:27.61,1:07:29.88,Default,,0000,0000,0000,,to r squared?
Dialogue: 0,1:07:29.88,1:07:32.87,Default,,0000,0000,0000,,Because didn't we say\Nearlier the domain
Dialogue: 0,1:07:32.87,1:07:34.85,Default,,0000,0000,0000,,is x squared plus y squared?
Dialogue: 0,1:07:34.85,1:07:37.55,Default,,0000,0000,0000,,Wouldn't that be r squared?
Dialogue: 0,1:07:37.55,1:07:38.34,Default,,0000,0000,0000,,MAGDALENA TODA: No.
Dialogue: 0,1:07:38.34,1:07:39.05,Default,,0000,0000,0000,,No, wait.
Dialogue: 0,1:07:39.05,1:07:40.21,Default,,0000,0000,0000,,This is r squared.
Dialogue: 0,1:07:40.21,1:07:40.83,Default,,0000,0000,0000,,STUDENT: Right.
Dialogue: 0,1:07:40.83,1:07:43.66,Default,,0000,0000,0000,,Why didn't we plug r\Nsquared into the 1 again.
Dialogue: 0,1:07:43.66,1:07:47.32,Default,,0000,0000,0000,,MAGDALENA TODA: And that means\Nr is between 0 and 1, right?
Dialogue: 0,1:07:47.32,1:07:47.99,Default,,0000,0000,0000,,STUDENT: Oh, OK.
Dialogue: 0,1:07:47.99,1:07:49.82,Default,,0000,0000,0000,,MAGDALENA TODA: r squared\Nbeing less than 1.
Dialogue: 0,1:07:49.82,1:07:51.77,Default,,0000,0000,0000,,That means r is between 0 and 1.
Dialogue: 0,1:07:51.77,1:07:52.97,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:07:52.97,1:07:56.57,Default,,0000,0000,0000,,And one last problem-- no.
Dialogue: 0,1:07:56.57,1:07:58.97,Default,,0000,0000,0000,,No last problem.
Dialogue: 0,1:07:58.97,1:08:00.77,Default,,0000,0000,0000,,We have barely 10 minutes.
Dialogue: 0,1:08:00.77,1:08:04.07,Default,,0000,0000,0000,,So you read from the book some.
Dialogue: 0,1:08:04.07,1:08:07.67,Default,,0000,0000,0000,,I will come back to this\Nsection, and I'll do review.
Dialogue: 0,1:08:07.67,1:08:10.97,Default,,0000,0000,0000,,Have a wonderful\Nspring break, and I'm
Dialogue: 0,1:08:10.97,1:08:13.67,Default,,0000,0000,0000,,going to see you after\Nspring break on Tuesday.
Dialogue: 0,1:08:13.67,1:08:16.42,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:08:16.42,1:08:18.16,Default,,0000,0000,0000,,