[Script Info] Title: [Events] Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text Dialogue: 0,0:00:00.00,0:00:04.39,Default,,0000,0000,0000,,PROFESSOR: I have\Nsome assignments Dialogue: 0,0:00:04.39,0:00:06.34,Default,,0000,0000,0000,,that I want to give you back. Dialogue: 0,0:00:06.34,0:00:09.51,Default,,0000,0000,0000,,And I'm just going\Nto put them here, Dialogue: 0,0:00:09.51,0:00:13.72,Default,,0000,0000,0000,,and I'll ask you to pick them\Nup as soon as we take a break. Dialogue: 0,0:00:13.72,0:00:16.85,Default,,0000,0000,0000,, Dialogue: 0,0:00:16.85,0:00:20.58,Default,,0000,0000,0000,,There are explanations there\Nhow they were computed in red. Dialogue: 0,0:00:20.58,0:00:23.48,Default,,0000,0000,0000,,If you have questions,\Nyou can as me Dialogue: 0,0:00:23.48,0:00:26.00,Default,,0000,0000,0000,,so I can ask my grader about it. Dialogue: 0,0:00:26.00,0:00:28.95,Default,,0000,0000,0000,, Dialogue: 0,0:00:28.95,0:00:33.58,Default,,0000,0000,0000,,Now, I promised you that\NI would move on today, Dialogue: 0,0:00:33.58,0:00:35.22,Default,,0000,0000,0000,,and that's what I'm going to do. Dialogue: 0,0:00:35.22,0:00:39.48,Default,,0000,0000,0000,,I'm moving on to something\Nthat you're gong to love. Dialogue: 0,0:00:39.48,0:00:47.84,Default,,0000,0000,0000,,[? Practically ?] chapter 12\Nis integration of functions Dialogue: 0,0:00:47.84,0:00:49.32,Default,,0000,0000,0000,,of several variables. Dialogue: 0,0:00:49.32,0:00:58.67,Default,,0000,0000,0000,, Dialogue: 0,0:00:58.67,0:01:01.32,Default,,0000,0000,0000,,And to warn you\Nwe're going to see Dialogue: 0,0:01:01.32,0:01:08.88,Default,,0000,0000,0000,,how we introduce introduction\Nto the double integral. Dialogue: 0,0:01:08.88,0:01:15.52,Default,,0000,0000,0000,, Dialogue: 0,0:01:15.52,0:01:17.39,Default,,0000,0000,0000,,But you will say, wait a minute. Dialogue: 0,0:01:17.39,0:01:22.48,Default,,0000,0000,0000,,I don't even know if I\Nremember the simple integral. Dialogue: 0,0:01:22.48,0:01:24.33,Default,,0000,0000,0000,,And that's why I'm here. Dialogue: 0,0:01:24.33,0:01:31.52,Default,,0000,0000,0000,,I want to remind you what the\Ndefinite integral was both Dialogue: 0,0:01:31.52,0:01:35.42,Default,,0000,0000,0000,,as a formal definition-- let's\Ndo it as a formal definition Dialogue: 0,0:01:35.42,0:01:39.93,Default,,0000,0000,0000,,first, then come up with a\Ngeometric interpretation based Dialogue: 0,0:01:39.93,0:01:40.47,Default,,0000,0000,0000,,on that. Dialogue: 0,0:01:40.47,0:01:45.00,Default,,0000,0000,0000,,And finally write\Ndown the definition Dialogue: 0,0:01:45.00,0:01:49.38,Default,,0000,0000,0000,,and the fundamental\Ntheorem of calculus. Dialogue: 0,0:01:49.38,0:01:52.39,Default,,0000,0000,0000,,So assume you have a\Nfunction that's continuous. Dialogue: 0,0:01:52.39,0:01:56.31,Default,,0000,0000,0000,, Dialogue: 0,0:01:56.31,0:02:05.09,Default,,0000,0000,0000,,Continuous over a certain\Nintegral of a, b interval in R. Dialogue: 0,0:02:05.09,0:02:08.69,Default,,0000,0000,0000,,And you know that\Nin that case, you Dialogue: 0,0:02:08.69,0:02:23.04,Default,,0000,0000,0000,,can "define the\Ndefinite integral of f Dialogue: 0,0:02:23.04,0:02:29.48,Default,,0000,0000,0000,,of x from or between a and b." Dialogue: 0,0:02:29.48,0:02:35.30,Default,,0000,0000,0000,,And as the notation is denoted,\Nby integral from a to b f of x Dialogue: 0,0:02:35.30,0:02:35.80,Default,,0000,0000,0000,,dx. Dialogue: 0,0:02:35.80,0:02:42.49,Default,,0000,0000,0000,, Dialogue: 0,0:02:42.49,0:02:46.10,Default,,0000,0000,0000,,Well, how do we define this? Dialogue: 0,0:02:46.10,0:02:47.34,Default,,0000,0000,0000,,This is just the notation. Dialogue: 0,0:02:47.34,0:02:51.14,Default,,0000,0000,0000,,How do we define it? Dialogue: 0,0:02:51.14,0:02:58.83,Default,,0000,0000,0000,,We have to have a set up, and\Nwe are thinking of a x, y frame. Dialogue: 0,0:02:58.83,0:03:02.19,Default,,0000,0000,0000,,You have a function,\Nf, that's continuous. Dialogue: 0,0:03:02.19,0:03:05.93,Default,,0000,0000,0000,, Dialogue: 0,0:03:05.93,0:03:07.75,Default,,0000,0000,0000,,And you are thinking,\Noh, wait a minute. Dialogue: 0,0:03:07.75,0:03:11.09,Default,,0000,0000,0000,,I would like to be\Nable to evaluate Dialogue: 0,0:03:11.09,0:03:12.47,Default,,0000,0000,0000,,the area under the integral. Dialogue: 0,0:03:12.47,0:03:16.59,Default,,0000,0000,0000,, Dialogue: 0,0:03:16.59,0:03:19.68,Default,,0000,0000,0000,,And if you ask your teacher\Nwhen you are in fourth grade, Dialogue: 0,0:03:19.68,0:03:22.48,Default,,0000,0000,0000,,your teacher will say, well,\NI can give you some graphing Dialogue: 0,0:03:22.48,0:03:23.55,Default,,0000,0000,0000,,paper. Dialogue: 0,0:03:23.55,0:03:25.41,Default,,0000,0000,0000,,And with that\Ngraphing paper, you Dialogue: 0,0:03:25.41,0:03:35.55,Default,,0000,0000,0000,,can eventually approximate\Nyour area like that. Dialogue: 0,0:03:35.55,0:03:42.79,Default,,0000,0000,0000,,Sort of what you get here is\Nlike you draw a horizontal Dialogue: 0,0:03:42.79,0:03:46.52,Default,,0000,0000,0000,,so that the little part\Nabove the horizontal Dialogue: 0,0:03:46.52,0:03:49.10,Default,,0000,0000,0000,,cancels out with the little\Npart below the horizontal. Dialogue: 0,0:03:49.10,0:03:51.49,Default,,0000,0000,0000,,So more or less,\Nthe pink rectangle Dialogue: 0,0:03:51.49,0:03:56.19,Default,,0000,0000,0000,,is a good approximation\Nof the first slice. Dialogue: 0,0:03:56.19,0:03:59.66,Default,,0000,0000,0000,,But you say yeah, but the first\Nslice is a curvilinear slice. Dialogue: 0,0:03:59.66,0:04:03.08,Default,,0000,0000,0000,,Yes, but we make it\Nlike a stop function. Dialogue: 0,0:04:03.08,0:04:06.77,Default,,0000,0000,0000,,So then you say, OK,\Nhow about this fellow? Dialogue: 0,0:04:06.77,0:04:11.23,Default,,0000,0000,0000,,I'm going to approximate\Nit in a similar way, Dialogue: 0,0:04:11.23,0:04:15.03,Default,,0000,0000,0000,,and I'm going to have a bunch\Nof rectangles on this graphing Dialogue: 0,0:04:15.03,0:04:15.53,Default,,0000,0000,0000,,paper. Dialogue: 0,0:04:15.53,0:04:18.45,Default,,0000,0000,0000,,And I'm going to\Ncompute their areas, Dialogue: 0,0:04:18.45,0:04:20.76,Default,,0000,0000,0000,,and I'm going to come up\Nwith an approximation, Dialogue: 0,0:04:20.76,0:04:23.87,Default,,0000,0000,0000,,and I'll give it to my\Nfourth grade teacher. Dialogue: 0,0:04:23.87,0:04:26.83,Default,,0000,0000,0000,,And that's what we\Ndid in fourth grade, Dialogue: 0,0:04:26.83,0:04:29.38,Default,,0000,0000,0000,,but this is not fourth grade. Dialogue: 0,0:04:29.38,0:04:32.88,Default,,0000,0000,0000,,And actually, it's\Nvery relevant to us Dialogue: 0,0:04:32.88,0:04:35.61,Default,,0000,0000,0000,,that this has\Napplications to our life, Dialogue: 0,0:04:35.61,0:04:38.63,Default,,0000,0000,0000,,to our digital world,\Nthat people did not Dialogue: 0,0:04:38.63,0:04:44.49,Default,,0000,0000,0000,,understand when Riemann\Nintroduced the Riemann sum. Dialogue: 0,0:04:44.49,0:04:49.27,Default,,0000,0000,0000,,They thought, OK, the idea\Nmakes sense that practically we Dialogue: 0,0:04:49.27,0:04:54.22,Default,,0000,0000,0000,,have a huge picture\Nhere, and I'm Dialogue: 0,0:04:54.22,0:04:59.43,Default,,0000,0000,0000,,taking a and b and a function\Nthat's continuous over a and b. Dialogue: 0,0:04:59.43,0:05:02.18,Default,,0000,0000,0000,,And then I say I'm\Ngoing to split this Dialogue: 0,0:05:02.18,0:05:08.20,Default,,0000,0000,0000,,into a equidistant intervals. Dialogue: 0,0:05:08.20,0:05:10.93,Default,,0000,0000,0000,,I don't know how\Nmany I want, but let Dialogue: 0,0:05:10.93,0:05:12.09,Default,,0000,0000,0000,,me make them eight of them. Dialogue: 0,0:05:12.09,0:05:12.63,Default,,0000,0000,0000,,I don't know. Dialogue: 0,0:05:12.63,0:05:14.31,Default,,0000,0000,0000,,They have to have\Nthe same length. Dialogue: 0,0:05:14.31,0:05:17.41,Default,,0000,0000,0000,,And I'll call this delta x. Dialogue: 0,0:05:17.41,0:05:18.51,Default,,0000,0000,0000,,It has to be the same. Dialogue: 0,0:05:18.51,0:05:21.78,Default,,0000,0000,0000,,And, you guys, please forgive\Nme for the horrible picture. Dialogue: 0,0:05:21.78,0:05:25.99,Default,,0000,0000,0000,,They don't look like\Nthe same step, delta x, Dialogue: 0,0:05:25.99,0:05:28.63,Default,,0000,0000,0000,,but it should be the same. Dialogue: 0,0:05:28.63,0:05:32.64,Default,,0000,0000,0000,,In each of them I\Narbitrarily, say it again, Dialogue: 0,0:05:32.64,0:05:39.34,Default,,0000,0000,0000,,Magdalena, arbitrarily pick\Nx1 star, and another point, Dialogue: 0,0:05:39.34,0:05:44.92,Default,,0000,0000,0000,,x2 star wherever I want inside. Dialogue: 0,0:05:44.92,0:05:47.58,Default,,0000,0000,0000,,I'm just getting [INAUDIBLE]. Dialogue: 0,0:05:47.58,0:05:51.30,Default,,0000,0000,0000,,X4 star, and this is x8 star. Dialogue: 0,0:05:51.30,0:05:54.13,Default,,0000,0000,0000,,But let's say that in general\NI don't know they are 8. Dialogue: 0,0:05:54.13,0:05:56.27,Default,,0000,0000,0000,,They could be n. Dialogue: 0,0:05:56.27,0:05:57.18,Default,,0000,0000,0000,,xn star. Dialogue: 0,0:05:57.18,0:05:59.89,Default,,0000,0000,0000,,And passing to the\Nlimit with respect Dialogue: 0,0:05:59.89,0:06:02.67,Default,,0000,0000,0000,,to n going to infinity,\Nwhat am I going to get? Dialogue: 0,0:06:02.67,0:06:06.80,Default,,0000,0000,0000,,Well, in the first\Ncam I'm going up, Dialogue: 0,0:06:06.80,0:06:08.86,Default,,0000,0000,0000,,and I'm hitting\Nat what altitude? Dialogue: 0,0:06:08.86,0:06:13.15,Default,,0000,0000,0000,,I'm hitting at the altitude\Ncalled f of x1 star. Dialogue: 0,0:06:13.15,0:06:17.24,Default,,0000,0000,0000,,And that's going to be the\Nheight of this-- what is this? Dialogue: 0,0:06:17.24,0:06:17.82,Default,,0000,0000,0000,,Strip? Dialogue: 0,0:06:17.82,0:06:18.35,Default,,0000,0000,0000,,Right? Dialogue: 0,0:06:18.35,0:06:21.36,Default,,0000,0000,0000,,Or rectangle. Dialogue: 0,0:06:21.36,0:06:21.86,Default,,0000,0000,0000,,OK. Dialogue: 0,0:06:21.86,0:06:24.69,Default,,0000,0000,0000,,And I'm going to do\Nthe same with green Dialogue: 0,0:06:24.69,0:06:26.99,Default,,0000,0000,0000,,for the second rectangle. Dialogue: 0,0:06:26.99,0:06:32.17,Default,,0000,0000,0000,,I'll pick x2 star, and\Nthen that doesn't work. Dialogue: 0,0:06:32.17,0:06:33.10,Default,,0000,0000,0000,,And I'll take this. Dialogue: 0,0:06:33.10,0:06:34.96,Default,,0000,0000,0000,,Let's see if I can do\Nthe light green one, Dialogue: 0,0:06:34.96,0:06:36.36,Default,,0000,0000,0000,,because spring is here. Dialogue: 0,0:06:36.36,0:06:37.29,Default,,0000,0000,0000,,Let's see. Dialogue: 0,0:06:37.29,0:06:38.88,Default,,0000,0000,0000,,That's beautiful. Dialogue: 0,0:06:38.88,0:06:40.66,Default,,0000,0000,0000,,I go up. Dialogue: 0,0:06:40.66,0:06:44.98,Default,,0000,0000,0000,,I hit here at x2 star. Dialogue: 0,0:06:44.98,0:06:48.29,Default,,0000,0000,0000,,I get f of x2 star. Dialogue: 0,0:06:48.29,0:06:50.31,Default,,0000,0000,0000,,And so on and so forth. Dialogue: 0,0:06:50.31,0:06:53.48,Default,,0000,0000,0000,, Dialogue: 0,0:06:53.48,0:06:57.88,Default,,0000,0000,0000,,Until I get to, let's say,\Nthe last of the Mohicans. Dialogue: 0,0:06:57.88,0:07:00.61,Default,,0000,0000,0000,,This will be xn minus\N1, and this is going Dialogue: 0,0:07:00.61,0:07:06.19,Default,,0000,0000,0000,,to be xn star, the purple guy. Dialogue: 0,0:07:06.19,0:07:07.95,Default,,0000,0000,0000,,And this is going\Nto be the height Dialogue: 0,0:07:07.95,0:07:12.28,Default,,0000,0000,0000,,of that last of the Mohicans. Dialogue: 0,0:07:12.28,0:07:19.44,Default,,0000,0000,0000,,So when I compute the sum, I\Ncall that approximating sum Dialogue: 0,0:07:19.44,0:07:23.49,Default,,0000,0000,0000,,or Riemann approximating sum,\Nbecause Riemann had nothing Dialogue: 0,0:07:23.49,0:07:25.88,Default,,0000,0000,0000,,better to do than invent it. Dialogue: 0,0:07:25.88,0:07:27.88,Default,,0000,0000,0000,,He didn't even know\Nthat we are going Dialogue: 0,0:07:27.88,0:07:32.85,Default,,0000,0000,0000,,to get pixels that are in\Nlarger and larger quantities. Dialogue: 0,0:07:32.85,0:07:36.04,Default,,0000,0000,0000,,Like, we get 3,000 by 900. Dialogue: 0,0:07:36.04,0:07:41.44,Default,,0000,0000,0000,,He didn't know we are going to\Nhave all those digital gadgets. Dialogue: 0,0:07:41.44,0:07:45.64,Default,,0000,0000,0000,,But passing to the\Nlimit practically should Dialogue: 0,0:07:45.64,0:07:49.33,Default,,0000,0000,0000,,be easier to understand\Nfor teenagers now Dialogue: 0,0:07:49.33,0:07:53.30,Default,,0000,0000,0000,,age, because it's like\Nmaking the number of pixels Dialogue: 0,0:07:53.30,0:07:57.88,Default,,0000,0000,0000,,larger and larger, and the\Npixels practically invisible. Dialogue: 0,0:07:57.88,0:08:01.71,Default,,0000,0000,0000,,Remember, I mean, I don't\Nknow, those old TVs, Dialogue: 0,0:08:01.71,0:08:04.24,Default,,0000,0000,0000,,color TVs where you could\Nstill see the squares? Dialogue: 0,0:08:04.24,0:08:05.15,Default,,0000,0000,0000,,STUDENT: Mm-hm. Dialogue: 0,0:08:05.15,0:08:06.07,Default,,0000,0000,0000,,PROFESSOR: Well, yeah. Dialogue: 0,0:08:06.07,0:08:07.98,Default,,0000,0000,0000,,When you were little. Dialogue: 0,0:08:07.98,0:08:10.85,Default,,0000,0000,0000,,But I remember them\Nmuch better than you. Dialogue: 0,0:08:10.85,0:08:13.97,Default,,0000,0000,0000,,And, yes, as the number\Nof pixels will increase, Dialogue: 0,0:08:13.97,0:08:18.19,Default,,0000,0000,0000,,that means I'm taking the limit\Nand going larger and larger. Dialogue: 0,0:08:18.19,0:08:20.61,Default,,0000,0000,0000,,That means\Npractically limitless. Dialogue: 0,0:08:20.61,0:08:22.72,Default,,0000,0000,0000,,Infinity will give\Nme an ideal image. Dialogue: 0,0:08:22.72,0:08:27.12,Default,,0000,0000,0000,,My eye will be as if I could see\Nthe image that's a curvilinear Dialogue: 0,0:08:27.12,0:08:31.01,Default,,0000,0000,0000,,image as a real person. Dialogue: 0,0:08:31.01,0:08:35.04,Default,,0000,0000,0000,,And, of course, the\Nquality of our movies Dialogue: 0,0:08:35.04,0:08:36.06,Default,,0000,0000,0000,,really increased a lot. Dialogue: 0,0:08:36.06,0:08:41.21,Default,,0000,0000,0000,,And this is what I'm\Ntrying to emphasize here. Dialogue: 0,0:08:41.21,0:08:47.12,Default,,0000,0000,0000,,So you have f of x1 star delta\Nx plus the last rectangle Dialogue: 0,0:08:47.12,0:08:51.58,Default,,0000,0000,0000,,area, f of xn star delta x. Dialogue: 0,0:08:51.58,0:08:56.20,Default,,0000,0000,0000,,Well, as a mathematician,\NI don't write it like that. Dialogue: 0,0:08:56.20,0:08:58.77,Default,,0000,0000,0000,,How do I write it\Nas a mathematician? Dialogue: 0,0:08:58.77,0:09:00.68,Default,,0000,0000,0000,,Well, we are funny people. Dialogue: 0,0:09:00.68,0:09:02.15,Default,,0000,0000,0000,,We like Greek. Dialogue: 0,0:09:02.15,0:09:03.05,Default,,0000,0000,0000,,It's all Greek to me. Dialogue: 0,0:09:03.05,0:09:16.42,Default,,0000,0000,0000,,So we go sum and from-- no. k\Nfrom 1 to n, f of x sub k star. Dialogue: 0,0:09:16.42,0:09:23.74,Default,,0000,0000,0000,,So I have k from 1 to n exactly\Nan rectangles area to add. Dialogue: 0,0:09:23.74,0:09:25.68,Default,,0000,0000,0000,,And this is going to\Nbe [INAUDIBLE], which Dialogue: 0,0:09:25.68,0:09:28.05,Default,,0000,0000,0000,,is the same everywhere. Dialogue: 0,0:09:28.05,0:09:35.73,Default,,0000,0000,0000,,In that case, I made\Nthe partition is equal. Dialogue: 0,0:09:35.73,0:09:39.19,Default,,0000,0000,0000,,So practically I have\Nthe same distance. Dialogue: 0,0:09:39.19,0:09:41.10,Default,,0000,0000,0000,,And what is this\Nlimit? [? Lim ?] Dialogue: 0,0:09:41.10,0:09:45.75,Default,,0000,0000,0000,,is going to be exactly integral\Nfrom a to b of f of x dx. Dialogue: 0,0:09:45.75,0:09:48.82,Default,,0000,0000,0000,,And I make a smile here,\Nand I say I'm very happy. Dialogue: 0,0:09:48.82,0:09:55.43,Default,,0000,0000,0000,,This is as a meaning is\Nthe area under the graph. Dialogue: 0,0:09:55.43,0:09:57.51,Default,,0000,0000,0000,,If-- well, I didn't\Nsay something. Dialogue: 0,0:09:57.51,0:10:01.39,Default,,0000,0000,0000,,If I want it to be\Npositive, otherwise it's Dialogue: 0,0:10:01.39,0:10:04.40,Default,,0000,0000,0000,,getting not to be the\Narea under the graph. Dialogue: 0,0:10:04.40,0:10:08.35,Default,,0000,0000,0000,,The integral will still\Nbe defined like that. Dialogue: 0,0:10:08.35,0:10:12.11,Default,,0000,0000,0000,,But what's going to happen if\NI have, for example, half of it Dialogue: 0,0:10:12.11,0:10:15.38,Default,,0000,0000,0000,,above and half of it below? Dialogue: 0,0:10:15.38,0:10:18.18,Default,,0000,0000,0000,,I'm going to get this,\Nand I'm going to get that. Dialogue: 0,0:10:18.18,0:10:23.25,Default,,0000,0000,0000,,And when I add them, I'm going\Nto get a negative answer, Dialogue: 0,0:10:23.25,0:10:26.50,Default,,0000,0000,0000,,because this is a negative\Narea, and that's a positive area Dialogue: 0,0:10:26.50,0:10:28.68,Default,,0000,0000,0000,,and they try to\Nannihilate each other. Dialogue: 0,0:10:28.68,0:10:32.01,Default,,0000,0000,0000,,But this guy under\Nthe water is stronger, Dialogue: 0,0:10:32.01,0:10:35.93,Default,,0000,0000,0000,,like an iceberg that's\N20% on tip of the water, Dialogue: 0,0:10:35.93,0:10:39.08,Default,,0000,0000,0000,,80% of the iceberg\Nis under the water. Dialogue: 0,0:10:39.08,0:10:39.83,Default,,0000,0000,0000,,So the same thing. Dialogue: 0,0:10:39.83,0:10:45.40,Default,,0000,0000,0000,,I'm going to get a negative\Nanswer in volume [INAUDIBLE]. Dialogue: 0,0:10:45.40,0:10:45.98,Default,,0000,0000,0000,,OK. Dialogue: 0,0:10:45.98,0:10:49.08,Default,,0000,0000,0000,,Now, we remember that\Nvery well, but now we Dialogue: 0,0:10:49.08,0:10:54.65,Default,,0000,0000,0000,,have to generalize this\Nthingy to something else. Dialogue: 0,0:10:54.65,0:10:57.47,Default,,0000,0000,0000,, Dialogue: 0,0:10:57.47,0:11:03.21,Default,,0000,0000,0000,,And I will give you\Na curvilinear domain. Dialogue: 0,0:11:03.21,0:11:04.04,Default,,0000,0000,0000,,Where shall I erase? Dialogue: 0,0:11:04.04,0:11:07.29,Default,,0000,0000,0000,,I don't know. Dialogue: 0,0:11:07.29,0:11:09.39,Default,,0000,0000,0000,,Here. Dialogue: 0,0:11:09.39,0:11:12.50,Default,,0000,0000,0000,,What if somebody gives you\Nthe image of a potatoe-- well, Dialogue: 0,0:11:12.50,0:11:13.38,Default,,0000,0000,0000,,I don't know. Dialogue: 0,0:11:13.38,0:11:14.65,Default,,0000,0000,0000,,Something. Dialogue: 0,0:11:14.65,0:11:15.69,Default,,0000,0000,0000,,A blob. Dialogue: 0,0:11:15.69,0:11:24.86,Default,,0000,0000,0000,,Some nice curvilinear domain--\Nand says, you know what? Dialogue: 0,0:11:24.86,0:11:29.90,Default,,0000,0000,0000,,I want to approximate the area\Nof this image, curvilinear Dialogue: 0,0:11:29.90,0:11:35.14,Default,,0000,0000,0000,,image, to the best\Nof my abilities. Dialogue: 0,0:11:35.14,0:11:42.37,Default,,0000,0000,0000,,And compute it, and eventually I\Nhave some weighted sum of that. Dialogue: 0,0:11:42.37,0:11:52.34,Default,,0000,0000,0000,,So if one would have\Nto compute the area, Dialogue: 0,0:11:52.34,0:11:55.97,Default,,0000,0000,0000,,it wouldn't be so hard,\Nbecause we would say, Dialogue: 0,0:11:55.97,0:12:05.12,Default,,0000,0000,0000,,OK, I have to\N"partition this domain Dialogue: 0,0:12:05.12,0:12:19.77,Default,,0000,0000,0000,,into small sections using\Na rectangular partition Dialogue: 0,0:12:19.77,0:12:31.45,Default,,0000,0000,0000,,or square partition." Dialogue: 0,0:12:31.45,0:12:32.10,Default,,0000,0000,0000,,And how? Dialogue: 0,0:12:32.10,0:12:34.80,Default,,0000,0000,0000,,Well, I'm going to--\Nyou have to imagine Dialogue: 0,0:12:34.80,0:12:41.19,Default,,0000,0000,0000,,that I have a bunch\Nof a grid, and I'm Dialogue: 0,0:12:41.19,0:12:43.18,Default,,0000,0000,0000,,partitioning the whole thing. Dialogue: 0,0:12:43.18,0:12:53.63,Default,,0000,0000,0000,, Dialogue: 0,0:12:53.63,0:12:55.73,Default,,0000,0000,0000,,And you say, wait a minute. Dialogue: 0,0:12:55.73,0:12:56.53,Default,,0000,0000,0000,,Wait a minute. Dialogue: 0,0:12:56.53,0:12:57.69,Default,,0000,0000,0000,,It's not so easy. Dialogue: 0,0:12:57.69,0:13:01.50,Default,,0000,0000,0000,,I mean, they are not all\Nthe same area, Magdalena. Dialogue: 0,0:13:01.50,0:13:05.84,Default,,0000,0000,0000,,Even if you tried to make these\Nequidistant in both directions, Dialogue: 0,0:13:05.84,0:13:07.88,Default,,0000,0000,0000,,look at this guy. Dialogue: 0,0:13:07.88,0:13:09.07,Default,,0000,0000,0000,,Look at that guy. Dialogue: 0,0:13:09.07,0:13:10.81,Default,,0000,0000,0000,,He's much bigger than that. Dialogue: 0,0:13:10.81,0:13:14.08,Default,,0000,0000,0000,,Look at this small\Nguy, and so on. Dialogue: 0,0:13:14.08,0:13:26.97,Default,,0000,0000,0000,,So we have to imagine that we\Nlook at the so-called normal Dialogue: 0,0:13:26.97,0:13:27.55,Default,,0000,0000,0000,,the partition. Dialogue: 0,0:13:27.55,0:13:34.07,Default,,0000,0000,0000,, Dialogue: 0,0:13:34.07,0:13:37.48,Default,,0000,0000,0000,,And let's say in the normal,\Nor the length of the partition, Dialogue: 0,0:13:37.48,0:13:38.99,Default,,0000,0000,0000,,is denoted like that. Dialogue: 0,0:13:38.99,0:13:41.03,Default,,0000,0000,0000,,We have to give that a meaning. Dialogue: 0,0:13:41.03,0:13:51.21,Default,,0000,0000,0000,,Well, let's say "this\Nis the highest diameter Dialogue: 0,0:13:51.21,0:14:04.15,Default,,0000,0000,0000,,for all subdomains\Nin the picture." Dialogue: 0,0:14:04.15,0:14:06.04,Default,,0000,0000,0000,,And you say, wait a minute. Dialogue: 0,0:14:06.04,0:14:08.09,Default,,0000,0000,0000,,But these subdomains\Nshould have names. Dialogue: 0,0:14:08.09,0:14:11.69,Default,,0000,0000,0000,,Well, they don't have names,\Nbut assume they have areas. Dialogue: 0,0:14:11.69,0:14:16.38,Default,,0000,0000,0000,,This would be-- I have to\Nfind a way to denote them Dialogue: 0,0:14:16.38,0:14:18.43,Default,,0000,0000,0000,,and be orderly. Dialogue: 0,0:14:18.43,0:14:32.34,Default,,0000,0000,0000,,A1, A2, A3, A4, A5, AN,\NAM, AN, stuff like that. Dialogue: 0,0:14:32.34,0:14:38.44,Default,,0000,0000,0000,,So practically I'm looking\Nat the highest diameter. Dialogue: 0,0:14:38.44,0:14:43.29,Default,,0000,0000,0000,,When I have a domain, I\Nlook at the largest instance Dialogue: 0,0:14:43.29,0:14:44.90,Default,,0000,0000,0000,,inside that domain. Dialogue: 0,0:14:44.90,0:14:47.05,Default,,0000,0000,0000,,So what would be the diameter? Dialogue: 0,0:14:47.05,0:14:50.11,Default,,0000,0000,0000,,The largest distance between\Ntwo points in that domain. Dialogue: 0,0:14:50.11,0:14:52.26,Default,,0000,0000,0000,,I'll call that the diameter. Dialogue: 0,0:14:52.26,0:14:52.76,Default,,0000,0000,0000,,OK. Dialogue: 0,0:14:52.76,0:14:57.55,Default,,0000,0000,0000,,I want that diameter to\Ngo got 0 in the limit. Dialogue: 0,0:14:57.55,0:15:03.36,Default,,0000,0000,0000,,So I want this partition\Nto go to 0 in the limit. Dialogue: 0,0:15:03.36,0:15:05.75,Default,,0000,0000,0000,,And that means I'm\N"shrinking" the pixels. Dialogue: 0,0:15:05.75,0:15:08.71,Default,,0000,0000,0000,, Dialogue: 0,0:15:08.71,0:15:11.18,Default,,0000,0000,0000,,"Shrinking" in\Nquotes, the pixels. Dialogue: 0,0:15:11.18,0:15:17.12,Default,,0000,0000,0000,, Dialogue: 0,0:15:17.12,0:15:20.71,Default,,0000,0000,0000,,How would I mimic\Nwhat I did here? Dialogue: 0,0:15:20.71,0:15:23.44,Default,,0000,0000,0000,,Well, it would be\Neasier to get the area. Dialogue: 0,0:15:23.44,0:15:29.41,Default,,0000,0000,0000,,In this case, I would have\Nsome sort of A sum limit. Dialogue: 0,0:15:29.41,0:15:30.10,Default,,0000,0000,0000,,I'm sorry. Dialogue: 0,0:15:30.10,0:15:36.34,Default,,0000,0000,0000,,The curvilinear\Narea of the domain. Dialogue: 0,0:15:36.34,0:15:40.57,Default,,0000,0000,0000,,Let's call it-- what\Ndo you want to call it? Dialogue: 0,0:15:40.57,0:15:45.30,Default,,0000,0000,0000,,D for domain--\Ninside the domain. Dialogue: 0,0:15:45.30,0:15:46.21,Default,,0000,0000,0000,,OK? Dialogue: 0,0:15:46.21,0:15:48.48,Default,,0000,0000,0000,,This whole thing would be what? Dialogue: 0,0:15:48.48,0:16:01.24,Default,,0000,0000,0000,,Would be limit of summation of,\Nlet's say, limit of what kind? Dialogue: 0,0:16:01.24,0:16:04.00,Default,,0000,0000,0000,,k from 1 to n. Dialogue: 0,0:16:04.00,0:16:06.27,Default,,0000,0000,0000,,Limit n goes to infinity. Dialogue: 0,0:16:06.27,0:16:16.27,Default,,0000,0000,0000,,K from 1 to n of\Nthese tiny A sub k's, Dialogue: 0,0:16:16.27,0:16:17.40,Default,,0000,0000,0000,,areas of the subdomain. Dialogue: 0,0:16:17.40,0:16:24.52,Default,,0000,0000,0000,, Dialogue: 0,0:16:24.52,0:16:25.29,Default,,0000,0000,0000,,Wait a minute. Dialogue: 0,0:16:25.29,0:16:29.97,Default,,0000,0000,0000,,But you say, but what if\NI want something else? Dialogue: 0,0:16:29.97,0:16:34.08,Default,,0000,0000,0000,,Like, I'm going to\Nbuild some geography. Dialogue: 0,0:16:34.08,0:16:35.11,Default,,0000,0000,0000,,This is the domain. Dialogue: 0,0:16:35.11,0:16:38.56,Default,,0000,0000,0000,,That's something like\Non a map, and I'm Dialogue: 0,0:16:38.56,0:16:40.67,Default,,0000,0000,0000,,going to build a\Nmountain on top of it. Dialogue: 0,0:16:40.67,0:16:43.40,Default,,0000,0000,0000,,I'll take some Play-Do,\NI'll take some Play-Do, Dialogue: 0,0:16:43.40,0:16:46.27,Default,,0000,0000,0000,,and I'm going to\Nmodel some geography. Dialogue: 0,0:16:46.27,0:16:47.39,Default,,0000,0000,0000,,And you say, wait a minute. Dialogue: 0,0:16:47.39,0:16:49.39,Default,,0000,0000,0000,,Do you make mountains? Dialogue: 0,0:16:49.39,0:16:52.15,Default,,0000,0000,0000,,I'm afraid to make Rocky\NMountains, because they Dialogue: 0,0:16:52.15,0:16:55.85,Default,,0000,0000,0000,,may have points where the\Nfunction is not smooth. Dialogue: 0,0:16:55.85,0:16:58.31,Default,,0000,0000,0000,,If I don't have\Nderivative at the peak, Dialogue: 0,0:16:58.31,0:17:01.26,Default,,0000,0000,0000,,them I'm in trouble, in general. Dialogue: 0,0:17:01.26,0:17:03.02,Default,,0000,0000,0000,,Although you say,\Nwell, but the function Dialogue: 0,0:17:03.02,0:17:04.74,Default,,0000,0000,0000,,has to be only continuous. Dialogue: 0,0:17:04.74,0:17:05.26,Default,,0000,0000,0000,,I know. Dialogue: 0,0:17:05.26,0:17:05.76,Default,,0000,0000,0000,,I know. Dialogue: 0,0:17:05.76,0:17:09.98,Default,,0000,0000,0000,,But I don't want any kind\Nof really nasty singularity Dialogue: 0,0:17:09.98,0:17:12.30,Default,,0000,0000,0000,,where I can have a\Ncrack in the mountain Dialogue: 0,0:17:12.30,0:17:15.83,Default,,0000,0000,0000,,or a well or\Nsomething like that. Dialogue: 0,0:17:15.83,0:17:18.59,Default,,0000,0000,0000,,So I assume the\Ngeography to be smooth, Dialogue: 0,0:17:18.59,0:17:21.42,Default,,0000,0000,0000,,the function of\N[INAUDIBLE] is continuous, Dialogue: 0,0:17:21.42,0:17:23.31,Default,,0000,0000,0000,,and the picture\Nshould look something Dialogue: 0,0:17:23.31,0:17:27.65,Default,,0000,0000,0000,,like-- let's see\Nif I can do that. Dialogue: 0,0:17:27.65,0:17:33.96,Default,,0000,0000,0000,, Dialogue: 0,0:17:33.96,0:17:38.17,Default,,0000,0000,0000,,The projection, the\Nshadow of this geography, Dialogue: 0,0:17:38.17,0:17:43.40,Default,,0000,0000,0000,,would be the domain, [? D. ?]\NAnd this is equal, f of x what? Dialogue: 0,0:17:43.40,0:17:44.55,Default,,0000,0000,0000,,You say, what? Dialogue: 0,0:17:44.55,0:17:46.38,Default,,0000,0000,0000,,Magdalena, I don't understand. Dialogue: 0,0:17:46.38,0:17:51.72,Default,,0000,0000,0000,,The exact shadow of this fellow\Nwhere I have the sun on top Dialogue: 0,0:17:51.72,0:17:54.36,Default,,0000,0000,0000,,here-- that's the sun. Dialogue: 0,0:17:54.36,0:17:59.33,Default,,0000,0000,0000,,Spring is coming-- the shade\Nis the plain, or domain, x, y. Dialogue: 0,0:17:59.33,0:18:03.28,Default,,0000,0000,0000,,I take all my points in x, y. Dialogue: 0,0:18:03.28,0:18:05.63,Default,,0000,0000,0000,,I mean, I take really\Nall my points in x, y, Dialogue: 0,0:18:05.63,0:18:10.36,Default,,0000,0000,0000,,and the value of the altitude\Non this geography at the point Dialogue: 0,0:18:10.36,0:18:13.68,Default,,0000,0000,0000,,x, y would be z\Nequals f of x, y. Dialogue: 0,0:18:13.68,0:18:20.78,Default,,0000,0000,0000,,And somebody's asking me, OK,\Nif this would be a can of Coke, Dialogue: 0,0:18:20.78,0:18:23.90,Default,,0000,0000,0000,,it would be easy to\Ncompute the volume, right? Dialogue: 0,0:18:23.90,0:18:27.68,Default,,0000,0000,0000,,Practically you have a\Nconstant altitude everywhere, Dialogue: 0,0:18:27.68,0:18:30.15,Default,,0000,0000,0000,,and you have the area of\Nthe base times the height, Dialogue: 0,0:18:30.15,0:18:32.75,Default,,0000,0000,0000,,and that's your volume. Dialogue: 0,0:18:32.75,0:18:39.46,Default,,0000,0000,0000,,But what if somebody asks you to\Nfind the volume under the hat? Dialogue: 0,0:18:39.46,0:18:47.12,Default,,0000,0000,0000,,"Find the volume\Nundo this graph." Dialogue: 0,0:18:47.12,0:18:51.03,Default,,0000,0000,0000,,STUDENT: I would take it\Nmore as two functions. Dialogue: 0,0:18:51.03,0:18:53.47,Default,,0000,0000,0000,,So the top line would\Nbe the one function, Dialogue: 0,0:18:53.47,0:18:55.43,Default,,0000,0000,0000,,and the bottom line would\Nbe another function. Dialogue: 0,0:18:55.43,0:18:58.78,Default,,0000,0000,0000,,So if you take the volume of the\Ntop function minus the volume Dialogue: 0,0:18:58.78,0:19:00.32,Default,,0000,0000,0000,,of the bottom\Nfunction, it'd give you Dialogue: 0,0:19:00.32,0:19:02.78,Default,,0000,0000,0000,,the total volume of the object. Dialogue: 0,0:19:02.78,0:19:05.57,Default,,0000,0000,0000,,PROFESSOR: And actually,\NI want the total volume Dialogue: 0,0:19:05.57,0:19:07.66,Default,,0000,0000,0000,,above the sea level. Dialogue: 0,0:19:07.66,0:19:12.75,Default,,0000,0000,0000,,So I'm going to--\Nsometimes I can take it up Dialogue: 0,0:19:12.75,0:19:16.12,Default,,0000,0000,0000,,to a certain level where-- let's\Nsay the mountain is up to here, Dialogue: 0,0:19:16.12,0:19:18.49,Default,,0000,0000,0000,,and I want it only up to here. Dialogue: 0,0:19:18.49,0:19:22.22,Default,,0000,0000,0000,,So I want everything,\Nincluding the-- the walls Dialogue: 0,0:19:22.22,0:19:24.14,Default,,0000,0000,0000,,would be cylindrical. Dialogue: 0,0:19:24.14,0:19:24.72,Default,,0000,0000,0000,,STUDENT: Yeah. Dialogue: 0,0:19:24.72,0:19:26.14,Default,,0000,0000,0000,,PROFESSOR: If I\Nwant all the volume, Dialogue: 0,0:19:26.14,0:19:27.76,Default,,0000,0000,0000,,that's going to be\Na little bit easier. Dialogue: 0,0:19:27.76,0:19:29.47,Default,,0000,0000,0000,,Let's see why. Dialogue: 0,0:19:29.47,0:19:31.70,Default,,0000,0000,0000,,I will have limit. Dialogue: 0,0:19:31.70,0:19:35.63,Default,,0000,0000,0000,,The idea is, as you\Nsaid very well, limit. Dialogue: 0,0:19:35.63,0:19:37.10,Default,,0000,0000,0000,,n goes to infinity. Dialogue: 0,0:19:37.10,0:19:42.15,Default,,0000,0000,0000,,A sum k from 1 to n. Dialogue: 0,0:19:42.15,0:19:44.88,Default,,0000,0000,0000,,And what kind of\Npartition can I build? Dialogue: 0,0:19:44.88,0:19:47.64,Default,,0000,0000,0000,,I'll take the\Nline, and I'll say, Dialogue: 0,0:19:47.64,0:19:53.03,Default,,0000,0000,0000,,I'll build myself\Na partition with a, Dialogue: 0,0:19:53.03,0:19:57.51,Default,,0000,0000,0000,,let's say, the\Ntypical domain, AK. Dialogue: 0,0:19:57.51,0:20:00.96,Default,,0000,0000,0000,,I have A1, A2 A3, A4, AK, AN. Dialogue: 0,0:20:00.96,0:20:03.47,Default,,0000,0000,0000,,How may of those little domains? Dialogue: 0,0:20:03.47,0:20:04.46,Default,,0000,0000,0000,,AN. Dialogue: 0,0:20:04.46,0:20:07.94,Default,,0000,0000,0000,,That will be all the\Nlittle subdomains Dialogue: 0,0:20:07.94,0:20:12.41,Default,,0000,0000,0000,,inside the green curve. Dialogue: 0,0:20:12.41,0:20:14.51,Default,,0000,0000,0000,,The green loop. Dialogue: 0,0:20:14.51,0:20:16.66,Default,,0000,0000,0000,,In that case, what do I do? Dialogue: 0,0:20:16.66,0:20:24.26,Default,,0000,0000,0000,,For each of these guys, I go\Nup, and I go, oh, my god, this Dialogue: 0,0:20:24.26,0:20:27.38,Default,,0000,0000,0000,,looks like a skyscraper,\Nbut the corners, Dialogue: 0,0:20:27.38,0:20:29.08,Default,,0000,0000,0000,,when I go through\Nthis surface, are Dialogue: 0,0:20:29.08,0:20:30.90,Default,,0000,0000,0000,,in the different dimensions. Dialogue: 0,0:20:30.90,0:20:32.43,Default,,0000,0000,0000,,What am I going to do? Dialogue: 0,0:20:32.43,0:20:34.76,Default,,0000,0000,0000,,That forces me to\Nbuild a skyscraper Dialogue: 0,0:20:34.76,0:20:38.57,Default,,0000,0000,0000,,by thinking I take a\Npoint in the domain, Dialogue: 0,0:20:38.57,0:20:45.11,Default,,0000,0000,0000,,I go up until that hits the\Nsurface, pinches the surface, Dialogue: 0,0:20:45.11,0:20:47.62,Default,,0000,0000,0000,,and this is the\Naltitude that I'm going Dialogue: 0,0:20:47.62,0:20:50.28,Default,,0000,0000,0000,,to select for my skyscraper. Dialogue: 0,0:20:50.28,0:20:54.49,Default,,0000,0000,0000,,And here I'm going to have\Nanother skyscraper, and here Dialogue: 0,0:20:54.49,0:20:57.80,Default,,0000,0000,0000,,another one and another one,\Nso practically it's dense. Dialogue: 0,0:20:57.80,0:21:02.45,Default,,0000,0000,0000,,I have a skyscraper next\Nto the other or a less like Dialogue: 0,0:21:02.45,0:21:02.96,Default,,0000,0000,0000,,[INAUDIBLE]. Dialogue: 0,0:21:02.96,0:21:06.55,Default,,0000,0000,0000,,Not so many gaps\Nin certain areas. Dialogue: 0,0:21:06.55,0:21:12.64,Default,,0000,0000,0000,,So I'm going to say\Nf of x kappa star. Dialogue: 0,0:21:12.64,0:21:18.68,Default,,0000,0000,0000,,Now those would be the\Naltitudes of the buildings. Dialogue: 0,0:21:18.68,0:21:20.92,Default,,0000,0000,0000,,Magdalena, you don't\Nknow how to spell. Dialogue: 0,0:21:20.92,0:21:27.97,Default,,0000,0000,0000,,Altitudes of the buildings. Dialogue: 0,0:21:27.97,0:21:31.86,Default,,0000,0000,0000,, Dialogue: 0,0:21:31.86,0:21:32.70,Default,,0000,0000,0000,,What are they? Dialogue: 0,0:21:32.70,0:21:34.06,Default,,0000,0000,0000,,Parallel [INAUDIBLE] by P's. Dialogue: 0,0:21:34.06,0:21:36.23,Default,,0000,0000,0000,,Can you say parallel by P? Dialogue: 0,0:21:36.23,0:21:37.04,Default,,0000,0000,0000,,OK. Dialogue: 0,0:21:37.04,0:21:40.72,Default,,0000,0000,0000,,[INAUDIBLE] what. Dialogue: 0,0:21:40.72,0:21:49.72,Default,,0000,0000,0000,,Ak where Ak will be the basis\Nof the area of the basis. Dialogue: 0,0:21:49.72,0:21:51.67,Default,,0000,0000,0000,,is of my building. Dialogue: 0,0:21:51.67,0:21:54.86,Default,,0000,0000,0000,, Dialogue: 0,0:21:54.86,0:21:55.49,Default,,0000,0000,0000,,OK. Dialogue: 0,0:21:55.49,0:21:59.29,Default,,0000,0000,0000,,The green part will\Nbe the flat area Dialogue: 0,0:21:59.29,0:22:03.58,Default,,0000,0000,0000,,of the floor of the skyscraper. Dialogue: 0,0:22:03.58,0:22:05.60,Default,,0000,0000,0000,,Is this hard? Dialogue: 0,0:22:05.60,0:22:06.83,Default,,0000,0000,0000,,Gosh, yes. Dialogue: 0,0:22:06.83,0:22:12.71,Default,,0000,0000,0000,,If you want to do it by\Nhand and take the limit Dialogue: 0,0:22:12.71,0:22:15.62,Default,,0000,0000,0000,,you would really kill\Nyourself in the process. Dialogue: 0,0:22:15.62,0:22:17.42,Default,,0000,0000,0000,,This is how you introduce it. Dialogue: 0,0:22:17.42,0:22:21.90,Default,,0000,0000,0000,,You can prove this limit exists,\Nand you can prove that limits Dialogue: 0,0:22:21.90,0:22:32.80,Default,,0000,0000,0000,,exist and will be the volume of\Nthe region under the geography Dialogue: 0,0:22:32.80,0:22:38.76,Default,,0000,0000,0000,,z equals f of x,y and\Nabove the sea level. Dialogue: 0,0:22:38.76,0:22:43.23,Default,,0000,0000,0000,, Dialogue: 0,0:22:43.23,0:22:46.78,Default,,0000,0000,0000,,The seal level\Nmeaning z equals z. Dialogue: 0,0:22:46.78,0:22:49.72,Default,,0000,0000,0000,,STUDENT: What's under a of k? Dialogue: 0,0:22:49.72,0:22:50.30,Default,,0000,0000,0000,,PROFESSOR: Ak. Dialogue: 0,0:22:50.30,0:22:51.47,Default,,0000,0000,0000,,STUDENT: What is [INAUDIBLE] Dialogue: 0,0:22:51.47,0:22:54.08,Default,,0000,0000,0000,,PROFESSOR: Volume of the region. Dialogue: 0,0:22:54.08,0:22:55.74,Default,,0000,0000,0000,,STUDENT: Oh, I know,\Nlike what under it? Dialogue: 0,0:22:55.74,0:22:56.41,Default,,0000,0000,0000,,PROFESSOR: Here? Dialogue: 0,0:22:56.41,0:22:57.26,Default,,0000,0000,0000,,STUDENT: No, up. Dialogue: 0,0:22:57.26,0:22:58.05,Default,,0000,0000,0000,,PROFESSOR: Here? Dialogue: 0,0:22:58.05,0:22:58.59,Default,,0000,0000,0000,,STUDENT: Yes. Dialogue: 0,0:22:58.59,0:23:00.60,Default,,0000,0000,0000,,PROFESSOR: Area of the\Nbasis of a building. Dialogue: 0,0:23:00.60,0:23:01.56,Default,,0000,0000,0000,,STUDENT: Oh, the basis. Dialogue: 0,0:23:01.56,0:23:04.07,Default,,0000,0000,0000,,PROFESSOR: So practically\Nthis green thingy Dialogue: 0,0:23:04.07,0:23:10.73,Default,,0000,0000,0000,,is a basis like the base rate. Dialogue: 0,0:23:10.73,0:23:13.40,Default,,0000,0000,0000,,How large is the basement\Nof that building. Dialogue: 0,0:23:13.40,0:23:16.24,Default,,0000,0000,0000,,Ak. Dialogue: 0,0:23:16.24,0:23:18.47,Default,,0000,0000,0000,,Now how am I going\Nto write this? Dialogue: 0,0:23:18.47,0:23:19.40,Default,,0000,0000,0000,,This is something new. Dialogue: 0,0:23:19.40,0:23:27.28,Default,,0000,0000,0000,,We have to invent a notion\Nfor it, and since it's Ak, Dialogue: 0,0:23:27.28,0:23:31.39,Default,,0000,0000,0000,,looks more or less like\Na square or a rectangle. Dialogue: 0,0:23:31.39,0:23:35.12,Default,,0000,0000,0000,,You think, well, wouldn't--\NOK, if it's a rectangle, Dialogue: 0,0:23:35.12,0:23:38.41,Default,,0000,0000,0000,,I know I'm going to get\Ndelta x and delta y right? Dialogue: 0,0:23:38.41,0:23:41.54,Default,,0000,0000,0000,,The width times the height,\Nwhatever those two dimensions. Dialogue: 0,0:23:41.54,0:23:42.40,Default,,0000,0000,0000,,It makes sense. Dialogue: 0,0:23:42.40,0:23:44.73,Default,,0000,0000,0000,,But what if I have\Nthis domain that's Dialogue: 0,0:23:44.73,0:23:47.27,Default,,0000,0000,0000,,curvilinear or that\Ndomain or that domain. Dialogue: 0,0:23:47.27,0:23:49.69,Default,,0000,0000,0000,,Of course, the diameter\Nof such a domain Dialogue: 0,0:23:49.69,0:23:54.00,Default,,0000,0000,0000,,is less than the diameter of the\Npartition, so I'm very happy. Dialogue: 0,0:23:54.00,0:23:55.87,Default,,0000,0000,0000,,The highest diameter,\Nsay I can get it here, Dialogue: 0,0:23:55.87,0:23:59.42,Default,,0000,0000,0000,,and this is shrinking\Nto zero, and pixels Dialogue: 0,0:23:59.42,0:24:01.58,Default,,0000,0000,0000,,are shrinking to zero. Dialogue: 0,0:24:01.58,0:24:05.41,Default,,0000,0000,0000,,But what am I going to\Ndo about those guys? Dialogue: 0,0:24:05.41,0:24:09.72,Default,,0000,0000,0000,,Well, you can assume that\NI am still approximating Dialogue: 0,0:24:09.72,0:24:14.75,Default,,0000,0000,0000,,with some squares and as\Nthe pixels are getting Dialogue: 0,0:24:14.75,0:24:17.32,Default,,0000,0000,0000,,to be many, many,\Nmany more, it doesn't Dialogue: 0,0:24:17.32,0:24:19.52,Default,,0000,0000,0000,,matter that I'm doing this. Dialogue: 0,0:24:19.52,0:24:21.80,Default,,0000,0000,0000,,Let me show you what I'm doing. Dialogue: 0,0:24:21.80,0:24:29.07,Default,,0000,0000,0000,,So on the floor, on the-- this\Nis the city floor, whatever. Dialogue: 0,0:24:29.07,0:24:32.18,Default,,0000,0000,0000,,What we do in practice,\Nwe approximate that Dialogue: 0,0:24:32.18,0:24:42.27,Default,,0000,0000,0000,,like on the graphing paper\Nwith tiny square domains, Dialogue: 0,0:24:42.27,0:24:48.80,Default,,0000,0000,0000,,and we call them delta Ak will\Nbe delta Sk times delta Yk, Dialogue: 0,0:24:48.80,0:24:53.64,Default,,0000,0000,0000,,and I tried to make it a uniform\Npartition as much as I can. Dialogue: 0,0:24:53.64,0:24:56.23,Default,,0000,0000,0000,,Now as the number of\Npixels goes to infinity Dialogue: 0,0:24:56.23,0:24:59.36,Default,,0000,0000,0000,,and those pixels will\Nbecome smaller and smaller, Dialogue: 0,0:24:59.36,0:25:04.24,Default,,0000,0000,0000,,it doesn't there that the actual\Ncontour of your Riemann sum Dialogue: 0,0:25:04.24,0:25:07.09,Default,,0000,0000,0000,,will look like graphing paper. Dialogue: 0,0:25:07.09,0:25:10.41,Default,,0000,0000,0000,,It will get refined, more\Nrefined, more refined, smoother Dialogue: 0,0:25:10.41,0:25:12.71,Default,,0000,0000,0000,,and smoother, and\Nit's going to be Dialogue: 0,0:25:12.71,0:25:17.89,Default,,0000,0000,0000,,really close to the ideal\Nimage, which is a curve. Dialogue: 0,0:25:17.89,0:25:20.34,Default,,0000,0000,0000,,So as that end goes\Nto infinity, you're Dialogue: 0,0:25:20.34,0:25:24.94,Default,,0000,0000,0000,,not going to see this-- what is\Nthis called-- zig zag thingy. Dialogue: 0,0:25:24.94,0:25:25.81,Default,,0000,0000,0000,,Not anymore. Dialogue: 0,0:25:25.81,0:25:31.95,Default,,0000,0000,0000,,The zig zag thingy will go into\Nthe limit to the green curve. Dialogue: 0,0:25:31.95,0:25:34.57,Default,,0000,0000,0000,,This is what the\Npixels are about. Dialogue: 0,0:25:34.57,0:25:38.30,Default,,0000,0000,0000,,This is how our\Nlife changed a lot. Dialogue: 0,0:25:38.30,0:25:39.07,Default,,0000,0000,0000,,OK? Dialogue: 0,0:25:39.07,0:25:39.84,Default,,0000,0000,0000,,All right. Dialogue: 0,0:25:39.84,0:25:41.85,Default,,0000,0000,0000,,Now good. Dialogue: 0,0:25:41.85,0:25:45.02,Default,,0000,0000,0000,,How am I going\Ncompute this thing? Dialogue: 0,0:25:45.02,0:25:47.73,Default,,0000,0000,0000,, Dialogue: 0,0:25:47.73,0:25:52.43,Default,,0000,0000,0000,,Well, I don't know, but let\Nme give it a name first. Dialogue: 0,0:25:52.43,0:25:56.13,Default,,0000,0000,0000,,It's going to be double\Nintegral over-- what Dialogue: 0,0:25:56.13,0:25:58.96,Default,,0000,0000,0000,,do want the floor to be called? Dialogue: 0,0:25:58.96,0:26:02.27,Default,,0000,0000,0000,, Dialogue: 0,0:26:02.27,0:26:04.44,Default,,0000,0000,0000,,We called d domain before. Dialogue: 0,0:26:04.44,0:26:06.46,Default,,0000,0000,0000,,What should I call this? Dialogue: 0,0:26:06.46,0:26:09.87,Default,,0000,0000,0000,,Big D. Not round. Dialogue: 0,0:26:09.87,0:26:13.87,Default,,0000,0000,0000,,Over D. That's the\Nfloor, the foundation Dialogue: 0,0:26:13.87,0:26:16.66,Default,,0000,0000,0000,,of the whole city-- of\Nthe whole area of the city Dialogue: 0,0:26:16.66,0:26:18.05,Default,,0000,0000,0000,,that I'm looking at. Dialogue: 0,0:26:18.05,0:26:28.19,Default,,0000,0000,0000,,Then I have f of xy,\Nda, and what is this? Dialogue: 0,0:26:28.19,0:26:29.64,Default,,0000,0000,0000,,This is exactly that. Dialogue: 0,0:26:29.64,0:26:35.37,Default,,0000,0000,0000,,It's the limit of sum of the--\Nwhat is the difference here? Dialogue: 0,0:26:35.37,0:26:37.04,Default,,0000,0000,0000,,You say, wait a\Nminute, Magdalena, Dialogue: 0,0:26:37.04,0:26:40.16,Default,,0000,0000,0000,,but I think I don't\Nunderstand what you did. Dialogue: 0,0:26:40.16,0:26:43.93,Default,,0000,0000,0000,,You tried to copy the\Nconcept from here, Dialogue: 0,0:26:43.93,0:26:47.63,Default,,0000,0000,0000,,but you forgot you have a\Nfunction of two variables. Dialogue: 0,0:26:47.63,0:26:52.56,Default,,0000,0000,0000,,In that case, this mister,\Nwhoever it is that goes up Dialogue: 0,0:26:52.56,0:26:57.82,Default,,0000,0000,0000,,is not xk, it's XkYk. Dialogue: 0,0:26:57.82,0:27:02.06,Default,,0000,0000,0000,,So I have two variables--\Ndoesn't change anything Dialogue: 0,0:27:02.06,0:27:03.88,Default,,0000,0000,0000,,for the couple. Dialogue: 0,0:27:03.88,0:27:08.19,Default,,0000,0000,0000,,This couple represents a\Npoint on the skyscraper Dialogue: 0,0:27:08.19,0:27:16.32,Default,,0000,0000,0000,,so that when I go up, I hit the\Nroof with this exact altitude. Dialogue: 0,0:27:16.32,0:27:19.67,Default,,0000,0000,0000,,So what is the double integral\Nof a continuous function Dialogue: 0,0:27:19.67,0:27:26.48,Default,,0000,0000,0000,,f of x and y, two variables,\Nwith respect to area level. Dialogue: 0,0:27:26.48,0:27:32.78,Default,,0000,0000,0000,,Well, it's going to be just\Nthe limit of this huge thing. Dialogue: 0,0:27:32.78,0:27:37.67,Default,,0000,0000,0000,,In fact, it's how\Ndo we compute it? Dialogue: 0,0:27:37.67,0:27:40.58,Default,,0000,0000,0000,,Let's see how we\Ncompute it in practice. Dialogue: 0,0:27:40.58,0:27:42.82,Default,,0000,0000,0000,,It shouldn't be a big deal. Dialogue: 0,0:27:42.82,0:27:56.08,Default,,0000,0000,0000,, Dialogue: 0,0:27:56.08,0:27:57.97,Default,,0000,0000,0000,,What if I have a\Nrectangular domain, Dialogue: 0,0:27:57.97,0:28:00.86,Default,,0000,0000,0000,,and that's going to\Nmake my life easier. Dialogue: 0,0:28:00.86,0:28:05.84,Default,,0000,0000,0000,,I'm going to have a\Nrectangular domain in plane, Dialogue: 0,0:28:05.84,0:28:07.88,Default,,0000,0000,0000,,and which one is the x-axis? Dialogue: 0,0:28:07.88,0:28:09.55,Default,,0000,0000,0000,,This one. Dialogue: 0,0:28:09.55,0:28:14.75,Default,,0000,0000,0000,,From A to B, I have the x\Nmoving between a and Mr. y Dialogue: 0,0:28:14.75,0:28:19.55,Default,,0000,0000,0000,,says, I'm going to\Nbe between c and d. Dialogue: 0,0:28:19.55,0:28:23.18,Default,,0000,0000,0000,,C is here, and d is here. Dialogue: 0,0:28:23.18,0:28:28.48,Default,,0000,0000,0000,,So this is going to be\Nthe so-called rectangle Dialogue: 0,0:28:28.48,0:28:37.15,Default,,0000,0000,0000,,a, b cross c, d meaning\Nthe set of all the pairs-- Dialogue: 0,0:28:37.15,0:28:42.02,Default,,0000,0000,0000,,or the couples xy-- inside\Nit, what does it mean? Dialogue: 0,0:28:42.02,0:28:45.56,Default,,0000,0000,0000,,x, y you playing with\Nthe property there. Dialogue: 0,0:28:45.56,0:28:48.51,Default,,0000,0000,0000,,X is between a and b, thank god. Dialogue: 0,0:28:48.51,0:28:50.21,Default,,0000,0000,0000,,It's easy. Dialogue: 0,0:28:50.21,0:28:53.53,Default,,0000,0000,0000,,And y must be between\Nc and d, also easy. Dialogue: 0,0:28:53.53,0:28:57.88,Default,,0000,0000,0000,,A, b, c, d are fixed real\Nnumbers in this order. Dialogue: 0,0:28:57.88,0:29:02.05,Default,,0000,0000,0000,,A is less than b, and c is less. Dialogue: 0,0:29:02.05,0:29:05.03,Default,,0000,0000,0000,,And we have this\Ngeography on top, Dialogue: 0,0:29:05.03,0:29:09.08,Default,,0000,0000,0000,,and I will tell you\Nwhat it looks like. Dialogue: 0,0:29:09.08,0:29:13.72,Default,,0000,0000,0000,,I'm going to try and draw\Nsome beautiful geography. Dialogue: 0,0:29:13.72,0:29:19.62,Default,,0000,0000,0000,,And now I'm thinking\Nof my son, who is 10. Dialogue: 0,0:29:19.62,0:29:23.59,Default,,0000,0000,0000,,He played with this kind of toy\Nthat was exactly this color, Dialogue: 0,0:29:23.59,0:29:25.80,Default,,0000,0000,0000,,lime, and it had needles. Dialogue: 0,0:29:25.80,0:29:27.96,Default,,0000,0000,0000,,Do you guys remember that toy? Dialogue: 0,0:29:27.96,0:29:30.91,Default,,0000,0000,0000,,I am sure you're young\Nenough to remember that. Dialogue: 0,0:29:30.91,0:29:35.33,Default,,0000,0000,0000,,You have your palm like that,\Nand you see this square thingy, Dialogue: 0,0:29:35.33,0:29:37.43,Default,,0000,0000,0000,,and it's all made\Nof needles that Dialogue: 0,0:29:37.43,0:29:40.81,Default,,0000,0000,0000,,look like thin,\Ntiny skyscrapers, Dialogue: 0,0:29:40.81,0:29:47.40,Default,,0000,0000,0000,,and you push through and all\Nthose needles go up and take Dialogue: 0,0:29:47.40,0:29:49.66,Default,,0000,0000,0000,,the shape of your hand. Dialogue: 0,0:29:49.66,0:29:51.92,Default,,0000,0000,0000,,And of course, he would\Nput it on his face, Dialogue: 0,0:29:51.92,0:29:54.35,Default,,0000,0000,0000,,and you could see\Nhis face and so on. Dialogue: 0,0:29:54.35,0:29:56.10,Default,,0000,0000,0000,,But what is that? Dialogue: 0,0:29:56.10,0:29:59.89,Default,,0000,0000,0000,,That's exactly the Riemann\Nsum, the Riemann approximation, Dialogue: 0,0:29:59.89,0:30:02.20,Default,,0000,0000,0000,,because if you think of\Nall those needles or tiny-- Dialogue: 0,0:30:02.20,0:30:07.30,Default,,0000,0000,0000,,what are they, like\Nthe tiny skyscrapers-- Dialogue: 0,0:30:07.30,0:30:11.61,Default,,0000,0000,0000,,the sum of the them approximates\Nthe curvilinear shape. Dialogue: 0,0:30:11.61,0:30:15.89,Default,,0000,0000,0000,,If you put that over your face,\Nyour face is nice and smooth, Dialogue: 0,0:30:15.89,0:30:19.19,Default,,0000,0000,0000,,curvilinear except for\Na few single areas, Dialogue: 0,0:30:19.19,0:30:24.60,Default,,0000,0000,0000,,but if you actually\Nlook at that needle Dialogue: 0,0:30:24.60,0:30:27.64,Default,,0000,0000,0000,,thingy that is\Ngiving the figure, Dialogue: 0,0:30:27.64,0:30:30.06,Default,,0000,0000,0000,,you recognize the figure. Dialogue: 0,0:30:30.06,0:30:33.56,Default,,0000,0000,0000,,It's like a pattern recognition,\Nbut it's not your face. Dialogue: 0,0:30:33.56,0:30:34.91,Default,,0000,0000,0000,,I mean it is and it's not. Dialogue: 0,0:30:34.91,0:30:39.22,Default,,0000,0000,0000,,It's an approximation of\Nyour face, a very rough face. Dialogue: 0,0:30:39.22,0:30:42.14,Default,,0000,0000,0000,,You have to take that\Nrough model of your face Dialogue: 0,0:30:42.14,0:30:43.85,Default,,0000,0000,0000,,and smooth it out. Dialogue: 0,0:30:43.85,0:30:44.45,Default,,0000,0000,0000,,How? Dialogue: 0,0:30:44.45,0:30:49.93,Default,,0000,0000,0000,,By passing to the\Nlimit, and this is what Dialogue: 0,0:30:49.93,0:30:52.24,Default,,0000,0000,0000,,animation is doing actually. Dialogue: 0,0:30:52.24,0:30:55.92,Default,,0000,0000,0000,,On top of that you want this\Nto have some other properties-- Dialogue: 0,0:30:55.92,0:31:01.25,Default,,0000,0000,0000,,illumination of some sort--\Nlight coming from what angle. Dialogue: 0,0:31:01.25,0:31:04.91,Default,,0000,0000,0000,,That is all rendering\Ntechniques are actually Dialogue: 0,0:31:04.91,0:31:06.81,Default,,0000,0000,0000,,applied mathematics. Dialogue: 0,0:31:06.81,0:31:09.55,Default,,0000,0000,0000,,In animation, the\Npeople who programmed Dialogue: 0,0:31:09.55,0:31:12.94,Default,,0000,0000,0000,,Toy Story-- that\Nwas a long time ago, Dialogue: 0,0:31:12.94,0:31:16.76,Default,,0000,0000,0000,,but everything that\Ncame after Toy Story 2 Dialogue: 0,0:31:16.76,0:31:20.55,Default,,0000,0000,0000,,was based on mathematical\Nrendering techniques. Dialogue: 0,0:31:20.55,0:31:23.82,Default,,0000,0000,0000,,Everything based on\Nthe notion of length. Dialogue: 0,0:31:23.82,0:31:25.08,Default,,0000,0000,0000,,All right. Dialogue: 0,0:31:25.08,0:31:28.01,Default,,0000,0000,0000,,So the way we compute\Nthis in practice Dialogue: 0,0:31:28.01,0:31:31.08,Default,,0000,0000,0000,,is going to be very simple,\Nbecause you're going to think, Dialogue: 0,0:31:31.08,0:31:33.33,Default,,0000,0000,0000,,how am I going to do the\Nrectangle for the rectangles? Dialogue: 0,0:31:33.33,0:31:35.80,Default,,0000,0000,0000,,That'll be very easy. Dialogue: 0,0:31:35.80,0:31:43.57,Default,,0000,0000,0000,,I split the rectangle perfectly\Ninto other tiny rectangle. Dialogue: 0,0:31:43.57,0:31:46.89,Default,,0000,0000,0000,,Every rectangle will\Nhave the same dimension. Dialogue: 0,0:31:46.89,0:31:48.67,Default,,0000,0000,0000,,Delta x and delta y. Dialogue: 0,0:31:48.67,0:31:51.52,Default,,0000,0000,0000,, Dialogue: 0,0:31:51.52,0:31:53.04,Default,,0000,0000,0000,,Does it makes sense? Dialogue: 0,0:31:53.04,0:31:55.92,Default,,0000,0000,0000,,So practically when\NI go to the limit, Dialogue: 0,0:31:55.92,0:32:02.22,Default,,0000,0000,0000,,I have summation f\Nof xk star, yk star Dialogue: 0,0:32:02.22,0:32:06.84,Default,,0000,0000,0000,,inside the delta x\Ndelta y delta Magdalena, Dialogue: 0,0:32:06.84,0:32:11.53,Default,,0000,0000,0000,,the same kind of displacement\Nwhen I take k from 1 to n, Dialogue: 0,0:32:11.53,0:32:16.17,Default,,0000,0000,0000,,and I pass to the limit\Naccording to the partition, Dialogue: 0,0:32:16.17,0:32:17.84,Default,,0000,0000,0000,,what's going to happen? Dialogue: 0,0:32:17.84,0:32:21.24,Default,,0000,0000,0000,,These guys, according\Nto Mr. Linux, Dialogue: 0,0:32:21.24,0:32:26.12,Default,,0000,0000,0000,,will go to be infinitesimal\Nelements, dx, dy. Dialogue: 0,0:32:26.12,0:32:29.53,Default,,0000,0000,0000,,This whole thing will\Ngo to double integral Dialogue: 0,0:32:29.53,0:32:36.66,Default,,0000,0000,0000,,of f of x and y,\Nand Mr. y says, OK Dialogue: 0,0:32:36.66,0:32:39.33,Default,,0000,0000,0000,,it's like you want him to\Nintegrate him one at a time. Dialogue: 0,0:32:39.33,0:32:43.61,Default,,0000,0000,0000,,This is actually something that\Nwe are going to see in a second Dialogue: 0,0:32:43.61,0:32:45.03,Default,,0000,0000,0000,,and verify it. Dialogue: 0,0:32:45.03,0:32:50.42,Default,,0000,0000,0000,,X goes between a and b,\Nand y goes between c and d, Dialogue: 0,0:32:50.42,0:32:55.01,Default,,0000,0000,0000,,and this is an application\Nof a big theorem called Dialogue: 0,0:32:55.01,0:33:01.45,Default,,0000,0000,0000,,Fubini's Theorem that\Nsays, wait a minute, Dialogue: 0,0:33:01.45,0:33:06.83,Default,,0000,0000,0000,,if you do it like this over\Na rectangle a,b cross c,d, Dialogue: 0,0:33:06.83,0:33:12.36,Default,,0000,0000,0000,,you're double integral can\Nbe written as three things. Dialogue: 0,0:33:12.36,0:33:18.21,Default,,0000,0000,0000,,Double integral over your\Nsquare domain f of x,y dA, Dialogue: 0,0:33:18.21,0:33:22.14,Default,,0000,0000,0000,,or you integral from c to\Nd, integral from a to b, Dialogue: 0,0:33:22.14,0:33:28.17,Default,,0000,0000,0000,,f of x,y dx dy, or you\Ncan also swap the order, Dialogue: 0,0:33:28.17,0:33:32.14,Default,,0000,0000,0000,,because you say, well, you can\Ndo the integration with respect Dialogue: 0,0:33:32.14,0:33:34.77,Default,,0000,0000,0000,,to y first. Dialogue: 0,0:33:34.77,0:33:36.80,Default,,0000,0000,0000,,Nobody stops you\Nfrom doing that, Dialogue: 0,0:33:36.80,0:33:40.83,Default,,0000,0000,0000,,and y has to be\Nbetween what and what? Dialogue: 0,0:33:40.83,0:33:41.54,Default,,0000,0000,0000,,STUDENT: C and d. Dialogue: 0,0:33:41.54,0:33:42.92,Default,,0000,0000,0000,,PROFESSOR: C and d, thank you. Dialogue: 0,0:33:42.92,0:33:47.19,Default,,0000,0000,0000,,And then whatever you get,\Nyou get to integrate that Dialogue: 0,0:33:47.19,0:33:52.17,Default,,0000,0000,0000,,with respect to x from a to b. Dialogue: 0,0:33:52.17,0:33:56.28,Default,,0000,0000,0000,,So no matter in what\Norder you do it, Dialogue: 0,0:33:56.28,0:33:59.20,Default,,0000,0000,0000,,you'll get the same thing. Dialogue: 0,0:33:59.20,0:34:02.99,Default,,0000,0000,0000,,Let's see an easy example,\Nand you'll say, well, Dialogue: 0,0:34:02.99,0:34:06.32,Default,,0000,0000,0000,,start with some [INAUDIBLE]\Nexample, Magdalena, Dialogue: 0,0:34:06.32,0:34:08.61,Default,,0000,0000,0000,,because we are just\Nstarting, and that's Dialogue: 0,0:34:08.61,0:34:10.09,Default,,0000,0000,0000,,exactly what I'm going to. Dialogue: 0,0:34:10.09,0:34:11.64,Default,,0000,0000,0000,,I will just misbehave. Dialogue: 0,0:34:11.64,0:34:14.63,Default,,0000,0000,0000,,I'm not going to go by the book. Dialogue: 0,0:34:14.63,0:34:19.41,Default,,0000,0000,0000,,And I will say I'm going\Nby whatever I want to go. Dialogue: 0,0:34:19.41,0:34:27.59,Default,,0000,0000,0000,,X is between 0, 2, and\Ny is between 0 and 2 Dialogue: 0,0:34:27.59,0:34:32.97,Default,,0000,0000,0000,,and 3-- this is 2, this\Nis 3-- and my domain Dialogue: 0,0:34:32.97,0:34:38.63,Default,,0000,0000,0000,,will be the rectangle\N0, 2 times 0, 3. Dialogue: 0,0:34:38.63,0:34:42.16,Default,,0000,0000,0000,,This is neat on the floor. Dialogue: 0,0:34:42.16,0:34:59.94,Default,,0000,0000,0000,,Compute the volume of the\Nbox of basis d and height 5. Dialogue: 0,0:34:59.94,0:35:02.07,Default,,0000,0000,0000,,Can I draw that? Dialogue: 0,0:35:02.07,0:35:03.61,Default,,0000,0000,0000,,It gets out of the picture. Dialogue: 0,0:35:03.61,0:35:04.57,Default,,0000,0000,0000,,I'm just kidding. Dialogue: 0,0:35:04.57,0:35:07.83,Default,,0000,0000,0000,,This is 5, and that's\Nsort of the box. Dialogue: 0,0:35:07.83,0:35:10.69,Default,,0000,0000,0000,, Dialogue: 0,0:35:10.69,0:35:13.59,Default,,0000,0000,0000,,And you say, wait a minute, I\Nknow that from third grade-- Dialogue: 0,0:35:13.59,0:35:16.04,Default,,0000,0000,0000,,I mean, first grade, whenever. Dialogue: 0,0:35:16.04,0:35:17.23,Default,,0000,0000,0000,,How do we do that? Dialogue: 0,0:35:17.23,0:35:21.34,Default,,0000,0000,0000,,We go 2 units times\N3 units that's Dialogue: 0,0:35:21.34,0:35:25.46,Default,,0000,0000,0000,,going to be 6 square inches\Non the bottom of the box, Dialogue: 0,0:35:25.46,0:35:27.59,Default,,0000,0000,0000,,and then times 5. Dialogue: 0,0:35:27.59,0:35:31.44,Default,,0000,0000,0000,,So the volume has to be\N2 times 3 times 5, which Dialogue: 0,0:35:31.44,0:35:35.49,Default,,0000,0000,0000,,is 30 square inches. Dialogue: 0,0:35:35.49,0:35:37.09,Default,,0000,0000,0000,,I don't care what it is. Dialogue: 0,0:35:37.09,0:35:39.37,Default,,0000,0000,0000,,I'm a mathematician, right? Dialogue: 0,0:35:39.37,0:35:39.87,Default,,0000,0000,0000,,OK. Dialogue: 0,0:35:39.87,0:35:44.00,Default,,0000,0000,0000,,How does somebody who just\Nlearned Tonelli's-- Fubini Dialogue: 0,0:35:44.00,0:35:46.96,Default,,0000,0000,0000,,Tonelli's Theorem\Ndo the problem. Dialogue: 0,0:35:46.96,0:35:49.18,Default,,0000,0000,0000,,That person will\Nsay, wait a minute, Dialogue: 0,0:35:49.18,0:35:54.80,Default,,0000,0000,0000,,now I know that the\Nfunction is going to be z Dialogue: 0,0:35:54.80,0:36:00.43,Default,,0000,0000,0000,,equals f of xy, which in\Nthis case happens to be cost. Dialogue: 0,0:36:00.43,0:36:05.02,Default,,0000,0000,0000,,According to what you told us,\Nthe theorem you claim Magdalena Dialogue: 0,0:36:05.02,0:36:07.13,Default,,0000,0000,0000,,proved to this theorem,\Nbut there is a sketch Dialogue: 0,0:36:07.13,0:36:08.95,Default,,0000,0000,0000,,of the proof in the book. Dialogue: 0,0:36:08.95,0:36:13.36,Default,,0000,0000,0000,,According to this,\Nthe double integral Dialogue: 0,0:36:13.36,0:36:21.40,Default,,0000,0000,0000,,that you have over the\Ndomain d, and this is dA. Dialogue: 0,0:36:21.40,0:36:30.53,Default,,0000,0000,0000,,DA will be called element of\Narea, which is also dx dy. Dialogue: 0,0:36:30.53,0:36:34.51,Default,,0000,0000,0000,,This can be solved in\Ntwo different ways. Dialogue: 0,0:36:34.51,0:36:38.28,Default,,0000,0000,0000,,You take integral\Nfrom-- where is x going? Dialogue: 0,0:36:38.28,0:36:41.97,Default,,0000,0000,0000,,Do we want to do it\Nfirst in x or in y? Dialogue: 0,0:36:41.97,0:36:44.92,Default,,0000,0000,0000,,If we put dy dx, that means\Nwe integrate with respect Dialogue: 0,0:36:44.92,0:36:49.25,Default,,0000,0000,0000,,to y first, and y\Ngoes between 0 and 3, Dialogue: 0,0:36:49.25,0:36:53.20,Default,,0000,0000,0000,,so I have to pay attention\Nto the limits of integration. Dialogue: 0,0:36:53.20,0:36:56.29,Default,,0000,0000,0000,,And then x between\N0 and 2 and again Dialogue: 0,0:36:56.29,0:36:58.85,Default,,0000,0000,0000,,I have to pay attention to\Nthe limits of integration Dialogue: 0,0:36:58.85,0:37:03.12,Default,,0000,0000,0000,,all the time and,\Nhere, who is my f? Dialogue: 0,0:37:03.12,0:37:06.67,Default,,0000,0000,0000,,Is the altitude 5 that's\Nconstant in my case? Dialogue: 0,0:37:06.67,0:37:08.50,Default,,0000,0000,0000,,I'm not worried about it. Dialogue: 0,0:37:08.50,0:37:10.64,Default,,0000,0000,0000,,Let me see if I get 30? Dialogue: 0,0:37:10.64,0:37:16.49,Default,,0000,0000,0000,,I'm just checking if this\Ntheorem was true or is just Dialogue: 0,0:37:16.49,0:37:20.98,Default,,0000,0000,0000,,something that you cannot apply. Dialogue: 0,0:37:20.98,0:37:25.62,Default,,0000,0000,0000,,How do you integrate\N5 with respect to y? Dialogue: 0,0:37:25.62,0:37:26.48,Default,,0000,0000,0000,,STUDENT: 5y. Dialogue: 0,0:37:26.48,0:37:27.52,Default,,0000,0000,0000,,PROFESSOR: 5y, very good. Dialogue: 0,0:37:27.52,0:37:34.12,Default,,0000,0000,0000,,So it's going to be 5y between\Ny equals 0 down and y equals 3 Dialogue: 0,0:37:34.12,0:37:38.74,Default,,0000,0000,0000,,up, and how much\Nis that 5y, we're Dialogue: 0,0:37:38.74,0:37:41.44,Default,,0000,0000,0000,,doing y equals 0 down\Nand y equals 3 up, Dialogue: 0,0:37:41.44,0:37:42.91,Default,,0000,0000,0000,,what number is that? Dialogue: 0,0:37:42.91,0:37:44.87,Default,,0000,0000,0000,,STUDENT: 25. Dialogue: 0,0:37:44.87,0:37:45.85,Default,,0000,0000,0000,,PROFESSOR: What? Dialogue: 0,0:37:45.85,0:37:46.83,Default,,0000,0000,0000,,STUDENT: 25. Dialogue: 0,0:37:46.83,0:37:47.81,Default,,0000,0000,0000,,PROFESSOR: 25? Dialogue: 0,0:37:47.81,0:37:49.77,Default,,0000,0000,0000,,STUDENT: One [INAUDIBLE] 15. Dialogue: 0,0:37:49.77,0:37:54.20,Default,,0000,0000,0000,,PROFESSOR: No, you did--\Nyou are thinking ahead. Dialogue: 0,0:37:54.20,0:38:00.01,Default,,0000,0000,0000,,So I go 5 times 3 minus\N5 times 0 equals 15. Dialogue: 0,0:38:00.01,0:38:04.11,Default,,0000,0000,0000,,So when I compute this\Nvariation of 5y between y Dialogue: 0,0:38:04.11,0:38:06.82,Default,,0000,0000,0000,,equals 3 and y equals\N0, I just block in Dialogue: 0,0:38:06.82,0:38:08.26,Default,,0000,0000,0000,,and make the difference. Dialogue: 0,0:38:08.26,0:38:09.73,Default,,0000,0000,0000,,Why do I do that? Dialogue: 0,0:38:09.73,0:38:15.79,Default,,0000,0000,0000,,It's the simplest application\Nof that FT, fundamental theorem. Dialogue: 0,0:38:15.79,0:38:19.48,Default,,0000,0000,0000,,The one that I did not\Nspecify in [INAUDIBLE]. Dialogue: 0,0:38:19.48,0:38:23.49,Default,,0000,0000,0000,,I should have specified when\NI have a g function that Dialogue: 0,0:38:23.49,0:38:28.19,Default,,0000,0000,0000,,is continuous between\Nalpha and beta, how do we Dialogue: 0,0:38:28.19,0:38:30.30,Default,,0000,0000,0000,,integrate with respect to x? Dialogue: 0,0:38:30.30,0:38:33.48,Default,,0000,0000,0000,,I get the antiderivative\Nof rule G. Let's call Dialogue: 0,0:38:33.48,0:38:37.14,Default,,0000,0000,0000,,that big G. Compute\Nit at the end points, Dialogue: 0,0:38:37.14,0:38:39.19,Default,,0000,0000,0000,,and I make the difference. Dialogue: 0,0:38:39.19,0:38:41.79,Default,,0000,0000,0000,,So I compute the\Nantiderivative at an endpoint-- Dialogue: 0,0:38:41.79,0:38:44.37,Default,,0000,0000,0000,,at the other endpoint-- then I'm\Ngoing to make the difference. Dialogue: 0,0:38:44.37,0:38:49.70,Default,,0000,0000,0000,,That's the same thing I do\Nhere, so 5 times 3 is 15, Dialogue: 0,0:38:49.70,0:38:54.41,Default,,0000,0000,0000,,5 times 0 is 0,\N15 minus 0 is 15. Dialogue: 0,0:38:54.41,0:38:56.03,Default,,0000,0000,0000,,I can keep moving. Dialogue: 0,0:38:56.03,0:38:59.10,Default,,0000,0000,0000,,Everything in the\Nparentheses is the number 15. Dialogue: 0,0:38:59.10,0:39:03.52,Default,,0000,0000,0000,,I copy and paste, and that\Nshould be a piece of cake. Dialogue: 0,0:39:03.52,0:39:07.01,Default,,0000,0000,0000,,What do I get? Dialogue: 0,0:39:07.01,0:39:09.51,Default,,0000,0000,0000,,STUDENT: 15. Dialogue: 0,0:39:09.51,0:39:15.56,Default,,0000,0000,0000,,PROFESSOR: I get 15\Ntimes x between 0 and 2. Dialogue: 0,0:39:15.56,0:39:16.91,Default,,0000,0000,0000,,Integral of 1 is x. Dialogue: 0,0:39:16.91,0:39:19.78,Default,,0000,0000,0000,,Integral of 1 is x\Nwith respect to x, Dialogue: 0,0:39:19.78,0:39:24.20,Default,,0000,0000,0000,,so I get 15 times 2, which\Nis 30, and you go, duh, Dialogue: 0,0:39:24.20,0:39:26.83,Default,,0000,0000,0000,,[INAUDIBLE]. Dialogue: 0,0:39:26.83,0:39:28.78,Default,,0000,0000,0000,,That was elementary mathematics. Dialogue: 0,0:39:28.78,0:39:32.05,Default,,0000,0000,0000,,Yes, you were lucky you\Nknew that volume of the box, Dialogue: 0,0:39:32.05,0:39:35.73,Default,,0000,0000,0000,,but what if somebody gave\Nyou a curvilinear area? Dialogue: 0,0:39:35.73,0:39:39.36,Default,,0000,0000,0000,,What if somebody gave you\Nsomething quite complicated? Dialogue: 0,0:39:39.36,0:39:40.70,Default,,0000,0000,0000,,What would you do? Dialogue: 0,0:39:40.70,0:39:43.45,Default,,0000,0000,0000,,You have know calculus. Dialogue: 0,0:39:43.45,0:39:45.92,Default,,0000,0000,0000,,That's your only chance. Dialogue: 0,0:39:45.92,0:39:51.72,Default,,0000,0000,0000,,If you don't calculus,\Nyou are dead meat. Dialogue: 0,0:39:51.72,0:40:00.68,Default,,0000,0000,0000,,So I'm saying, how\Nabout another problem. Dialogue: 0,0:40:00.68,0:40:04.50,Default,,0000,0000,0000,,That look like it's\Ncomplicated, but calculus Dialogue: 0,0:40:04.50,0:40:08.46,Default,,0000,0000,0000,,is something\N[INAUDIBLE] with that. Dialogue: 0,0:40:08.46,0:40:15.76,Default,,0000,0000,0000,,Suppose that I have a square\Nin the plane between-- this Dialogue: 0,0:40:15.76,0:40:19.91,Default,,0000,0000,0000,,is x and y-- do you\Nwant square 0,1 0,1 Dialogue: 0,0:40:19.91,0:40:22.63,Default,,0000,0000,0000,,or you want minus 1\Nto 1 minus 1 to 1. Dialogue: 0,0:40:22.63,0:40:26.16,Default,,0000,0000,0000,, Dialogue: 0,0:40:26.16,0:40:28.08,Default,,0000,0000,0000,,It doesn't matter. Dialogue: 0,0:40:28.08,0:40:32.97,Default,,0000,0000,0000,,Well, let's take minus\N1 to 1 and minus 1 to 1, Dialogue: 0,0:40:32.97,0:40:35.99,Default,,0000,0000,0000,,and I'll try to draw\Nas well as I can, Dialogue: 0,0:40:35.99,0:40:38.18,Default,,0000,0000,0000,,which I cannot but it's OK. Dialogue: 0,0:40:38.18,0:40:41.56,Default,,0000,0000,0000,,You will forgive me. Dialogue: 0,0:40:41.56,0:40:42.43,Default,,0000,0000,0000,,This is the floor. Dialogue: 0,0:40:42.43,0:40:45.62,Default,,0000,0000,0000,, Dialogue: 0,0:40:45.62,0:40:48.21,Default,,0000,0000,0000,,If I were just a\Nlittle tiny square Dialogue: 0,0:40:48.21,0:40:52.30,Default,,0000,0000,0000,,in this room plus the\Nequivalent square in that room Dialogue: 0,0:40:52.30,0:40:53.79,Default,,0000,0000,0000,,and that room and that room. Dialogue: 0,0:40:53.79,0:40:56.34,Default,,0000,0000,0000,,This is the origin. Dialogue: 0,0:40:56.34,0:40:57.65,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,0:40:57.65,0:41:00.03,Default,,0000,0000,0000,,So what you're\Nlooking at right now Dialogue: 0,0:41:00.03,0:41:05.42,Default,,0000,0000,0000,,is this square foot\Nof carpet that I have, Dialogue: 0,0:41:05.42,0:41:11.90,Default,,0000,0000,0000,,but I have another one here and\Nanother one behind the wall, Dialogue: 0,0:41:11.90,0:41:15.59,Default,,0000,0000,0000,,and so do I everything in mind? Dialogue: 0,0:41:15.59,0:41:21.13,Default,,0000,0000,0000,,X is between minus 1 and 1,\Ny is between minus 1 and 1. Dialogue: 0,0:41:21.13,0:41:24.52,Default,,0000,0000,0000,, Dialogue: 0,0:41:24.52,0:41:30.35,Default,,0000,0000,0000,,And somebody gives you z\Nto be a positive function, Dialogue: 0,0:41:30.35,0:41:36.08,Default,,0000,0000,0000,,continuous function, which\Nis x squared plus y squared. Dialogue: 0,0:41:36.08,0:41:37.42,Default,,0000,0000,0000,,And you go, already. Dialogue: 0,0:41:37.42,0:41:39.29,Default,,0000,0000,0000,,Oh, my god. Dialogue: 0,0:41:39.29,0:41:41.95,Default,,0000,0000,0000,,I already have this\Nkind of hard function. Dialogue: 0,0:41:41.95,0:41:44.35,Default,,0000,0000,0000,,It's not a hard thing to do. Dialogue: 0,0:41:44.35,0:41:45.38,Default,,0000,0000,0000,,Let's draw that. Dialogue: 0,0:41:45.38,0:41:47.99,Default,,0000,0000,0000,,What are we going to get? Dialogue: 0,0:41:47.99,0:41:55.98,Default,,0000,0000,0000,,Your favorite [INAUDIBLE]\Nthat goes like this. Dialogue: 0,0:41:55.98,0:41:59.78,Default,,0000,0000,0000,,And imagine what's\Ngoing to happen Dialogue: 0,0:41:59.78,0:42:03.40,Default,,0000,0000,0000,,with this is like a vase. Dialogue: 0,0:42:03.40,0:42:06.85,Default,,0000,0000,0000,,Inside, it has this\Ncircular paraboloid. Dialogue: 0,0:42:06.85,0:42:17.76,Default,,0000,0000,0000,,But the walls of this vase are--\NI cannot draw better than that. Dialogue: 0,0:42:17.76,0:42:25.31,Default,,0000,0000,0000,,So the walls of this\Nvase are squares. Dialogue: 0,0:42:25.31,0:42:30.04,Default,,0000,0000,0000,,And what you have inside is\Nthe carved circular paraboloid. Dialogue: 0,0:42:30.04,0:42:32.58,Default,,0000,0000,0000,, Dialogue: 0,0:42:32.58,0:42:45.33,Default,,0000,0000,0000,,Now I'm asking\Nyou, how do I find Dialogue: 0,0:42:45.33,0:43:02.46,Default,,0000,0000,0000,,volume of the body under and\Nabove D, which is minus 1, Dialogue: 0,0:43:02.46,0:43:03.51,Default,,0000,0000,0000,,1, minus 1, 1. Dialogue: 0,0:43:03.51,0:43:05.41,Default,,0000,0000,0000,,It's hard to draw that, right? Dialogue: 0,0:43:05.41,0:43:06.97,Default,,0000,0000,0000,,It's hard to draw. Dialogue: 0,0:43:06.97,0:43:09.81,Default,,0000,0000,0000,,So what do we do? Dialogue: 0,0:43:09.81,0:43:15.51,Default,,0000,0000,0000,, Dialogue: 0,0:43:15.51,0:43:16.59,Default,,0000,0000,0000,,We start imagining things. Dialogue: 0,0:43:16.59,0:43:20.06,Default,,0000,0000,0000,, Dialogue: 0,0:43:20.06,0:43:24.11,Default,,0000,0000,0000,,Actually, when you cut with\Na plane that is y equals 1, Dialogue: 0,0:43:24.11,0:43:27.87,Default,,0000,0000,0000,,you would get a parabola. Dialogue: 0,0:43:27.87,0:43:35.88,Default,,0000,0000,0000,,And so when you look at what the\Npicture is going to look like, Dialogue: 0,0:43:35.88,0:43:39.58,Default,,0000,0000,0000,,you're going to have\Na parabola like this, Dialogue: 0,0:43:39.58,0:43:41.79,Default,,0000,0000,0000,,a parabola like that,\Nexactly the same, Dialogue: 0,0:43:41.79,0:43:45.99,Default,,0000,0000,0000,,a parallel parabola like this\Nand a parabola like that. Dialogue: 0,0:43:45.99,0:43:49.27,Default,,0000,0000,0000,,Now I started drawing better. Dialogue: 0,0:43:49.27,0:43:51.58,Default,,0000,0000,0000,,And you say, how did you\Nstart drawing better? Dialogue: 0,0:43:51.58,0:43:53.37,Default,,0000,0000,0000,,Well, with a little\Nbit of practice. Dialogue: 0,0:43:53.37,0:43:59.56,Default,,0000,0000,0000,,Where are the maxima\Nof this thing? Dialogue: 0,0:43:59.56,0:44:00.51,Default,,0000,0000,0000,,At the corners. Dialogue: 0,0:44:00.51,0:44:01.20,Default,,0000,0000,0000,,Why is that? Dialogue: 0,0:44:01.20,0:44:05.65,Default,,0000,0000,0000,,Because at the corners,\Nyou get 1, 1 for both. Dialogue: 0,0:44:05.65,0:44:10.27,Default,,0000,0000,0000,,Of course, to do the absolute\Nextrema, minimum, maximum, Dialogue: 0,0:44:10.27,0:44:14.71,Default,,0000,0000,0000,,we would have to go back to\Nsection 11.7 and do the thing. Dialogue: 0,0:44:14.71,0:44:19.13,Default,,0000,0000,0000,,But practically, it's easy\Nto see that at the corners, Dialogue: 0,0:44:19.13,0:44:23.05,Default,,0000,0000,0000,,you have the height 2 because\Nthis is the point 1, 1. Dialogue: 0,0:44:23.05,0:44:28.93,Default,,0000,0000,0000,,And the same height, 2 and 2\Nand 2, are at every corner. Dialogue: 0,0:44:28.93,0:44:31.87,Default,,0000,0000,0000,,That would be the\Nmaximum that you have. Dialogue: 0,0:44:31.87,0:44:39.34,Default,,0000,0000,0000,,So you have 1 minus 1 and so\Non-- minus 1, 1, and minus 1, Dialogue: 0,0:44:39.34,0:44:42.39,Default,,0000,0000,0000,,minus 1, who is behind\Nme, minus 1, minus 1. Dialogue: 0,0:44:42.39,0:44:46.65,Default,,0000,0000,0000,,That goes all the way to 2. Dialogue: 0,0:44:46.65,0:44:51.24,Default,,0000,0000,0000,,So it's hard to do an\Napproximation with a three Dialogue: 0,0:44:51.24,0:44:53.09,Default,,0000,0000,0000,,dimensional model. Dialogue: 0,0:44:53.09,0:44:54.56,Default,,0000,0000,0000,,Thank god there is calculus. Dialogue: 0,0:44:54.56,0:44:59.11,Default,,0000,0000,0000,,So you say integral of x\Nsquared plus y squared, Dialogue: 0,0:44:59.11,0:45:05.93,Default,,0000,0000,0000,,as simple as that, da over the\Ndomain, D, which is minus 1, Dialogue: 0,0:45:05.93,0:45:07.82,Default,,0000,0000,0000,,1, minus 1, 1. Dialogue: 0,0:45:07.82,0:45:10.37,Default,,0000,0000,0000,,How do you write it\Naccording to the theorem Dialogue: 0,0:45:10.37,0:45:13.39,Default,,0000,0000,0000,,that I told you\Nabout, Fubini-Tonelli? Dialogue: 0,0:45:13.39,0:45:19.64,Default,,0000,0000,0000,,Then you have integral integral\Nx squared plus y squared dy dx. Dialogue: 0,0:45:19.64,0:45:22.43,Default,,0000,0000,0000,, Dialogue: 0,0:45:22.43,0:45:25.21,Default,,0000,0000,0000,,Doesn't matter which\None I'm taking. Dialogue: 0,0:45:25.21,0:45:26.62,Default,,0000,0000,0000,,I can do dy dx. Dialogue: 0,0:45:26.62,0:45:27.86,Default,,0000,0000,0000,,I can do dx dy. Dialogue: 0,0:45:27.86,0:45:31.22,Default,,0000,0000,0000,,I just have to pay\Nattention to the endpoints. Dialogue: 0,0:45:31.22,0:45:33.36,Default,,0000,0000,0000,,Lucky for you the\Nendpoints are the same. Dialogue: 0,0:45:33.36,0:45:35.46,Default,,0000,0000,0000,,y is between minus 1 and 1. Dialogue: 0,0:45:35.46,0:45:37.44,Default,,0000,0000,0000,,x is between minus 1 and 1. Dialogue: 0,0:45:37.44,0:45:40.75,Default,,0000,0000,0000,, Dialogue: 0,0:45:40.75,0:45:44.72,Default,,0000,0000,0000,,I wouldn't known how to compute\Nthe volume of this vase made Dialogue: 0,0:45:44.72,0:45:46.09,Default,,0000,0000,0000,,of marble or made\Nof whatever you Dialogue: 0,0:45:46.09,0:45:53.84,Default,,0000,0000,0000,,want to make it unless I knew\Nto compute this integral. Dialogue: 0,0:45:53.84,0:45:58.58,Default,,0000,0000,0000,,Now you have to help me\Nbecause it's not hard Dialogue: 0,0:45:58.58,0:46:03.90,Default,,0000,0000,0000,,but it's not easy either, so we\Nneed a little bit of attention. Dialogue: 0,0:46:03.90,0:46:05.86,Default,,0000,0000,0000,,We always start from the\Ninside to the outside. Dialogue: 0,0:46:05.86,0:46:10.56,Default,,0000,0000,0000,,The outer person has to be just\Nneglected for the time being Dialogue: 0,0:46:10.56,0:46:14.58,Default,,0000,0000,0000,,and I focus all my attention\Nto this integration. Dialogue: 0,0:46:14.58,0:46:18.06,Default,,0000,0000,0000,,And when I integrate\Nwith respect to y, Dialogue: 0,0:46:18.06,0:46:20.45,Default,,0000,0000,0000,,y is the variable for me. Dialogue: 0,0:46:20.45,0:46:22.80,Default,,0000,0000,0000,,Nothing else exists\Nfor the time being, Dialogue: 0,0:46:22.80,0:46:27.54,Default,,0000,0000,0000,,but y being a variable,\Nx being like a constant. Dialogue: 0,0:46:27.54,0:46:29.99,Default,,0000,0000,0000,,So when you integrate x\Nsquared plus y squared Dialogue: 0,0:46:29.99,0:46:34.51,Default,,0000,0000,0000,,with respect to y, you have\Nto pay attention a little bit. Dialogue: 0,0:46:34.51,0:46:39.79,Default,,0000,0000,0000,,It's about the same if you\Nhad 7 squared plus y squared. Dialogue: 0,0:46:39.79,0:46:43.41,Default,,0000,0000,0000,,So this x squared\Nis like a constant. Dialogue: 0,0:46:43.41,0:46:45.36,Default,,0000,0000,0000,,So what do you get inside? Dialogue: 0,0:46:45.36,0:46:47.36,Default,,0000,0000,0000,,Let's apply the fundamental\Ntheorem of calculus. Dialogue: 0,0:46:47.36,0:46:48.29,Default,,0000,0000,0000,,STUDENT: x squared y. Dialogue: 0,0:46:48.29,0:46:49.25,Default,,0000,0000,0000,,PROFESSOR: x squared y. Dialogue: 0,0:46:49.25,0:46:49.96,Default,,0000,0000,0000,,Excellent. Dialogue: 0,0:46:49.96,0:46:51.96,Default,,0000,0000,0000,,I'm very proud of you. Dialogue: 0,0:46:51.96,0:46:52.67,Default,,0000,0000,0000,,Plus? Dialogue: 0,0:46:52.67,0:46:53.80,Default,,0000,0000,0000,,STUDENT: y cubed over 3. Dialogue: 0,0:46:53.80,0:46:55.08,Default,,0000,0000,0000,,PROFESSOR: y cubed over three. Dialogue: 0,0:46:55.08,0:46:57.46,Default,,0000,0000,0000,,Again, I'm proud of you. Dialogue: 0,0:46:57.46,0:47:03.31,Default,,0000,0000,0000,,Evaluated between y equals\Nminus 1 down, y equals 1 up. Dialogue: 0,0:47:03.31,0:47:07.02,Default,,0000,0000,0000,,And I will do the math later\Nbecause I'm getting tired. Dialogue: 0,0:47:07.02,0:47:09.73,Default,,0000,0000,0000,, Dialogue: 0,0:47:09.73,0:47:11.79,Default,,0000,0000,0000,,Now let's do the math. Dialogue: 0,0:47:11.79,0:47:13.28,Default,,0000,0000,0000,,I don't know what\NI'm going to get. Dialogue: 0,0:47:13.28,0:47:18.93,Default,,0000,0000,0000,,I get minus 1 to 1, a\Nbig bracket, and dx. Dialogue: 0,0:47:18.93,0:47:21.93,Default,,0000,0000,0000,,And in this big bracket, I\Nhave to do the difference Dialogue: 0,0:47:21.93,0:47:23.27,Default,,0000,0000,0000,,between two values. Dialogue: 0,0:47:23.27,0:47:26.92,Default,,0000,0000,0000,,So I put two parentheses. Dialogue: 0,0:47:26.92,0:47:29.96,Default,,0000,0000,0000,,When y equals 1, I\Nget x squared 1-- Dialogue: 0,0:47:29.96,0:47:33.80,Default,,0000,0000,0000,,I'm not going to write\Nthat down-- plus 1 cubed Dialogue: 0,0:47:33.80,0:47:36.51,Default,,0000,0000,0000,,over 3, 1/3. Dialogue: 0,0:47:36.51,0:47:41.92,Default,,0000,0000,0000,,I'm done with evaluating\Nthis sausage thingy at 1. Dialogue: 0,0:47:41.92,0:47:44.31,Default,,0000,0000,0000,,It's an expression\Nthat I evaluate. Dialogue: 0,0:47:44.31,0:47:46.58,Default,,0000,0000,0000,,It could be a lot longer. Dialogue: 0,0:47:46.58,0:47:49.13,Default,,0000,0000,0000,,I'm not planning to give you\Nlong expressions in the midterm Dialogue: 0,0:47:49.13,0:47:51.94,Default,,0000,0000,0000,,because you're going to\Nmake algebra mistakes, Dialogue: 0,0:47:51.94,0:47:55.27,Default,,0000,0000,0000,,and that's not what I want. Dialogue: 0,0:47:55.27,0:48:01.15,Default,,0000,0000,0000,,For minus 1, what do we\Nhave Minus x squared. Dialogue: 0,0:48:01.15,0:48:04.50,Default,,0000,0000,0000,,What is y equals minus\N1 plugged in here? Dialogue: 0,0:48:04.50,0:48:05.46,Default,,0000,0000,0000,,Minus 1/3. Dialogue: 0,0:48:05.46,0:48:09.09,Default,,0000,0000,0000,, Dialogue: 0,0:48:09.09,0:48:10.52,Default,,0000,0000,0000,,I have to pay attention. Dialogue: 0,0:48:10.52,0:48:15.65,Default,,0000,0000,0000,,You realize that if I mess\Nup a sign, it's all done. Dialogue: 0,0:48:15.65,0:48:21.22,Default,,0000,0000,0000,,So in this case, I say, but\Nthis I have minus, minus. Dialogue: 0,0:48:21.22,0:48:24.18,Default,,0000,0000,0000,,A minus in front of\Na minus is a plus, Dialogue: 0,0:48:24.18,0:48:30.87,Default,,0000,0000,0000,,so I'm practically doubling\Nthe x squared plus 1/3 Dialogue: 0,0:48:30.87,0:48:33.93,Default,,0000,0000,0000,,and taking it\Nbetween minus 1 and 1 Dialogue: 0,0:48:33.93,0:48:36.68,Default,,0000,0000,0000,,and just with respect to x. Dialogue: 0,0:48:36.68,0:48:38.16,Default,,0000,0000,0000,,So you say, wait a minute. Dialogue: 0,0:48:38.16,0:48:38.97,Default,,0000,0000,0000,,But that's easy. Dialogue: 0,0:48:38.97,0:48:41.12,Default,,0000,0000,0000,,I've done that when\NI was in Calc 1. Dialogue: 0,0:48:41.12,0:48:41.86,Default,,0000,0000,0000,,Of course. Dialogue: 0,0:48:41.86,0:48:47.04,Default,,0000,0000,0000,,This is the nice part that\Nyou get, a simple integral Dialogue: 0,0:48:47.04,0:48:51.77,Default,,0000,0000,0000,,from the ones in Calc 1. Dialogue: 0,0:48:51.77,0:48:56.53,Default,,0000,0000,0000,,Let's solve this one and find\Nout what the area will be. Dialogue: 0,0:48:56.53,0:48:59.17,Default,,0000,0000,0000,,What do we get? Dialogue: 0,0:48:59.17,0:48:59.98,Default,,0000,0000,0000,,Is it hard? Dialogue: 0,0:48:59.98,0:49:00.92,Default,,0000,0000,0000,,No. Dialogue: 0,0:49:00.92,0:49:02.15,Default,,0000,0000,0000,,Kick Mr. 2 out. Dialogue: 0,0:49:02.15,0:49:04.97,Default,,0000,0000,0000,,He's just messing\Nup with your life. Dialogue: 0,0:49:04.97,0:49:06.15,Default,,0000,0000,0000,,Kick him out. Dialogue: 0,0:49:06.15,0:49:08.50,Default,,0000,0000,0000,,2, out. Dialogue: 0,0:49:08.50,0:49:11.90,Default,,0000,0000,0000,,And then integral of\Nx squared plus 1/3 Dialogue: 0,0:49:11.90,0:49:15.72,Default,,0000,0000,0000,,is going to be x\Ncubed over 3 plus-- Dialogue: 0,0:49:15.72,0:49:16.69,Default,,0000,0000,0000,,STUDENT: x over 3. Dialogue: 0,0:49:16.69,0:49:18.62,Default,,0000,0000,0000,,PROFESSOR: x over 3, very good. Dialogue: 0,0:49:18.62,0:49:22.96,Default,,0000,0000,0000,,Evaluated between x equals\Nminus 1 down, x equals 1 up. Dialogue: 0,0:49:22.96,0:49:26.36,Default,,0000,0000,0000,, Dialogue: 0,0:49:26.36,0:49:27.73,Default,,0000,0000,0000,,Let's see what we get. Dialogue: 0,0:49:27.73,0:49:30.94,Default,,0000,0000,0000,,2 times bracket. Dialogue: 0,0:49:30.94,0:49:33.56,Default,,0000,0000,0000,,I'll put a parentheses\Nfor the first fractions, Dialogue: 0,0:49:33.56,0:49:36.98,Default,,0000,0000,0000,,and another minus, and\Nanother parentheses. Dialogue: 0,0:49:36.98,0:49:42.07,Default,,0000,0000,0000,,What's the first edition\Nof fractions that I get? Dialogue: 0,0:49:42.07,0:49:44.44,Default,,0000,0000,0000,,1/3 plus 1/3. Dialogue: 0,0:49:44.44,0:49:47.36,Default,,0000,0000,0000,,I'll put 2/3 because I'm lazy. Dialogue: 0,0:49:47.36,0:49:49.22,Default,,0000,0000,0000,,Then minus what? Dialogue: 0,0:49:49.22,0:49:51.31,Default,,0000,0000,0000,,STUDENT: Minus 1/3. Dialogue: 0,0:49:51.31,0:49:56.37,Default,,0000,0000,0000,,PROFESSOR: Minus 1/3\Nminus 1/3, minus 2/3. Dialogue: 0,0:49:56.37,0:50:01.12,Default,,0000,0000,0000,,And now I should be able to\Nnot beat around the bush. Dialogue: 0,0:50:01.12,0:50:04.44,Default,,0000,0000,0000,,Tell me what the answer\Nwill be in the end. Dialogue: 0,0:50:04.44,0:50:06.27,Default,,0000,0000,0000,,STUDENT: 8/3. Dialogue: 0,0:50:06.27,0:50:08.25,Default,,0000,0000,0000,,PROFESSOR: 8/3. Dialogue: 0,0:50:08.25,0:50:09.79,Default,,0000,0000,0000,,Does that make sense? Dialogue: 0,0:50:09.79,0:50:12.43,Default,,0000,0000,0000,,When you do that in\Nmath, you should always Dialogue: 0,0:50:12.43,0:50:16.89,Default,,0000,0000,0000,,think-- one of the famous\Nprofessors at Harvard Dialogue: 0,0:50:16.89,0:50:21.65,Default,,0000,0000,0000,,was saying one time\Nshe asked the students, Dialogue: 0,0:50:21.65,0:50:23.80,Default,,0000,0000,0000,,how many hours of\Nlife do we have have Dialogue: 0,0:50:23.80,0:50:25.77,Default,,0000,0000,0000,,in one day, blah, blah, blah? Dialogue: 0,0:50:25.77,0:50:30.28,Default,,0000,0000,0000,,And many students\Ncame up with 36, 37. Dialogue: 0,0:50:30.28,0:50:35.56,Default,,0000,0000,0000,,So always make sure that the\Nanswer you get makes sense. Dialogue: 0,0:50:35.56,0:50:37.89,Default,,0000,0000,0000,,This is part of a cube, right? Dialogue: 0,0:50:37.89,0:50:42.92,Default,,0000,0000,0000,,It's like carved in a\Ncube or a rectangle. Dialogue: 0,0:50:42.92,0:50:46.49,Default,,0000,0000,0000,, Dialogue: 0,0:50:46.49,0:50:48.57,Default,,0000,0000,0000,,Now, what's the height? Dialogue: 0,0:50:48.57,0:50:53.24,Default,,0000,0000,0000,,If this were to go\Nup all the way to 2, Dialogue: 0,0:50:53.24,0:50:58.61,Default,,0000,0000,0000,,it would be 2, 2, and 2. Dialogue: 0,0:50:58.61,0:51:04.43,Default,,0000,0000,0000,,2 times 2 times 2 equals 8,\Nand what we got is 8 over 3. Dialogue: 0,0:51:04.43,0:51:08.83,Default,,0000,0000,0000,,Now, using our imagination,\Nit makes sense. Dialogue: 0,0:51:08.83,0:51:11.38,Default,,0000,0000,0000,,If I got a 16, I\Nwould say, oh my god. Dialogue: 0,0:51:11.38,0:51:12.22,Default,,0000,0000,0000,,No, no, no, no. Dialogue: 0,0:51:12.22,0:51:14.19,Default,,0000,0000,0000,,What is that? Dialogue: 0,0:51:14.19,0:51:17.97,Default,,0000,0000,0000,,So a little bit, I would think,\Ndoes this make sense or not? Dialogue: 0,0:51:17.97,0:51:21.71,Default,,0000,0000,0000,, Dialogue: 0,0:51:21.71,0:51:24.40,Default,,0000,0000,0000,,Let's do one more,\Na similar one. Dialogue: 0,0:51:24.40,0:51:28.18,Default,,0000,0000,0000,,Now I'm going to count\Non you a little bit more. Dialogue: 0,0:51:28.18,0:51:39.25,Default,,0000,0000,0000,, Dialogue: 0,0:51:39.25,0:51:41.23,Default,,0000,0000,0000,,STUDENT: Professor,\Ndid you calculate that Dialogue: 0,0:51:41.23,0:51:45.56,Default,,0000,0000,0000,,by just doing a quarter, and\Nthen just multiplying it by 4? Dialogue: 0,0:51:45.56,0:51:47.10,Default,,0000,0000,0000,,Because then that\Nwould just leave us Dialogue: 0,0:51:47.10,0:51:48.79,Default,,0000,0000,0000,,with zeroes [INAUDIBLE]. Dialogue: 0,0:51:48.79,0:51:50.71,Default,,0000,0000,0000,,PROFESSOR: You mean in\Nthat particular figure? Dialogue: 0,0:51:50.71,0:51:51.21,Default,,0000,0000,0000,,Yeah. Dialogue: 0,0:51:51.21,0:51:54.17,Default,,0000,0000,0000,,STUDENT: Yeah, because it\Nwas perfectly [INAUDIBLE]. Dialogue: 0,0:51:54.17,0:51:54.88,Default,,0000,0000,0000,,PROFESSOR: Yeah. Dialogue: 0,0:51:54.88,0:51:56.66,Default,,0000,0000,0000,,It's nice. Dialogue: 0,0:51:56.66,0:52:02.86,Default,,0000,0000,0000,,It's a little bit related\Nto some other problems that Dialogue: 0,0:52:02.86,0:52:03.65,Default,,0000,0000,0000,,come from pyramids. Dialogue: 0,0:52:03.65,0:52:06.51,Default,,0000,0000,0000,, Dialogue: 0,0:52:06.51,0:52:16.25,Default,,0000,0000,0000,,By the way, how can you compute\Nthe volume of a square pyramid? Dialogue: 0,0:52:16.25,0:52:21.47,Default,,0000,0000,0000,, Dialogue: 0,0:52:21.47,0:52:26.36,Default,,0000,0000,0000,,Suppose that you have\Nthe same problem. Dialogue: 0,0:52:26.36,0:52:30.61,Default,,0000,0000,0000,,Minus 1 to 1 for x and y. Dialogue: 0,0:52:30.61,0:52:34.76,Default,,0000,0000,0000,,Minus 1 to 1, minus 1 to 1. Dialogue: 0,0:52:34.76,0:53:04.56,Default,,0000,0000,0000,,Let's say the pyramid would\Nhave the something like that. Dialogue: 0,0:53:04.56,0:53:06.65,Default,,0000,0000,0000,,What would be the volume\Nof such a pyramid? Dialogue: 0,0:53:06.65,0:53:10.47,Default,,0000,0000,0000,, Dialogue: 0,0:53:10.47,0:53:12.86,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,0:53:12.86,0:53:17.96,Default,,0000,0000,0000,,PROFESSOR: The height\Nis h for extra credit. Dialogue: 0,0:53:17.96,0:53:32.64,Default,,0000,0000,0000,,Can you compute the\Nvolume of this pyramid Dialogue: 0,0:53:32.64,0:53:33.98,Default,,0000,0000,0000,,using double integrals? Dialogue: 0,0:53:33.98,0:53:40.92,Default,,0000,0000,0000,, Dialogue: 0,0:53:40.92,0:53:48.62,Default,,0000,0000,0000,,Say the height is h and the\Nbases is the square minus 1, Dialogue: 0,0:53:48.62,0:53:51.74,Default,,0000,0000,0000,,1, minus 1, 1. Dialogue: 0,0:53:51.74,0:53:54.42,Default,,0000,0000,0000,,I'm sure it can be\Ndone, but you know-- Dialogue: 0,0:53:54.42,0:53:58.06,Default,,0000,0000,0000,,now I'm testing what you\Nremember in terms of geometry Dialogue: 0,0:53:58.06,0:54:00.69,Default,,0000,0000,0000,,because we will deal\Nwith geometry a lot Dialogue: 0,0:54:00.69,0:54:03.10,Default,,0000,0000,0000,,in volumes and areas. Dialogue: 0,0:54:03.10,0:54:07.48,Default,,0000,0000,0000,,So how do you do that\Nin general, guys? Dialogue: 0,0:54:07.48,0:54:09.95,Default,,0000,0000,0000,,STUDENT: 1/3 [INAUDIBLE]. Dialogue: 0,0:54:09.95,0:54:14.26,Default,,0000,0000,0000,,PROFESSOR: 1/3 the\Nheight times the area Dialogue: 0,0:54:14.26,0:54:18.54,Default,,0000,0000,0000,,of the bases, which is what? Dialogue: 0,0:54:18.54,0:54:20.49,Default,,0000,0000,0000,,2 times 2. Dialogue: 0,0:54:20.49,0:54:28.29,Default,,0000,0000,0000,,2 times 2, 3, over 3, 4/3 h. Dialogue: 0,0:54:28.29,0:54:30.07,Default,,0000,0000,0000,,Can you prove that\Nwith calculus? Dialogue: 0,0:54:30.07,0:54:31.11,Default,,0000,0000,0000,,That's all I'm saying. Dialogue: 0,0:54:31.11,0:54:33.66,Default,,0000,0000,0000,,One point extra credit. Dialogue: 0,0:54:33.66,0:54:36.30,Default,,0000,0000,0000,,Can you prove that\Nwith calculus? Dialogue: 0,0:54:36.30,0:54:40.61,Default,,0000,0000,0000,,Actually, you would have\Nto use what you learned. Dialogue: 0,0:54:40.61,0:54:44.53,Default,,0000,0000,0000,,You can use Calc 2 as well. Dialogue: 0,0:54:44.53,0:54:46.71,Default,,0000,0000,0000,,Do you guys remember\Nthat there were Dialogue: 0,0:54:46.71,0:54:53.27,Default,,0000,0000,0000,,some cross-sectional areas, like\Nthis would be made of cheese, Dialogue: 0,0:54:53.27,0:54:56.75,Default,,0000,0000,0000,,and you come with a vertical\Nknife and cut cross sections. Dialogue: 0,0:54:56.75,0:54:57.91,Default,,0000,0000,0000,,They go like that. Dialogue: 0,0:54:57.91,0:54:59.30,Default,,0000,0000,0000,,But that's awfully hard. Dialogue: 0,0:54:59.30,0:55:02.100,Default,,0000,0000,0000,,Maybe you can do it differently\Nwith Calc 3 instead of Calc 2. Dialogue: 0,0:55:02.100,0:55:07.65,Default,,0000,0000,0000,, Dialogue: 0,0:55:07.65,0:55:10.12,Default,,0000,0000,0000,,Let's pick one from\Nthe book as well. Dialogue: 0,0:55:10.12,0:55:31.10,Default,,0000,0000,0000,, Dialogue: 0,0:55:31.10,0:55:33.18,Default,,0000,0000,0000,,OK. Dialogue: 0,0:55:33.18,0:55:38.93,Default,,0000,0000,0000,,So the same idea of using\Nthe Fubini-Tonelli argument Dialogue: 0,0:55:38.93,0:55:45.50,Default,,0000,0000,0000,,and have an iterative-- evaluate\Nthe following double integral Dialogue: 0,0:55:45.50,0:55:48.82,Default,,0000,0000,0000,,over the rectangle\Nof vertices 0, 0-- Dialogue: 0,0:55:48.82,0:55:52.12,Default,,0000,0000,0000,,write it down-- 3,\N0, 3, 2, and 0, 2. Dialogue: 0,0:55:52.12,0:56:01.55,Default,,0000,0000,0000,,So on the bases, you have a\Nrectangle of vertices 3, 0, 0, Dialogue: 0,0:56:01.55,0:56:14.37,Default,,0000,0000,0000,,0, 3, 2, and 0, 2. Dialogue: 0,0:56:14.37,0:56:18.65,Default,,0000,0000,0000,,And then somebody\Ntells you, find us Dialogue: 0,0:56:18.65,0:56:29.35,Default,,0000,0000,0000,,the double integral\Nof 2 minus y da Dialogue: 0,0:56:29.35,0:56:35.85,Default,,0000,0000,0000,,over r where r represents the\Nrectangle that we talked about. Dialogue: 0,0:56:35.85,0:56:37.73,Default,,0000,0000,0000,,This is exactly [INAUDIBLE]. Dialogue: 0,0:56:37.73,0:56:42.44,Default,,0000,0000,0000,, Dialogue: 0,0:56:42.44,0:56:45.30,Default,,0000,0000,0000,,And the answer we\Nshould get is 6. Dialogue: 0,0:56:45.30,0:56:48.92,Default,,0000,0000,0000,,And I'm saying on top of\Nwhat we said in the book, Dialogue: 0,0:56:48.92,0:56:52.98,Default,,0000,0000,0000,,can you give a geometric\Ninterpretation? Dialogue: 0,0:56:52.98,0:56:55.19,Default,,0000,0000,0000,,Does this have a\Ngeometric interpretation Dialogue: 0,0:56:55.19,0:56:57.16,Default,,0000,0000,0000,,you can think of or not? Dialogue: 0,0:56:57.16,0:57:01.37,Default,,0000,0000,0000,, Dialogue: 0,0:57:01.37,0:57:04.19,Default,,0000,0000,0000,,Well, first of all,\Nwhat is this animal? Dialogue: 0,0:57:04.19,0:57:07.05,Default,,0000,0000,0000,,According to the Fubini\Ntheorem, this animal Dialogue: 0,0:57:07.05,0:57:14.16,Default,,0000,0000,0000,,will have to be-- I have\Nit over a rectangle, Dialogue: 0,0:57:14.16,0:57:18.11,Default,,0000,0000,0000,,so assume x will be\Nbetween a and b, y Dialogue: 0,0:57:18.11,0:57:22.05,Default,,0000,0000,0000,,will be between c and d. Dialogue: 0,0:57:22.05,0:57:24.87,Default,,0000,0000,0000,,I have to figure\Nout who those are. Dialogue: 0,0:57:24.87,0:57:31.66,Default,,0000,0000,0000,,2 minus y and dy dx. Dialogue: 0,0:57:31.66,0:57:35.68,Default,,0000,0000,0000,, Dialogue: 0,0:57:35.68,0:57:37.75,Default,,0000,0000,0000,,Where is y between? Dialogue: 0,0:57:37.75,0:57:39.54,Default,,0000,0000,0000,,I should draw the\Npicture for the rectangle Dialogue: 0,0:57:39.54,0:57:42.76,Default,,0000,0000,0000,,because otherwise, it's\Nnot so easy to see. Dialogue: 0,0:57:42.76,0:57:50.90,Default,,0000,0000,0000,,I have 0, 0 here, 3, 0 here, 3,\N2 over here, shouldn't be hard. Dialogue: 0,0:57:50.90,0:57:53.22,Default,,0000,0000,0000,,So this is going to be 0, 2. Dialogue: 0,0:57:53.22,0:57:57.46,Default,,0000,0000,0000,,That's the y-axis and\Nthat's the x-axis. Dialogue: 0,0:57:57.46,0:58:00.93,Default,,0000,0000,0000,,Let's see if we can see it. Dialogue: 0,0:58:00.93,0:58:05.24,Default,,0000,0000,0000,,And what is the meaning\Nof the 6, I'm asking you? Dialogue: 0,0:58:05.24,0:58:07.15,Default,,0000,0000,0000,,I don't know. Dialogue: 0,0:58:07.15,0:58:11.38,Default,,0000,0000,0000,,x should be between\N0 and 3, right? Dialogue: 0,0:58:11.38,0:58:15.33,Default,,0000,0000,0000,,y should be between\N0 and 2, right? Dialogue: 0,0:58:15.33,0:58:16.89,Default,,0000,0000,0000,,Now you are experts in this. Dialogue: 0,0:58:16.89,0:58:20.92,Default,,0000,0000,0000,,We've done this twice, and\Nyou already know how to do it. Dialogue: 0,0:58:20.92,0:58:23.16,Default,,0000,0000,0000,,Integral from 0 to 3. Dialogue: 0,0:58:23.16,0:58:27.04,Default,,0000,0000,0000,,Then I take that,\Nand that's going Dialogue: 0,0:58:27.04,0:58:39.27,Default,,0000,0000,0000,,to be 2y minus y\Nsquared over 2 between y Dialogue: 0,0:58:39.27,0:58:43.46,Default,,0000,0000,0000,,equals 0 down and\Ny equals 2 up dx. Dialogue: 0,0:58:43.46,0:58:48.04,Default,,0000,0000,0000,, Dialogue: 0,0:58:48.04,0:58:54.69,Default,,0000,0000,0000,,That means integral from 0\Nto 3, bracket minus bracket Dialogue: 0,0:58:54.69,0:58:58.69,Default,,0000,0000,0000,,to make my life easier, dx. Dialogue: 0,0:58:58.69,0:59:02.45,Default,,0000,0000,0000,,Now, there is no x, thank god. Dialogue: 0,0:59:02.45,0:59:04.62,Default,,0000,0000,0000,,So that means I'm going\Nto have a constant Dialogue: 0,0:59:04.62,0:59:09.92,Default,,0000,0000,0000,,minus another constant, which\Nmeans I go 4 minus 4 over 2. Dialogue: 0,0:59:09.92,0:59:12.95,Default,,0000,0000,0000,,2, right? Dialogue: 0,0:59:12.95,0:59:18.43,Default,,0000,0000,0000,,The other one, for 0, I get 0. Dialogue: 0,0:59:18.43,0:59:20.60,Default,,0000,0000,0000,,I'm very happy I get 0\Nbecause in that case, Dialogue: 0,0:59:20.60,0:59:25.17,Default,,0000,0000,0000,,it's obvious that I get\N2 times 3, which is 6. Dialogue: 0,0:59:25.17,0:59:29.41,Default,,0000,0000,0000,,So I got what the book\Nsaid I'm going to get. Dialogue: 0,0:59:29.41,0:59:32.13,Default,,0000,0000,0000,,But do I have a geometric\Ninterpretation of that? Dialogue: 0,0:59:32.13,0:59:37.16,Default,,0000,0000,0000,,I would like to see\Nif anybody can-- Dialogue: 0,0:59:37.16,0:59:41.14,Default,,0000,0000,0000,,I'm going to give you a\Nbreak in a few minues-- Dialogue: 0,0:59:41.14,0:59:45.97,Default,,0000,0000,0000,,if anybody can think of a\Ngeometric interpretation. Dialogue: 0,0:59:45.97,0:59:52.63,Default,,0000,0000,0000,,What is this f of xy if I were\Nto interpret this as a graph? Dialogue: 0,0:59:52.63,0:59:55.10,Default,,0000,0000,0000,,x equals f of x and y. Dialogue: 0,0:59:55.10,0:59:55.76,Default,,0000,0000,0000,,Is this-- Dialogue: 0,0:59:55.76,0:59:57.52,Default,,0000,0000,0000,,STUDENT: 2 minus y. Dialogue: 0,0:59:57.52,1:00:04.55,Default,,0000,0000,0000,,PROFESSOR: So z equals 2\Nminus y is a plane, right? Dialogue: 0,1:00:04.55,1:00:08.14,Default,,0000,0000,0000,,STUDENT: Yes, but then you have\Nthe parabola is going down. Dialogue: 0,1:00:08.14,1:00:11.21,Default,,0000,0000,0000,,PROFESSOR: And how do I get\Nto draw this plane the best? Dialogue: 0,1:00:11.21,1:00:13.88,Default,,0000,0000,0000,,Because there are\Nmany ways to do it. Dialogue: 0,1:00:13.88,1:00:16.70,Default,,0000,0000,0000,,I look at this wall. Dialogue: 0,1:00:16.70,1:00:19.12,Default,,0000,0000,0000,,The y-axis is this. Dialogue: 0,1:00:19.12,1:00:21.09,Default,,0000,0000,0000,,The z-axis is the vertical line. Dialogue: 0,1:00:21.09,1:00:23.41,Default,,0000,0000,0000,,So I'm looking at this plane. Dialogue: 0,1:00:23.41,1:00:27.63,Default,,0000,0000,0000,,y plus z must be equal to 2. Dialogue: 0,1:00:27.63,1:00:29.85,Default,,0000,0000,0000,,So when is y plus z equal to 2? Dialogue: 0,1:00:29.85,1:00:34.15,Default,,0000,0000,0000,,When I am on a\Nline in the plane. Dialogue: 0,1:00:34.15,1:00:39.15,Default,,0000,0000,0000,,I'm going to draw that line\Nwith pink because I like pink. Dialogue: 0,1:00:39.15,1:00:41.24,Default,,0000,0000,0000,,This is y plus z equals 2. Dialogue: 0,1:00:41.24,1:00:44.48,Default,,0000,0000,0000,, Dialogue: 0,1:00:44.48,1:00:50.48,Default,,0000,0000,0000,,And imagine this line will be\Nshifted by parallelism as it Dialogue: 0,1:00:50.48,1:00:54.94,Default,,0000,0000,0000,,comes towards you on all these\Nother parallel vertical planes Dialogue: 0,1:00:54.94,1:00:57.72,Default,,0000,0000,0000,,that are parallel to the board. Dialogue: 0,1:00:57.72,1:01:04.77,Default,,0000,0000,0000,,So I'm going to have an\Nentire plane like that, Dialogue: 0,1:01:04.77,1:01:09.30,Default,,0000,0000,0000,,and I'm going to stop here. Dialogue: 0,1:01:09.30,1:01:13.10,Default,,0000,0000,0000,,When I'm in the plane\Nthat's called x equals 3-- Dialogue: 0,1:01:13.10,1:01:15.39,Default,,0000,0000,0000,,this is the plane\Ncalled x equals Dialogue: 0,1:01:15.39,1:01:20.16,Default,,0000,0000,0000,,3-- I have exactly this\Ntriangle, this [INAUDIBLE]. Dialogue: 0,1:01:20.16,1:01:23.69,Default,,0000,0000,0000,,It's in the plane\Nthat faces me here. Dialogue: 0,1:01:23.69,1:01:26.06,Default,,0000,0000,0000,,I don't know if\Nyou realize that. Dialogue: 0,1:01:26.06,1:01:30.72,Default,,0000,0000,0000,,I'll help you make a\Nhouse or something nice. Dialogue: 0,1:01:30.72,1:01:32.65,Default,,0000,0000,0000,,I think I'm getting hungry. Dialogue: 0,1:01:32.65,1:01:35.65,Default,,0000,0000,0000,,I imagine this again as\Nbeing a piece of cheese, Dialogue: 0,1:01:35.65,1:01:39.75,Default,,0000,0000,0000,,or it looks even like a piece\Nof cake would be with layers. Dialogue: 0,1:01:39.75,1:01:42.85,Default,,0000,0000,0000,, Dialogue: 0,1:01:42.85,1:01:47.56,Default,,0000,0000,0000,,So our question is, if\Nwe didn't know calculus Dialogue: 0,1:01:47.56,1:01:51.21,Default,,0000,0000,0000,,but we knew how to draw\Nthis, and somebody gave you Dialogue: 0,1:01:51.21,1:01:53.79,Default,,0000,0000,0000,,this at the GRE\Nor whatever exam, Dialogue: 0,1:01:53.79,1:01:55.62,Default,,0000,0000,0000,,how could you have done\Nit without calculus? Dialogue: 0,1:01:55.62,1:02:00.48,Default,,0000,0000,0000,,Just by cheating and\Npretending, I know how to do it, Dialogue: 0,1:02:00.48,1:02:02.92,Default,,0000,0000,0000,,but you've never done a\Ndouble integral in your life. Dialogue: 0,1:02:02.92,1:02:06.18,Default,,0000,0000,0000,,So I know it's a volume. Dialogue: 0,1:02:06.18,1:02:08.66,Default,,0000,0000,0000,,How do I get the volume? Dialogue: 0,1:02:08.66,1:02:10.15,Default,,0000,0000,0000,,What kind of geometric\Nbody is that? Dialogue: 0,1:02:10.15,1:02:11.64,Default,,0000,0000,0000,,STUDENT: A triangle. Dialogue: 0,1:02:11.64,1:02:13.63,Default,,0000,0000,0000,,STUDENT: It's a\Ntriangular prism. Dialogue: 0,1:02:13.63,1:02:15.62,Default,,0000,0000,0000,,PROFESSOR: It's a\Ntriangular prism. Dialogue: 0,1:02:15.62,1:02:16.12,Default,,0000,0000,0000,,Good. Dialogue: 0,1:02:16.12,1:02:19.62,Default,,0000,0000,0000,,And a triangular prism\Nhas what volume formula? Dialogue: 0,1:02:19.62,1:02:20.75,Default,,0000,0000,0000,,STUDENT: Base times height. Dialogue: 0,1:02:20.75,1:02:22.45,Default,,0000,0000,0000,,PROFESSOR: Base\Ntimes the height. Dialogue: 0,1:02:22.45,1:02:25.93,Default,,0000,0000,0000,,And the height has what area? Dialogue: 0,1:02:25.93,1:02:27.44,Default,,0000,0000,0000,,Let's see. Dialogue: 0,1:02:27.44,1:02:30.22,Default,,0000,0000,0000,,The base would be that, right? Dialogue: 0,1:02:30.22,1:02:34.17,Default,,0000,0000,0000,,And the height would be 3. Dialogue: 0,1:02:34.17,1:02:36.15,Default,,0000,0000,0000,,Am I right or not? Dialogue: 0,1:02:36.15,1:02:37.58,Default,,0000,0000,0000,,The height would be 3. Dialogue: 0,1:02:37.58,1:02:38.12,Default,,0000,0000,0000,,This is not-- Dialogue: 0,1:02:38.12,1:02:38.99,Default,,0000,0000,0000,,STUDENT: It's 2. Dialogue: 0,1:02:38.99,1:02:39.49,Default,,0000,0000,0000,,Yeah. Dialogue: 0,1:02:39.49,1:02:40.32,Default,,0000,0000,0000,,STUDENT: No, it's 3. Dialogue: 0,1:02:40.32,1:02:41.91,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NFrom here to here? Dialogue: 0,1:02:41.91,1:02:42.45,Default,,0000,0000,0000,,STUDENT: 3. Dialogue: 0,1:02:42.45,1:02:43.58,Default,,0000,0000,0000,,DR. MAGDALENA TODA: It's 3. Dialogue: 0,1:02:43.58,1:02:46.86,Default,,0000,0000,0000,,So how much is that? Dialogue: 0,1:02:46.86,1:02:47.69,Default,,0000,0000,0000,,How much-- OK. Dialogue: 0,1:02:47.69,1:02:50.01,Default,,0000,0000,0000,,From here to here is 2. Dialogue: 0,1:02:50.01,1:02:54.20,Default,,0000,0000,0000,,From here to here,\Nit's how much? Dialogue: 0,1:02:54.20,1:02:56.12,Default,,0000,0000,0000,,STUDENT: The height\Nis only-- I see-- Dialogue: 0,1:02:56.12,1:02:57.06,Default,,0000,0000,0000,,STUDENT: It's also 2. Dialogue: 0,1:02:57.06,1:02:59.44,Default,,0000,0000,0000,,DR. MAGDALENA TODA: It's\Nalso 2 because look at that. Dialogue: 0,1:02:59.44,1:03:01.82,Default,,0000,0000,0000,,It's an isosceles triangle. Dialogue: 0,1:03:01.82,1:03:03.73,Default,,0000,0000,0000,,This is 45 to 45. Dialogue: 0,1:03:03.73,1:03:05.25,Default,,0000,0000,0000,,So this is also 2. Dialogue: 0,1:03:05.25,1:03:08.86,Default,,0000,0000,0000,,2 to-- that's 90\Ndegrees, 45, 45. Dialogue: 0,1:03:08.86,1:03:09.36,Default,,0000,0000,0000,,OK. Dialogue: 0,1:03:09.36,1:03:13.22,Default,,0000,0000,0000,,So the area of the shaded purple\Ntriangle-- how much is that? Dialogue: 0,1:03:13.22,1:03:13.96,Default,,0000,0000,0000,,STUDENT: 2. Dialogue: 0,1:03:13.96,1:03:14.88,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 2. Dialogue: 0,1:03:14.88,1:03:17.12,Default,,0000,0000,0000,,2 times 2 over 2. Dialogue: 0,1:03:17.12,1:03:19.71,Default,,0000,0000,0000,,2 times 3 equals 6. Dialogue: 0,1:03:19.71,1:03:22.22,Default,,0000,0000,0000,,I don't need calculus. Dialogue: 0,1:03:22.22,1:03:24.17,Default,,0000,0000,0000,,In this case, I\Ndon't need calculus. Dialogue: 0,1:03:24.17,1:03:27.17,Default,,0000,0000,0000,,But when I have those\Nnasty curvilinear Dialogue: 0,1:03:27.17,1:03:31.96,Default,,0000,0000,0000,,z equals f of x, y, complicated\Nexpressions, I have no choice. Dialogue: 0,1:03:31.96,1:03:34.95,Default,,0000,0000,0000,,I have to do the\Ndouble integral. Dialogue: 0,1:03:34.95,1:03:37.95,Default,,0000,0000,0000,,But in this case, even if\NI didn't know how to do it, Dialogue: 0,1:03:37.95,1:03:39.20,Default,,0000,0000,0000,,I would still get the 6. Dialogue: 0,1:03:39.20,1:03:39.98,Default,,0000,0000,0000,,Yes, sir? Dialogue: 0,1:03:39.98,1:03:42.74,Default,,0000,0000,0000,,STUDENT: What if we\Ndid that on the exam? Dialogue: 0,1:03:42.74,1:03:44.32,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NWell, that's good. Dialogue: 0,1:03:44.32,1:03:45.74,Default,,0000,0000,0000,,I will then keep it in mind. Dialogue: 0,1:03:45.74,1:03:46.24,Default,,0000,0000,0000,,Yes. Dialogue: 0,1:03:46.24,1:03:48.56,Default,,0000,0000,0000,,It doesn't matter to me. Dialogue: 0,1:03:48.56,1:03:50.90,Default,,0000,0000,0000,,I have other colleagues who\Nreally care about the method Dialogue: 0,1:03:50.90,1:03:52.35,Default,,0000,0000,0000,,and start complaining. Dialogue: 0,1:03:52.35,1:03:55.56,Default,,0000,0000,0000,,I don't care how you\Nget to the answer Dialogue: 0,1:03:55.56,1:03:57.45,Default,,0000,0000,0000,,as long as you got\Nthe right answer. Dialogue: 0,1:03:57.45,1:03:59.40,Default,,0000,0000,0000,,Let me tell you my logic. Dialogue: 0,1:03:59.40,1:04:03.83,Default,,0000,0000,0000,,Suppose somebody hired you\Nthinking you're a good worker, Dialogue: 0,1:04:03.83,1:04:05.33,Default,,0000,0000,0000,,and you're smart and so on. Dialogue: 0,1:04:05.33,1:04:09.82,Default,,0000,0000,0000,,Would they care how you got to\Nthe solution of the problem? Dialogue: 0,1:04:09.82,1:04:14.22,Default,,0000,0000,0000,,As long as the problem\Nwas solved correctly, no. Dialogue: 0,1:04:14.22,1:04:18.25,Default,,0000,0000,0000,,And actually, the elementary\Nway is the fastest Dialogue: 0,1:04:18.25,1:04:20.12,Default,,0000,0000,0000,,because it's just 10 seconds. Dialogue: 0,1:04:20.12,1:04:20.92,Default,,0000,0000,0000,,You draw. Dialogue: 0,1:04:20.92,1:04:21.69,Default,,0000,0000,0000,,You imagine. Dialogue: 0,1:04:21.69,1:04:23.18,Default,,0000,0000,0000,,You know what it is. Dialogue: 0,1:04:23.18,1:04:28.31,Default,,0000,0000,0000,,So your boss will want you to\Nfind the fastest way to provide Dialogue: 0,1:04:28.31,1:04:29.23,Default,,0000,0000,0000,,the correct solution. Dialogue: 0,1:04:29.23,1:04:33.48,Default,,0000,0000,0000,,He's not going to\Ncare how you got that. Dialogue: 0,1:04:33.48,1:04:35.58,Default,,0000,0000,0000,,So no matter how\Nyou do it, as long Dialogue: 0,1:04:35.58,1:04:39.99,Default,,0000,0000,0000,,as you've got the right\Nanswer, I'm going to be happy. Dialogue: 0,1:04:39.99,1:04:49.23,Default,,0000,0000,0000,,I want to ask you to please\Ngo to page 927 in the book Dialogue: 0,1:04:49.23,1:04:50.38,Default,,0000,0000,0000,,and read. Dialogue: 0,1:04:50.38,1:04:52.58,Default,,0000,0000,0000,,It's only one page. Dialogue: 0,1:04:52.58,1:04:55.34,Default,,0000,0000,0000,,That whole end section, 12.1. Dialogue: 0,1:04:55.34,1:04:59.93,Default,,0000,0000,0000,,It's called an informal\Nargument for Fubini's theorem. Dialogue: 0,1:04:59.93,1:05:05.74,Default,,0000,0000,0000,,Practically, it's a proof of\NFubini's theorem, page 927. Dialogue: 0,1:05:05.74,1:05:08.64,Default,,0000,0000,0000,,And then I'm going to go\Nahead and start the homework Dialogue: 0,1:05:08.64,1:05:11.82,Default,,0000,0000,0000,,four, if you don't mind. Dialogue: 0,1:05:11.82,1:05:15.92,Default,,0000,0000,0000,,I'm going to go into WeBWork\Nand give you homework four. Dialogue: 0,1:05:15.92,1:05:18.58,Default,,0000,0000,0000,,And the first few\Nproblems that you Dialogue: 0,1:05:18.58,1:05:21.10,Default,,0000,0000,0000,,are going to be\Nexpected to solve Dialogue: 0,1:05:21.10,1:05:27.31,Default,,0000,0000,0000,,will be out of 12.1,\Nwhich is really easy. Dialogue: 0,1:05:27.31,1:05:28.79,Default,,0000,0000,0000,,I'll give you a\Nfew minutes back. Dialogue: 0,1:05:28.79,1:05:32.74,Default,,0000,0000,0000,,And we go on with 12.2,\Nand it's very similar. Dialogue: 0,1:05:32.74,1:05:34.71,Default,,0000,0000,0000,,You're going to like that. Dialogue: 0,1:05:34.71,1:05:39.64,Default,,0000,0000,0000,,And then we'll go home or\Nwherever we need to go. Dialogue: 0,1:05:39.64,1:05:42.10,Default,,0000,0000,0000,,So you have a few\Nminutes of a break. Dialogue: 0,1:05:42.10,1:05:45.56,Default,,0000,0000,0000,,Pick up your extra credits. Dialogue: 0,1:05:45.56,1:05:47.13,Default,,0000,0000,0000,,I'll call the names. Dialogue: 0,1:05:47.13,1:05:48.60,Default,,0000,0000,0000,,Lily. Dialogue: 0,1:05:48.60,1:05:52.02,Default,,0000,0000,0000,,You got a lot of points. Dialogue: 0,1:05:52.02,1:05:54.95,Default,,0000,0000,0000,,And [INAUDIBLE]. Dialogue: 0,1:05:54.95,1:05:56.91,Default,,0000,0000,0000,,And you have two separate ones. Dialogue: 0,1:05:56.91,1:05:58.36,Default,,0000,0000,0000,,Nathan. Dialogue: 0,1:05:58.36,1:05:58.86,Default,,0000,0000,0000,,Nathan? Dialogue: 0,1:05:58.86,1:06:02.31,Default,,0000,0000,0000,, Dialogue: 0,1:06:02.31,1:06:03.04,Default,,0000,0000,0000,,Rachel Smith. Dialogue: 0,1:06:03.04,1:06:05.73,Default,,0000,0000,0000,, Dialogue: 0,1:06:05.73,1:06:06.23,Default,,0000,0000,0000,,Austin. Dialogue: 0,1:06:06.23,1:06:09.28,Default,,0000,0000,0000,, Dialogue: 0,1:06:09.28,1:06:09.78,Default,,0000,0000,0000,,Thank you. Dialogue: 0,1:06:09.78,1:06:12.74,Default,,0000,0000,0000,, Dialogue: 0,1:06:12.74,1:06:13.72,Default,,0000,0000,0000,,Edgar. Dialogue: 0,1:06:13.72,1:06:16.18,Default,,0000,0000,0000,,[INAUDIBLE] Dialogue: 0,1:06:16.18,1:06:16.68,Default,,0000,0000,0000,,Aaron. Dialogue: 0,1:06:16.68,1:06:24.07,Default,,0000,0000,0000,, Dialogue: 0,1:06:24.07,1:06:24.57,Default,,0000,0000,0000,,Andre. Dialogue: 0,1:06:24.57,1:06:32.46,Default,,0000,0000,0000,, Dialogue: 0,1:06:32.46,1:06:35.41,Default,,0000,0000,0000,,Aaron. Dialogue: 0,1:06:35.41,1:06:35.91,Default,,0000,0000,0000,,Kasey. Dialogue: 0,1:06:35.91,1:06:39.85,Default,,0000,0000,0000,, Dialogue: 0,1:06:39.85,1:06:43.49,Default,,0000,0000,0000,,Kasey came up with\Na very good idea Dialogue: 0,1:06:43.49,1:06:47.53,Default,,0000,0000,0000,,that I will write\Na review sample. Dialogue: 0,1:06:47.53,1:06:48.62,Default,,0000,0000,0000,,Did I promise that? Dialogue: 0,1:06:48.62,1:06:52.20,Default,,0000,0000,0000,,A review sample for the midterm. Dialogue: 0,1:06:52.20,1:06:53.72,Default,,0000,0000,0000,,And so I said yes. Dialogue: 0,1:06:53.72,1:06:56.70,Default,,0000,0000,0000,, Dialogue: 0,1:06:56.70,1:07:01.16,Default,,0000,0000,0000,,Karen and Matthew. Dialogue: 0,1:07:01.16,1:07:07.60,Default,,0000,0000,0000,, Dialogue: 0,1:07:07.60,1:07:08.10,Default,,0000,0000,0000,,Reagan. Dialogue: 0,1:07:08.10,1:07:16.04,Default,,0000,0000,0000,, Dialogue: 0,1:07:16.04,1:07:17.91,Default,,0000,0000,0000,,Aaron. Dialogue: 0,1:07:17.91,1:07:20.51,Default,,0000,0000,0000,,When you submitted,\Nyou submitted. Dialogue: 0,1:07:20.51,1:07:21.19,Default,,0000,0000,0000,,Yeah. Dialogue: 0,1:07:21.19,1:07:21.86,Default,,0000,0000,0000,,And [INAUDIBLE]. Dialogue: 0,1:07:21.86,1:07:25.86,Default,,0000,0000,0000,, Dialogue: 0,1:07:25.86,1:07:26.86,Default,,0000,0000,0000,,here. Dialogue: 0,1:07:26.86,1:07:27.86,Default,,0000,0000,0000,,And I'm done. Dialogue: 0,1:07:27.86,1:07:46.36,Default,,0000,0000,0000,, Dialogue: 0,1:07:46.36,1:07:48.36,Default,,0000,0000,0000,,STUDENT: Did we\Nturn in [INAUDIBLE]? Dialogue: 0,1:07:48.36,1:07:49.86,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\NYes, absolutely. Dialogue: 0,1:07:49.86,1:08:08.86,Default,,0000,0000,0000,, Dialogue: 0,1:08:08.86,1:08:12.44,Default,,0000,0000,0000,,Now once we go over\N12.2, you will say, oh, Dialogue: 0,1:08:12.44,1:08:14.43,Default,,0000,0000,0000,,but I understand\Nthe Fubini theorem. Dialogue: 0,1:08:14.43,1:08:21.44,Default,,0000,0000,0000,, Dialogue: 0,1:08:21.44,1:08:23.93,Default,,0000,0000,0000,,I didn't know whether\Nthere's room for Fubini, Dialogue: 0,1:08:23.93,1:08:29.25,Default,,0000,0000,0000,,because once I cover the more\Ngeneral case, which is in 12.2, Dialogue: 0,1:08:29.25,1:08:33.58,Default,,0000,0000,0000,,you are going to understand\NWhy Fubini-Tonelli Dialogue: 0,1:08:33.58,1:08:36.97,Default,,0000,0000,0000,,works for rectangles. Dialogue: 0,1:08:36.97,1:08:47.22,Default,,0000,0000,0000,,So if I think of a domain\Nthat is of the following form, Dialogue: 0,1:08:47.22,1:08:54.21,Default,,0000,0000,0000,,in the x, y plane, I go x\Nis between and and b, right? Dialogue: 0,1:08:54.21,1:08:59.68,Default,,0000,0000,0000,,That's my favorite x. Dialogue: 0,1:08:59.68,1:09:02.34,Default,,0000,0000,0000,,So I take the pink\Nsegment, and I Dialogue: 0,1:09:02.34,1:09:05.05,Default,,0000,0000,0000,,say, everything that\Nhappens-- it's going Dialogue: 0,1:09:05.05,1:09:08.45,Default,,0000,0000,0000,,to happen on top of this world. Dialogue: 0,1:09:08.45,1:09:11.36,Default,,0000,0000,0000,,I have, let's say,\Ntwo functions. Dialogue: 0,1:09:11.36,1:09:14.10,Default,,0000,0000,0000,,To make my life easier, I'll\Nassume both of them [INAUDIBLE] Dialogue: 0,1:09:14.10,1:09:15.83,Default,,0000,0000,0000,,one bigger than the other. Dialogue: 0,1:09:15.83,1:09:23.88,Default,,0000,0000,0000,,But in case they are\Nnot both positive, Dialogue: 0,1:09:23.88,1:09:28.08,Default,,0000,0000,0000,,I just need f to be bigger\Nthan g for every point. Dialogue: 0,1:09:28.08,1:09:32.74,Default,,0000,0000,0000,,And the same argument\Nwill function. Dialogue: 0,1:09:32.74,1:09:38.89,Default,,0000,0000,0000,,This is f, continuous positive. Dialogue: 0,1:09:38.89,1:09:41.100,Default,,0000,0000,0000,,Then g, continuous\Npositive but smaller Dialogue: 0,1:09:41.100,1:09:44.91,Default,,0000,0000,0000,,in values than this one. Dialogue: 0,1:09:44.91,1:09:47.83,Default,,0000,0000,0000,, Dialogue: 0,1:09:47.83,1:09:48.80,Default,,0000,0000,0000,,Yes, sir? Dialogue: 0,1:09:48.80,1:09:51.19,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\N12.2 that we're starting? Dialogue: 0,1:09:51.19,1:09:52.23,Default,,0000,0000,0000,,DR. MAGDALENA TODA: 12.2. Dialogue: 0,1:09:52.23,1:09:56.00,Default,,0000,0000,0000,,And you are more organized\Nthan I am, and I appreciate it. Dialogue: 0,1:09:56.00,1:10:02.45,Default,,0000,0000,0000,,So integration over a\Nnon-rectangular domain. Dialogue: 0,1:10:02.45,1:10:06.93,Default,,0000,0000,0000,, Dialogue: 0,1:10:06.93,1:10:10.36,Default,,0000,0000,0000,,And we call this a\Ntype one because this Dialogue: 0,1:10:10.36,1:10:12.32,Default,,0000,0000,0000,,is what many books are using. Dialogue: 0,1:10:12.32,1:10:17.03,Default,,0000,0000,0000,,And this is that x is\Nbetween two fixed end points. Dialogue: 0,1:10:17.03,1:10:21.27,Default,,0000,0000,0000,,But y is between two\Nvariable end points. Dialogue: 0,1:10:21.27,1:10:24.30,Default,,0000,0000,0000,,So what's going to happen to y? Dialogue: 0,1:10:24.30,1:10:29.48,Default,,0000,0000,0000,,y is going to take\Nvalues between the lower, Dialogue: 0,1:10:29.48,1:10:34.75,Default,,0000,0000,0000,,the bottom one, which is\Ng of x, and the upper one, Dialogue: 0,1:10:34.75,1:10:37.06,Default,,0000,0000,0000,,which is f of x. Dialogue: 0,1:10:37.06,1:10:39.73,Default,,0000,0000,0000,,So this is how we\Ndefine the domain that's Dialogue: 0,1:10:39.73,1:10:45.23,Default,,0000,0000,0000,,shaded by me with black\Nshades, vertical strips here. Dialogue: 0,1:10:45.23,1:10:47.98,Default,,0000,0000,0000,,This is the domain. Dialogue: 0,1:10:47.98,1:10:56.24,Default,,0000,0000,0000,,Now you really do\Nnot need to prove Dialogue: 0,1:10:56.24,1:11:09.25,Default,,0000,0000,0000,,that double integral over\N1 dA over-- let's call Dialogue: 0,1:11:09.25,1:11:15.42,Default,,0000,0000,0000,,the domain D-- is what? Dialogue: 0,1:11:15.42,1:11:17.98,Default,,0000,0000,0000,, Dialogue: 0,1:11:17.98,1:11:27.78,Default,,0000,0000,0000,,Integral between f of x\Nminus g of x from a to b dx. Dialogue: 0,1:11:27.78,1:11:31.08,Default,,0000,0000,0000,, Dialogue: 0,1:11:31.08,1:11:32.09,Default,,0000,0000,0000,,And you say, what? Dialogue: 0,1:11:32.09,1:11:33.67,Default,,0000,0000,0000,,Magdalena, what are\Nyou trying to say? Dialogue: 0,1:11:33.67,1:11:34.76,Default,,0000,0000,0000,,OK. Dialogue: 0,1:11:34.76,1:11:37.33,Default,,0000,0000,0000,,Let's go back and\Nsay, what if somebody Dialogue: 0,1:11:37.33,1:11:41.37,Default,,0000,0000,0000,,would have asked you the\Nsame question in calculus 2? Dialogue: 0,1:11:41.37,1:11:44.64,Default,,0000,0000,0000,,Saying, guys I have a\Nquestion about the area Dialogue: 0,1:11:44.64,1:11:49.08,Default,,0000,0000,0000,,in the shaded strip,\Nvertical strip thing. Dialogue: 0,1:11:49.08,1:11:50.87,Default,,0000,0000,0000,,How are we going\Nto compute that? Dialogue: 0,1:11:50.87,1:11:53.59,Default,,0000,0000,0000,,And you would say,\Noh, I have an idea. Dialogue: 0,1:11:53.59,1:12:04.02,Default,,0000,0000,0000,,I take the area under the graph\Nf, and I shade that in orange. Dialogue: 0,1:12:04.02,1:12:05.92,Default,,0000,0000,0000,,And I know what that is. Dialogue: 0,1:12:05.92,1:12:07.50,Default,,0000,0000,0000,,So you would say, I\Nknow what that is. Dialogue: 0,1:12:07.50,1:12:08.99,Default,,0000,0000,0000,,That's going to be what? Dialogue: 0,1:12:08.99,1:12:13.28,Default,,0000,0000,0000,,Integral from a to be f of x dx. Dialogue: 0,1:12:13.28,1:12:16.89,Default,,0000,0000,0000,,Let's call that A1, right? Dialogue: 0,1:12:16.89,1:12:19.63,Default,,0000,0000,0000,,A1. Dialogue: 0,1:12:19.63,1:12:27.92,Default,,0000,0000,0000,,Then you go, minus the area\Nwith-- I'm just going to shade Dialogue: 0,1:12:27.92,1:12:32.13,Default,,0000,0000,0000,,that, brown strips under g. Dialogue: 0,1:12:32.13,1:12:35.05,Default,,0000,0000,0000,, Dialogue: 0,1:12:35.05,1:12:37.98,Default,,0000,0000,0000,,g of x dx. Dialogue: 0,1:12:37.98,1:12:39.32,Default,,0000,0000,0000,,And call that A2. Dialogue: 0,1:12:39.32,1:12:42.31,Default,,0000,0000,0000,, Dialogue: 0,1:12:42.31,1:12:45.30,Default,,0000,0000,0000,,A1 minus A2. Dialogue: 0,1:12:45.30,1:12:49.33,Default,,0000,0000,0000,,We know both of these\Nformulas from where? Dialogue: 0,1:12:49.33,1:12:52.90,Default,,0000,0000,0000,,Calc 1 because that's where\Nyou learned about the area Dialogue: 0,1:12:52.90,1:12:55.04,Default,,0000,0000,0000,,under the graph of a curve. Dialogue: 0,1:12:55.04,1:12:57.95,Default,,0000,0000,0000,,This is the area under\Nthe graph of a curve f. Dialogue: 0,1:12:57.95,1:13:00.88,Default,,0000,0000,0000,,This is the area under\Nthe graph of the curve g. Dialogue: 0,1:13:00.88,1:13:04.55,Default,,0000,0000,0000,,The black striped area\Nis their difference. Dialogue: 0,1:13:04.55,1:13:05.16,Default,,0000,0000,0000,,All right. Dialogue: 0,1:13:05.16,1:13:07.19,Default,,0000,0000,0000,,And so how much is that? Dialogue: 0,1:13:07.19,1:13:09.13,Default,,0000,0000,0000,,I'm sorry I put the wrong thing. Dialogue: 0,1:13:09.13,1:13:11.56,Default,,0000,0000,0000,,a, b. Dialogue: 0,1:13:11.56,1:13:13.53,Default,,0000,0000,0000,,That's going to be\Nintegral from a to b. Dialogue: 0,1:13:13.53,1:13:15.89,Default,,0000,0000,0000,,Now you say, wait,\Nwait, wait a minute. Dialogue: 0,1:13:15.89,1:13:17.13,Default,,0000,0000,0000,,Based on what? Dialogue: 0,1:13:17.13,1:13:20.26,Default,,0000,0000,0000,,Based on some sort of\Nadditivity property Dialogue: 0,1:13:20.26,1:13:23.64,Default,,0000,0000,0000,,of the integral of one\Nvariable, which says integral Dialogue: 0,1:13:23.64,1:13:27.02,Default,,0000,0000,0000,,from a to b of f plus g. Dialogue: 0,1:13:27.02,1:13:29.27,Default,,0000,0000,0000,,You can have f plus, minus g. Dialogue: 0,1:13:29.27,1:13:30.54,Default,,0000,0000,0000,,It doesn't matter. Dialogue: 0,1:13:30.54,1:13:31.94,Default,,0000,0000,0000,,dx. Dialogue: 0,1:13:31.94,1:13:37.88,Default,,0000,0000,0000,,You have integral from a to b f\Ndx plus integral from a to b g Dialogue: 0,1:13:37.88,1:13:39.18,Default,,0000,0000,0000,,dx. Dialogue: 0,1:13:39.18,1:13:42.39,Default,,0000,0000,0000,,It doesn't matter what. Dialogue: 0,1:13:42.39,1:13:46.05,Default,,0000,0000,0000,,You can have a linear\Ncombination of f and g. Dialogue: 0,1:13:46.05,1:13:46.91,Default,,0000,0000,0000,,Yes, Matthew? Dialogue: 0,1:13:46.91,1:13:49.18,Default,,0000,0000,0000,,MATTHEW: So this is\Njust for the domain? Dialogue: 0,1:13:49.18,1:13:52.79,Default,,0000,0000,0000,,So if you put it,\Nthat would be down. Dialogue: 0,1:13:52.79,1:13:55.53,Default,,0000,0000,0000,,So there might be\Nanother formula up here Dialogue: 0,1:13:55.53,1:13:57.35,Default,,0000,0000,0000,,that would be curved surface. Dialogue: 0,1:13:57.35,1:13:59.56,Default,,0000,0000,0000,,And this is the bottom,\Nso you're using integral Dialogue: 0,1:13:59.56,1:14:01.26,Default,,0000,0000,0000,,to find the base,\Nand then you're Dialogue: 0,1:14:01.26,1:14:03.71,Default,,0000,0000,0000,,going to plug that integral\Ninto the other integral. Dialogue: 0,1:14:03.71,1:14:06.08,Default,,0000,0000,0000,,DR. MAGDALENA TODA: So I'm\Njust using the property that's Dialogue: 0,1:14:06.08,1:14:10.66,Default,,0000,0000,0000,,called linearity of\Nthe simple integral, Dialogue: 0,1:14:10.66,1:14:14.66,Default,,0000,0000,0000,,meaning that if I have even\Na linear combination like af Dialogue: 0,1:14:14.66,1:14:22.25,Default,,0000,0000,0000,,plus bg, then a-- I have not a. Dialogue: 0,1:14:22.25,1:14:26.76,Default,,0000,0000,0000,,Let me call it big A and\Nbig B. Big A Af integral Dialogue: 0,1:14:26.76,1:14:29.36,Default,,0000,0000,0000,,of f plus big B integral of g. Dialogue: 0,1:14:29.36,1:14:30.61,Default,,0000,0000,0000,,You've learned that in Calc 2. Dialogue: 0,1:14:30.61,1:14:34.39,Default,,0000,0000,0000,,I'm doing this to apply it for\Nthese areas that are subtracted Dialogue: 0,1:14:34.39,1:14:36.25,Default,,0000,0000,0000,,from one another. Dialogue: 0,1:14:36.25,1:14:39.12,Default,,0000,0000,0000,,If I were to add, as you\Nsaid, I would put something Dialogue: 0,1:14:39.12,1:14:39.99,Default,,0000,0000,0000,,on top of that. Dialogue: 0,1:14:39.99,1:14:44.65,Default,,0000,0000,0000,,And then it would be like\Na superimposition onto it. Dialogue: 0,1:14:44.65,1:14:54.19,Default,,0000,0000,0000,,So I have integral from a to\Nb of f of x minus g of x dx. Dialogue: 0,1:14:54.19,1:14:56.84,Default,,0000,0000,0000,,And I claim that\Nthis is the same Dialogue: 0,1:14:56.84,1:15:06.90,Default,,0000,0000,0000,,as double integral of the\N1dA over the domain D. Dialogue: 0,1:15:06.90,1:15:10.17,Default,,0000,0000,0000,,How can you write\Nthat differently? Dialogue: 0,1:15:10.17,1:15:12.04,Default,,0000,0000,0000,,I'll tell you how you\Nwrite that differently. Dialogue: 0,1:15:12.04,1:15:19.15,Default,,0000,0000,0000,,Integral from a to b of\Nintegral from-- what's Dialogue: 0,1:15:19.15,1:15:21.34,Default,,0000,0000,0000,,the bottom value of Mr. Y? Dialogue: 0,1:15:21.34,1:15:23.88,Default,,0000,0000,0000,, Dialogue: 0,1:15:23.88,1:15:26.62,Default,,0000,0000,0000,,So Mr. X knows what he's doing. Dialogue: 0,1:15:26.62,1:15:28.62,Default,,0000,0000,0000,,He goes all the way from a to b. Dialogue: 0,1:15:28.62,1:15:31.33,Default,,0000,0000,0000,,The bottom value of y is g of x. Dialogue: 0,1:15:31.33,1:15:36.25,Default,,0000,0000,0000,,You go from the bottom value\Nof y g of x to the upper value Dialogue: 0,1:15:36.25,1:15:38.08,Default,,0000,0000,0000,,f of x. Dialogue: 0,1:15:38.08,1:15:42.32,Default,,0000,0000,0000,,And then you here put 1 and dy. Dialogue: 0,1:15:42.32,1:15:44.66,Default,,0000,0000,0000,,Is this the same thing? Dialogue: 0,1:15:44.66,1:15:46.44,Default,,0000,0000,0000,,You say, OK, I know this one. Dialogue: 0,1:15:46.44,1:15:49.29,Default,,0000,0000,0000,,I know this one from calc 2. Dialogue: 0,1:15:49.29,1:15:53.89,Default,,0000,0000,0000,,But Magdalena, the one\Nyou gave us is new. Dialogue: 0,1:15:53.89,1:15:55.45,Default,,0000,0000,0000,,It's new and not new, guys. Dialogue: 0,1:15:55.45,1:15:59.36,Default,,0000,0000,0000,,This is Fubini's\Ntheorem but generalized Dialogue: 0,1:15:59.36,1:16:01.46,Default,,0000,0000,0000,,to something that depends on x. Dialogue: 0,1:16:01.46,1:16:02.89,Default,,0000,0000,0000,,So how do I do that? Dialogue: 0,1:16:02.89,1:16:05.08,Default,,0000,0000,0000,,Integral of 1dy. Dialogue: 0,1:16:05.08,1:16:07.23,Default,,0000,0000,0000,,That's what? Dialogue: 0,1:16:07.23,1:16:11.99,Default,,0000,0000,0000,,That's y measured between two\Nvalues that don't depend on y. Dialogue: 0,1:16:11.99,1:16:16.67,Default,,0000,0000,0000,,They depend only on x, g of x on\Nthe bottom, f of x on the top. Dialogue: 0,1:16:16.67,1:16:20.16,Default,,0000,0000,0000,,So this is exactly the\Nintegral from a to b. Dialogue: 0,1:16:20.16,1:16:22.36,Default,,0000,0000,0000,,In terms of the\Nround parentheses, Dialogue: 0,1:16:22.36,1:16:25.83,Default,,0000,0000,0000,,I put-- what is y between\Nf of x and g of x? Dialogue: 0,1:16:25.83,1:16:30.00,Default,,0000,0000,0000,,f of x minus g of x dx. Dialogue: 0,1:16:30.00,1:16:34.47,Default,,0000,0000,0000,,So it is exactly the\Nsame thing from Calc 2 Dialogue: 0,1:16:34.47,1:16:36.41,Default,,0000,0000,0000,,expressed as a double integral. Dialogue: 0,1:16:36.41,1:16:42.22,Default,,0000,0000,0000,, Dialogue: 0,1:16:42.22,1:16:42.92,Default,,0000,0000,0000,,All right. Dialogue: 0,1:16:42.92,1:16:54.16,Default,,0000,0000,0000,,Now This is a type one\Nregion that we talked about. Dialogue: 0,1:16:54.16,1:16:59.60,Default,,0000,0000,0000,,A type two region is a\Nsimilar region, practically. Dialogue: 0,1:16:59.60,1:17:03.13,Default,,0000,0000,0000,,What you have to keep\Nin mind is they're both Dialogue: 0,1:17:03.13,1:17:05.77,Default,,0000,0000,0000,,given here as examples. Dialogue: 0,1:17:05.77,1:17:09.26,Default,,0000,0000,0000,,But the technique is\Nabsolutely the same. Dialogue: 0,1:17:09.26,1:17:13.10,Default,,0000,0000,0000,,If instead of\Ntaking this picture, Dialogue: 0,1:17:13.10,1:17:20.01,Default,,0000,0000,0000,,I would take y to move\Nbetween fixed values, Dialogue: 0,1:17:20.01,1:17:26.28,Default,,0000,0000,0000,,like y has to be between\Nc and d-- this is my y. Dialogue: 0,1:17:26.28,1:17:28.77,Default,,0000,0000,0000,,These are the fixed values. Dialogue: 0,1:17:28.77,1:17:33.24,Default,,0000,0000,0000,,And then give me\Nsome nice colors. Dialogue: 0,1:17:33.24,1:17:42.18,Default,,0000,0000,0000,,This curve and\Nthat curve-- OK, I Dialogue: 0,1:17:42.18,1:17:49.86,Default,,0000,0000,0000,,have to rotate my head because\Nthen this is going to be x. Dialogue: 0,1:17:49.86,1:17:51.78,Default,,0000,0000,0000,,This is going to be y. Dialogue: 0,1:17:51.78,1:17:57.00,Default,,0000,0000,0000,,And the blue thingy has\Nto be a function of y. Dialogue: 0,1:17:57.00,1:17:58.81,Default,,0000,0000,0000,,x is a function of y. Dialogue: 0,1:17:58.81,1:18:01.34,Default,,0000,0000,0000,,So how do I call that? Dialogue: 0,1:18:01.34,1:18:09.95,Default,,0000,0000,0000,,I have x or whatever\Nequals big F of y. Dialogue: 0,1:18:09.95,1:18:16.93,Default,,0000,0000,0000,,And here in the red one, I\Nhave x equals big G of y. Dialogue: 0,1:18:16.93,1:18:23.28,Default,,0000,0000,0000,,And how am I going to\Nevaluate the striped area? Dialogue: 0,1:18:23.28,1:18:30.55,Default,,0000,0000,0000,,Of course striped because I\Nhave again y is between c and d. Dialogue: 0,1:18:30.55,1:18:33.77,Default,,0000,0000,0000,,And what's moving is Mr. X. Dialogue: 0,1:18:33.77,1:18:37.48,Default,,0000,0000,0000,,And Mr. X refuses to\Nhave fixed variables. Dialogue: 0,1:18:37.48,1:18:41.96,Default,,0000,0000,0000,,Now he goes, I move from\Nthe bottom, which is G of y, Dialogue: 0,1:18:41.96,1:18:46.87,Default,,0000,0000,0000,,to the top, which is F of y. Dialogue: 0,1:18:46.87,1:18:50.80,Default,,0000,0000,0000,,How am I going to write\Nthe double integral Dialogue: 0,1:18:50.80,1:18:58.24,Default,,0000,0000,0000,,over this domain of\N1dA, where dA is dxdy. Dialogue: 0,1:18:58.24,1:19:00.38,Default,,0000,0000,0000,,Who's going to tell me? Dialogue: 0,1:19:00.38,1:19:05.14,Default,,0000,0000,0000,,Similarly, the same\Nreasoning as for this one. Dialogue: 0,1:19:05.14,1:19:10.31,Default,,0000,0000,0000,,I'm going to have the\Nintegral from what to what Dialogue: 0,1:19:10.31,1:19:12.19,Default,,0000,0000,0000,,of integral from what to what? Dialogue: 0,1:19:12.19,1:19:14.84,Default,,0000,0000,0000,,Who comes first, dx or dy? Dialogue: 0,1:19:14.84,1:19:15.52,Default,,0000,0000,0000,,STUDENT: dx. Dialogue: 0,1:19:15.52,1:19:16.94,Default,,0000,0000,0000,,DR. MAGDALENA TODA:\Ndx, very good. Dialogue: 0,1:19:16.94,1:19:18.61,Default,,0000,0000,0000,,And dy at the end. Dialogue: 0,1:19:18.61,1:19:23.13,Default,,0000,0000,0000,,So y will be between\Nc and d, and x Dialogue: 0,1:19:23.13,1:19:31.53,Default,,0000,0000,0000,,is going to be between\NG of y and F of y. Dialogue: 0,1:19:31.53,1:19:32.44,Default,,0000,0000,0000,,And here is y. Dialogue: 0,1:19:32.44,1:19:35.31,Default,,0000,0000,0000,, Dialogue: 0,1:19:35.31,1:19:38.51,Default,,0000,0000,0000,,How can I rewrite this integral? Dialogue: 0,1:19:38.51,1:19:40.01,Default,,0000,0000,0000,,Very easily. Dialogue: 0,1:19:40.01,1:19:46.07,Default,,0000,0000,0000,,The integral from c to\Nd of the guy on top, Dialogue: 0,1:19:46.07,1:19:54.09,Default,,0000,0000,0000,,the blue guy, F of y, minus the\Nguy on the bottom, G of y, dy. Dialogue: 0,1:19:54.09,1:20:00.17,Default,,0000,0000,0000,,Some people call the\Nvertical stip method Dialogue: 0,1:20:00.17,1:20:02.91,Default,,0000,0000,0000,,compared to the horizontal\Nstrip method, where Dialogue: 0,1:20:02.91,1:20:05.15,Default,,0000,0000,0000,,in this kind of\Nhorizontal strip method, Dialogue: 0,1:20:05.15,1:20:08.02,Default,,0000,0000,0000,,you just have to view\Nx as a function of y Dialogue: 0,1:20:08.02,1:20:11.73,Default,,0000,0000,0000,,and rotate your head and apply\Nthe same reasoning as before. Dialogue: 0,1:20:11.73,1:20:13.14,Default,,0000,0000,0000,,It's not a big deal. Dialogue: 0,1:20:13.14,1:20:15.90,Default,,0000,0000,0000,,You just need a little\Nbit of imagination, Dialogue: 0,1:20:15.90,1:20:20.46,Default,,0000,0000,0000,,and the result is the same. Dialogue: 0,1:20:20.46,1:20:24.62,Default,,0000,0000,0000,,An example that's\Nnot too hard-- I Dialogue: 0,1:20:24.62,1:20:26.45,Default,,0000,0000,0000,,want to give you\Nseveral examples. Dialogue: 0,1:20:26.45,1:20:29.45,Default,,0000,0000,0000,, Dialogue: 0,1:20:29.45,1:20:31.33,Default,,0000,0000,0000,,We have plenty of time. Dialogue: 0,1:20:31.33,1:20:36.05,Default,,0000,0000,0000,,Now it says, we have\Na triangular region. Dialogue: 0,1:20:36.05,1:20:40.98,Default,,0000,0000,0000,,And that is enclosed by lines\Ny equals 0, y equals 2x, Dialogue: 0,1:20:40.98,1:20:43.73,Default,,0000,0000,0000,,and x equals 1. Dialogue: 0,1:20:43.73,1:20:47.52,Default,,0000,0000,0000,,Let's see what that means\Nand be able to draw it. Dialogue: 0,1:20:47.52,1:20:51.39,Default,,0000,0000,0000,,It's very important to be\Nable to draw in this chapter. Dialogue: 0,1:20:51.39,1:20:54.77,Default,,0000,0000,0000,,If you're not, just\Nlearn how to draw, Dialogue: 0,1:20:54.77,1:20:56.69,Default,,0000,0000,0000,,and that will give\Nyou lots of ideas Dialogue: 0,1:20:56.69,1:20:58.39,Default,,0000,0000,0000,,on how to solve the problems. Dialogue: 0,1:20:58.39,1:21:18.27,Default,,0000,0000,0000,, Dialogue: 0,1:21:18.27,1:21:22.88,Default,,0000,0000,0000,,Chapter 12 is included\Ncompletely on the midterm. Dialogue: 0,1:21:22.88,1:21:25.37,Default,,0000,0000,0000,,So the midterm is\Non the 2nd of April. Dialogue: 0,1:21:25.37,1:21:29.85,Default,,0000,0000,0000,,For the midterm, we have chapter\N10, those three sections. Dialogue: 0,1:21:29.85,1:21:32.34,Default,,0000,0000,0000,,Then we have chapter\N11 completely, Dialogue: 0,1:21:32.34,1:21:40.19,Default,,0000,0000,0000,,and then we have chapter 12\Nnot completely, up to 12.6. Dialogue: 0,1:21:40.19,1:21:40.76,Default,,0000,0000,0000,,All right. Dialogue: 0,1:21:40.76,1:21:44.40,Default,,0000,0000,0000,,So what did I say? Dialogue: 0,1:21:44.40,1:21:49.14,Default,,0000,0000,0000,,I have a triangular region that\Nis obtained by intersecting Dialogue: 0,1:21:49.14,1:21:50.90,Default,,0000,0000,0000,,the following lines. Dialogue: 0,1:21:50.90,1:21:58.73,Default,,0000,0000,0000,,y equals 0, x equals\N1, and y equals 2x. Dialogue: 0,1:21:58.73,1:22:01.94,Default,,0000,0000,0000,,Can I draw them and\Nsee how they intersect? Dialogue: 0,1:22:01.94,1:22:03.34,Default,,0000,0000,0000,,It shouldn't be a big problem. Dialogue: 0,1:22:03.34,1:22:05.85,Default,,0000,0000,0000,,This is a line that\Npasses through the origin Dialogue: 0,1:22:05.85,1:22:07.99,Default,,0000,0000,0000,,and has slope 2. Dialogue: 0,1:22:07.99,1:22:10.98,Default,,0000,0000,0000,,So it should be\Nvery easy to draw. Dialogue: 0,1:22:10.98,1:22:18.27,Default,,0000,0000,0000,,At 1, x equals 1, the y will\Nbe 2 for this line of slope 2. Dialogue: 0,1:22:18.27,1:22:20.85,Default,,0000,0000,0000,,So I'll try to draw. Dialogue: 0,1:22:20.85,1:22:23.72,Default,,0000,0000,0000,,Does this look double to you? Dialogue: 0,1:22:23.72,1:22:29.26,Default,,0000,0000,0000,,So this is 2. Dialogue: 0,1:22:29.26,1:22:32.15,Default,,0000,0000,0000,,This is the point 1, 2. Dialogue: 0,1:22:32.15,1:22:35.22,Default,,0000,0000,0000,,And that's the line y equals 2x. Dialogue: 0,1:22:35.22,1:22:38.41,Default,,0000,0000,0000,,And that's the line y equals 0. Dialogue: 0,1:22:38.41,1:22:40.22,Default,,0000,0000,0000,,And that's the line x equals 1. Dialogue: 0,1:22:40.22,1:22:43.45,Default,,0000,0000,0000,,So can I shade this triangle? Dialogue: 0,1:22:43.45,1:22:47.56,Default,,0000,0000,0000,,Yeah, I can eventually,\Ndepending on what they ask me. Dialogue: 0,1:22:47.56,1:22:49.18,Default,,0000,0000,0000,,What do they ask me? Dialogue: 0,1:22:49.18,1:22:58.10,Default,,0000,0000,0000,,Find the double\Nintegral of x plus y dA Dialogue: 0,1:22:58.10,1:23:05.60,Default,,0000,0000,0000,,with respect to the area element\Nover T, T being the triangle. Dialogue: 0,1:23:05.60,1:23:09.51,Default,,0000,0000,0000,,So now I'm going to ask,\Ndid they say by what method? Dialogue: 0,1:23:09.51,1:23:12.65,Default,,0000,0000,0000,,Unfortunately, they say,\Ndo it by both methods. Dialogue: 0,1:23:12.65,1:23:17.19,Default,,0000,0000,0000,,That means both by x\Nintregration first and then Dialogue: 0,1:23:17.19,1:23:20.09,Default,,0000,0000,0000,,y integration and\Nthe other way around. Dialogue: 0,1:23:20.09,1:23:23.31,Default,,0000,0000,0000,,So they ask you to change\Nthe order of the integration Dialogue: 0,1:23:23.31,1:23:24.72,Default,,0000,0000,0000,,or do what? Dialogue: 0,1:23:24.72,1:23:27.27,Default,,0000,0000,0000,,Switch from vertical\Nstrip method Dialogue: 0,1:23:27.27,1:23:29.60,Default,,0000,0000,0000,,to horizontal strip method. Dialogue: 0,1:23:29.60,1:23:31.15,Default,,0000,0000,0000,,You should get the same answer. Dialogue: 0,1:23:31.15,1:23:34.16,Default,,0000,0000,0000,,That's a typical\Nfinal exam problem. Dialogue: 0,1:23:34.16,1:23:40.41,Default,,0000,0000,0000,,When we test you, if\Nyou are able to do this Dialogue: 0,1:23:40.41,1:23:43.31,Default,,0000,0000,0000,,through the vertical\Nstrip or horizontal Dialogue: 0,1:23:43.31,1:23:45.25,Default,,0000,0000,0000,,strip and change the\Norder of integration. Dialogue: 0,1:23:45.25,1:23:47.56,Default,,0000,0000,0000,,If I do it with the\Nvertical strip method, Dialogue: 0,1:23:47.56,1:23:52.01,Default,,0000,0000,0000,,who comes first,\Nthe dy or the dx? Dialogue: 0,1:23:52.01,1:23:53.31,Default,,0000,0000,0000,,Think a little bit. Dialogue: 0,1:23:53.31,1:23:55.61,Default,,0000,0000,0000,,Where do I put d--\NFubini [INAUDIBLE] Dialogue: 0,1:23:55.61,1:23:58.65,Default,,0000,0000,0000,,comes dy dx or dx dy? Dialogue: 0,1:23:58.65,1:23:59.60,Default,,0000,0000,0000,,STUDENT: dy. Dialogue: 0,1:23:59.60,1:24:01.55,Default,,0000,0000,0000,,PROFESSOR: dy dx. Dialogue: 0,1:24:01.55,1:24:04.36,Default,,0000,0000,0000,,So VSM. Dialogue: 0,1:24:04.36,1:24:06.56,Default,,0000,0000,0000,,You're going to laugh. Dialogue: 0,1:24:06.56,1:24:07.89,Default,,0000,0000,0000,,It's not written in the book. Dialogue: 0,1:24:07.89,1:24:10.81,Default,,0000,0000,0000,,It's like a childish name,\NVertical Strip Method, Dialogue: 0,1:24:10.81,1:24:12.62,Default,,0000,0000,0000,,meeting integration\Nwith respect to y Dialogue: 0,1:24:12.62,1:24:14.92,Default,,0000,0000,0000,,and then with respect to x. Dialogue: 0,1:24:14.92,1:24:17.73,Default,,0000,0000,0000,,It helped my students\Nthrough the last decade Dialogue: 0,1:24:17.73,1:24:19.59,Default,,0000,0000,0000,,to remember about\Nthe vertical strips. Dialogue: 0,1:24:19.59,1:24:25.49,Default,,0000,0000,0000,,And that's why I say something\Nthat's not using the book, VSM. Dialogue: 0,1:24:25.49,1:24:35.81,Default,,0000,0000,0000,,Now, I have integral from-- so\Nwho is Mr. X going from 0 to 1? Dialogue: 0,1:24:35.81,1:24:36.34,Default,,0000,0000,0000,,He's stable. Dialogue: 0,1:24:36.34,1:24:37.75,Default,,0000,0000,0000,,He's happy. Dialogue: 0,1:24:37.75,1:24:39.70,Default,,0000,0000,0000,,He's going between\Ntwo fixed values. Dialogue: 0,1:24:39.70,1:24:43.77,Default,,0000,0000,0000,,y goes between the\Nbottom line, which is 0. Dialogue: 0,1:24:43.77,1:24:44.65,Default,,0000,0000,0000,,We are lucky. Dialogue: 0,1:24:44.65,1:24:47.60,Default,,0000,0000,0000,,It's a really nice problem. Dialogue: 0,1:24:47.60,1:24:51.40,Default,,0000,0000,0000,,Going to y equals 2x. Dialogue: 0,1:24:51.40,1:24:54.02,Default,,0000,0000,0000,,So it's not hard at all. Dialogue: 0,1:24:54.02,1:24:59.23,Default,,0000,0000,0000,,And we have to integrate\Nthe function x plus y. Dialogue: 0,1:24:59.23,1:25:01.77,Default,,0000,0000,0000,,It should be a piece of cake. Dialogue: 0,1:25:01.77,1:25:06.60,Default,,0000,0000,0000,,Let's do this together because\Nyou've accumulated seniority Dialogue: 0,1:25:06.60,1:25:07.68,Default,,0000,0000,0000,,in this type of problem. Dialogue: 0,1:25:07.68,1:25:10.52,Default,,0000,0000,0000,, Dialogue: 0,1:25:10.52,1:25:12.44,Default,,0000,0000,0000,,What do I put inside? Dialogue: 0,1:25:12.44,1:25:14.58,Default,,0000,0000,0000,,What's integral of x\Nplus y with respect to y? Dialogue: 0,1:25:14.58,1:25:15.99,Default,,0000,0000,0000,,Is it hard? Dialogue: 0,1:25:15.99,1:25:19.27,Default,,0000,0000,0000,, Dialogue: 0,1:25:19.27,1:25:23.45,Default,,0000,0000,0000,,xy plus-- somebody tell me. Dialogue: 0,1:25:23.45,1:25:25.11,Default,,0000,0000,0000,,STUDENT: y squared. Dialogue: 0,1:25:25.11,1:25:29.16,Default,,0000,0000,0000,,PROFESSOR: y squared\Nover 2, between y Dialogue: 0,1:25:29.16,1:25:33.45,Default,,0000,0000,0000,,equals 0 on the bottom,\Ny equals 2x on top. Dialogue: 0,1:25:33.45,1:25:36.86,Default,,0000,0000,0000,,I have to be smart and\Nplug in the values y. Dialogue: 0,1:25:36.86,1:25:39.01,Default,,0000,0000,0000,,Otherwise, I'll never make it. Dialogue: 0,1:25:39.01,1:25:39.80,Default,,0000,0000,0000,,STUDENT: Professor? Dialogue: 0,1:25:39.80,1:25:40.90,Default,,0000,0000,0000,,PROFESSOR: Yes, sir? Dialogue: 0,1:25:40.90,1:25:43.31,Default,,0000,0000,0000,,STUDENT: Why did you take\N2x as the final value Dialogue: 0,1:25:43.31,1:25:45.31,Default,,0000,0000,0000,,because you have a\Nspecified triangle. Dialogue: 0,1:25:45.31,1:25:48.97,Default,,0000,0000,0000,,PROFESSOR: Because y\Nequals 2x is the expression Dialogue: 0,1:25:48.97,1:25:52.10,Default,,0000,0000,0000,,of the upper function. Dialogue: 0,1:25:52.10,1:25:54.78,Default,,0000,0000,0000,,The upper function is\Nthe line y equals 2x. Dialogue: 0,1:25:54.78,1:25:56.19,Default,,0000,0000,0000,,They provided that. Dialogue: 0,1:25:56.19,1:25:59.66,Default,,0000,0000,0000,,So from the bottom function\Nto the upper function, Dialogue: 0,1:25:59.66,1:26:02.38,Default,,0000,0000,0000,,the vertical strips go\Nbetween two functions. Dialogue: 0,1:26:02.38,1:26:05.28,Default,,0000,0000,0000,, Dialogue: 0,1:26:05.28,1:26:07.72,Default,,0000,0000,0000,,So when I plug in\Nhere y equals 2x, Dialogue: 0,1:26:07.72,1:26:09.72,Default,,0000,0000,0000,,I have to pay attention\Nto my algebra. Dialogue: 0,1:26:09.72,1:26:13.82,Default,,0000,0000,0000,,If I forget the 2, it's all\Nover for me, zero points. Dialogue: 0,1:26:13.82,1:26:16.26,Default,,0000,0000,0000,,Well, not zero points,\Nbut 10% credit. Dialogue: 0,1:26:16.26,1:26:20.09,Default,,0000,0000,0000,,I have no idea what I would\Nget, so I have to pay attention. Dialogue: 0,1:26:20.09,1:26:26.62,Default,,0000,0000,0000,,2x times x is 2x squared\Nplus 2x all squared-- guys, Dialogue: 0,1:26:26.62,1:26:30.52,Default,,0000,0000,0000,,keep an eye on me--\N4x squared over 2. Dialogue: 0,1:26:30.52,1:26:36.43,Default,,0000,0000,0000,,I put the first value\Nin a pink parentheses, Dialogue: 0,1:26:36.43,1:26:40.80,Default,,0000,0000,0000,,and then I move on to\Nthe line parentheses. Dialogue: 0,1:26:40.80,1:26:42.84,Default,,0000,0000,0000,,Evaluate it at 0. Dialogue: 0,1:26:42.84,1:26:44.51,Default,,0000,0000,0000,,That line is very lucky. Dialogue: 0,1:26:44.51,1:26:50.91,Default,,0000,0000,0000,,I get a 0 because y\Nequals 0 will give me 0. Dialogue: 0,1:26:50.91,1:26:54.18,Default,,0000,0000,0000,,What am I going to get here? Dialogue: 0,1:26:54.18,1:26:56.52,Default,,0000,0000,0000,,2x squared plus 2x squared. Dialogue: 0,1:26:56.52,1:26:57.02,Default,,0000,0000,0000,,Good. Dialogue: 0,1:26:57.02,1:26:59.41,Default,,0000,0000,0000,,What's 2x squared\Nplus 2x squared? Dialogue: 0,1:26:59.41,1:27:00.26,Default,,0000,0000,0000,,4x squared. Dialogue: 0,1:27:00.26,1:27:01.92,Default,,0000,0000,0000,,So a 4 goes out. Dialogue: 0,1:27:01.92,1:27:03.32,Default,,0000,0000,0000,,Kick him out. Dialogue: 0,1:27:03.32,1:27:06.19,Default,,0000,0000,0000,,Integral from 0\Nto 1 x squared dx. Dialogue: 0,1:27:06.19,1:27:08.05,Default,,0000,0000,0000,,Integral of x squared is? Dialogue: 0,1:27:08.05,1:27:11.65,Default,,0000,0000,0000,, Dialogue: 0,1:27:11.65,1:27:14.10,Default,,0000,0000,0000,,Integral of x squared is? Dialogue: 0,1:27:14.10,1:27:15.10,Default,,0000,0000,0000,,STUDENT: x cubed over 3. Dialogue: 0,1:27:15.10,1:27:16.18,Default,,0000,0000,0000,,PROFESSOR: x cubed over 3. Dialogue: 0,1:27:16.18,1:27:19.33,Default,,0000,0000,0000,,And if you take it\Nbetween 1 and 0, you get? Dialogue: 0,1:27:19.33,1:27:20.66,Default,,0000,0000,0000,,STUDENT: 1. Dialogue: 0,1:27:20.66,1:27:21.31,Default,,0000,0000,0000,,PROFESSOR: 1/3. Dialogue: 0,1:27:21.31,1:27:23.70,Default,,0000,0000,0000,,1/3 times 4 is 4/3. Dialogue: 0,1:27:23.70,1:27:26.82,Default,,0000,0000,0000,, Dialogue: 0,1:27:26.82,1:27:29.14,Default,,0000,0000,0000,,Suppose this is going to\Nhappen on the midterm, Dialogue: 0,1:27:29.14,1:27:32.40,Default,,0000,0000,0000,,and I'm asking you to do it\Nreversing the integration Dialogue: 0,1:27:32.40,1:27:33.93,Default,,0000,0000,0000,,order. Dialogue: 0,1:27:33.93,1:27:37.52,Default,,0000,0000,0000,,Then you are going to check\Nyour own work very beautifully Dialogue: 0,1:27:37.52,1:27:41.72,Default,,0000,0000,0000,,in the sense that\Nyou say, well, now Dialogue: 0,1:27:41.72,1:27:45.88,Default,,0000,0000,0000,,I'm going to see if I made\Na mistake in this one. Dialogue: 0,1:27:45.88,1:27:46.69,Default,,0000,0000,0000,,What do I do? Dialogue: 0,1:27:46.69,1:27:50.48,Default,,0000,0000,0000,,I erase the whole thing, and\Ninstead of vertical strips, Dialogue: 0,1:27:50.48,1:27:55.84,Default,,0000,0000,0000,,I'm going to put\Nhorizontal strips. Dialogue: 0,1:27:55.84,1:28:01.02,Default,,0000,0000,0000,,And you say, well, life is a\Nlittle bit harder in this case Dialogue: 0,1:28:01.02,1:28:04.53,Default,,0000,0000,0000,,because in this\Ncase, I have to look Dialogue: 0,1:28:04.53,1:28:10.85,Default,,0000,0000,0000,,at y between fixed\Nvalues, y between 0 and 1. Dialogue: 0,1:28:10.85,1:28:17.78,Default,,0000,0000,0000,,So y is between 0 and 1--\N0 and 2, fixed values. Dialogue: 0,1:28:17.78,1:28:22.84,Default,,0000,0000,0000,,And Mr. X says, I'm going\Nbetween two functions of y. Dialogue: 0,1:28:22.84,1:28:26.23,Default,,0000,0000,0000,,I don't know what those\Nfunctions of y are. Dialogue: 0,1:28:26.23,1:28:28.38,Default,,0000,0000,0000,,I'm puzzled. Dialogue: 0,1:28:28.38,1:28:30.60,Default,,0000,0000,0000,,You have to help\NMr. X know where Dialogue: 0,1:28:30.60,1:28:34.52,Default,,0000,0000,0000,,he's going because his life\Nright now is a little bit hard. Dialogue: 0,1:28:34.52,1:28:39.30,Default,,0000,0000,0000,,So what is the\Nfunction for the blue? Dialogue: 0,1:28:39.30,1:28:42.20,Default,,0000,0000,0000,, Dialogue: 0,1:28:42.20,1:28:44.16,Default,,0000,0000,0000,,Now he's not blue anymore. Dialogue: 0,1:28:44.16,1:28:45.14,Default,,0000,0000,0000,,He's brown. Dialogue: 0,1:28:45.14,1:28:47.84,Default,,0000,0000,0000,,x equals 1. Dialogue: 0,1:28:47.84,1:28:50.10,Default,,0000,0000,0000,,So he knows what\Nhe's going to be. Dialogue: 0,1:28:50.10,1:28:52.81,Default,,0000,0000,0000,,What is the x function\Nfor the red line Dialogue: 0,1:28:52.81,1:28:54.58,Default,,0000,0000,0000,,that [INAUDIBLE] asked about? Dialogue: 0,1:28:54.58,1:28:55.46,Default,,0000,0000,0000,,STUDENT: y over 2. Dialogue: 0,1:28:55.46,1:28:57.55,Default,,0000,0000,0000,,PROFESSOR: x must be y over 2. Dialogue: 0,1:28:57.55,1:29:01.01,Default,,0000,0000,0000,,It's the same thing, but I have\Nto express x in terms of y. Dialogue: 0,1:29:01.01,1:29:05.28,Default,,0000,0000,0000,,So I erase and I say\Nx equals y over 2. Dialogue: 0,1:29:05.28,1:29:06.70,Default,,0000,0000,0000,,Same thing. Dialogue: 0,1:29:06.70,1:29:11.14,Default,,0000,0000,0000,,So x has to be between what and\Nwhat, the bottom and the top? Dialogue: 0,1:29:11.14,1:29:13.68,Default,,0000,0000,0000,,Well, I turn my head. Dialogue: 0,1:29:13.68,1:29:19.39,Default,,0000,0000,0000,,The top must be x equals 1,\Nand the bottom one is y over 2. Dialogue: 0,1:29:19.39,1:29:24.98,Default,,0000,0000,0000,,That's the bottom one,\Nthe bottom value for x. Dialogue: 0,1:29:24.98,1:29:27.45,Default,,0000,0000,0000,,Now wish me luck because I\Nhave to get the same thing. Dialogue: 0,1:29:27.45,1:29:35.52,Default,,0000,0000,0000,,So integral from 0 to 2 of\Nintegral from y over 2 to 1. Dialogue: 0,1:29:35.52,1:29:37.51,Default,,0000,0000,0000,,Changing the order\Nof integration Dialogue: 0,1:29:37.51,1:29:40.65,Default,,0000,0000,0000,,doesn't change the\Nintegrand, which is exactly Dialogue: 0,1:29:40.65,1:29:43.41,Default,,0000,0000,0000,,the same function, f of xy. Dialogue: 0,1:29:43.41,1:29:46.81,Default,,0000,0000,0000,,This is the f function. Dialogue: 0,1:29:46.81,1:29:47.77,Default,,0000,0000,0000,,Then what changes? Dialogue: 0,1:29:47.77,1:29:48.81,Default,,0000,0000,0000,,The order of integration. Dialogue: 0,1:29:48.81,1:29:52.17,Default,,0000,0000,0000,,So I go dx first,\Ndy next and stop. Dialogue: 0,1:29:52.17,1:29:55.03,Default,,0000,0000,0000,, Dialogue: 0,1:29:55.03,1:30:00.23,Default,,0000,0000,0000,,I copy and paste the outer\Nones, and I focus my attention Dialogue: 0,1:30:00.23,1:30:05.60,Default,,0000,0000,0000,,to the red parentheses\Ninside, which I'm Dialogue: 0,1:30:05.60,1:30:07.89,Default,,0000,0000,0000,,going to copy and paste here. Dialogue: 0,1:30:07.89,1:30:12.48,Default,,0000,0000,0000,,I'll have to do some\Nmath very carefully. Dialogue: 0,1:30:12.48,1:30:13.50,Default,,0000,0000,0000,,So what do I have? Dialogue: 0,1:30:13.50,1:30:17.14,Default,,0000,0000,0000,,I have x plus y integrated\Nwith respect to x. Dialogue: 0,1:30:17.14,1:30:19.23,Default,,0000,0000,0000,,If I rush, it's a bad thing. Dialogue: 0,1:30:19.23,1:30:20.80,Default,,0000,0000,0000,,STUDENT: So that\Nwould be x squared. Dialogue: 0,1:30:20.80,1:30:21.67,Default,,0000,0000,0000,,PROFESSOR: x squared. Dialogue: 0,1:30:21.67,1:30:22.78,Default,,0000,0000,0000,,STUDENT: Over 2. Dialogue: 0,1:30:22.78,1:30:23.74,Default,,0000,0000,0000,,PROFESSOR: Over 2. Dialogue: 0,1:30:23.74,1:30:25.35,Default,,0000,0000,0000,,STUDENT: Plus xy. Dialogue: 0,1:30:25.35,1:30:29.61,Default,,0000,0000,0000,,PROFESSOR: Plus xy taken\Nbetween the following. Dialogue: 0,1:30:29.61,1:30:32.76,Default,,0000,0000,0000,,When x equals 1,\NI have it on top. Dialogue: 0,1:30:32.76,1:30:38.35,Default,,0000,0000,0000,,When x equals y over 2,\NI have it on the bottom. Dialogue: 0,1:30:38.35,1:30:39.69,Default,,0000,0000,0000,,OK. Dialogue: 0,1:30:39.69,1:30:42.58,Default,,0000,0000,0000,,This red thing, I'm a\Nlittle bit too lazy. Dialogue: 0,1:30:42.58,1:30:47.85,Default,,0000,0000,0000,,I'll copy and paste\Nit separately. Dialogue: 0,1:30:47.85,1:30:52.48,Default,,0000,0000,0000,,For the upper part, it's\Nreally easy to compute. Dialogue: 0,1:30:52.48,1:30:53.48,Default,,0000,0000,0000,,What do I get? Dialogue: 0,1:30:53.48,1:31:01.89,Default,,0000,0000,0000,,When x is 1, 1/2, 1/2\Nplus when x is 1, y. Dialogue: 0,1:31:01.89,1:31:07.12,Default,,0000,0000,0000,,Minus integral of--\Nwhen x is y over 2, Dialogue: 0,1:31:07.12,1:31:12.70,Default,,0000,0000,0000,,I get y squared over\N4 up here over 2. Dialogue: 0,1:31:12.70,1:31:19.36,Default,,0000,0000,0000,,So I should get y\Nsquared over 8 plus-- Dialogue: 0,1:31:19.36,1:31:21.78,Default,,0000,0000,0000,,I've got an x equals y over 2. Dialogue: 0,1:31:21.78,1:31:23.59,Default,,0000,0000,0000,,What do I get? Dialogue: 0,1:31:23.59,1:31:26.50,Default,,0000,0000,0000,,y squared over 2. Dialogue: 0,1:31:26.50,1:31:28.72,Default,,0000,0000,0000,,Is this hard? Dialogue: 0,1:31:28.72,1:31:31.63,Default,,0000,0000,0000,,It's very easy to make an\Nalgebra mistake on such Dialogue: 0,1:31:31.63,1:31:32.67,Default,,0000,0000,0000,,a problem, unfortunately. Dialogue: 0,1:31:32.67,1:31:37.02,Default,,0000,0000,0000,,I have y plus 1/2 plus what? Dialogue: 0,1:31:37.02,1:31:40.86,Default,,0000,0000,0000,,What is 1/2 plus 1/8? Dialogue: 0,1:31:40.86,1:31:41.82,Default,,0000,0000,0000,,STUDENT: 5/8. Dialogue: 0,1:31:41.82,1:31:48.52,Default,,0000,0000,0000,,PROFESSOR: 5 over 8\Nwith a minus y squared. Dialogue: 0,1:31:48.52,1:31:52.40,Default,,0000,0000,0000,, Dialogue: 0,1:31:52.40,1:31:54.27,Default,,0000,0000,0000,,So hopefully I did this right. Dialogue: 0,1:31:54.27,1:32:00.97,Default,,0000,0000,0000,,Now I'll go, OK, integral from\N0 to 2 of all of this animal, y Dialogue: 0,1:32:00.97,1:32:06.10,Default,,0000,0000,0000,,plus 1/2 minus 5\Nover 8, y squared. Dialogue: 0,1:32:06.10,1:32:10.66,Default,,0000,0000,0000,,What happens if I don't\Nget the right answer? Dialogue: 0,1:32:10.66,1:32:12.94,Default,,0000,0000,0000,,Then I go back and\Ncheck my work because I Dialogue: 0,1:32:12.94,1:32:14.98,Default,,0000,0000,0000,,know I'm supposed to get 4/3. Dialogue: 0,1:32:14.98,1:32:16.15,Default,,0000,0000,0000,,That was easy. Dialogue: 0,1:32:16.15,1:32:23.31,Default,,0000,0000,0000,,So what is integral of this\Nsausage, whatever it is? Dialogue: 0,1:32:23.31,1:32:29.85,Default,,0000,0000,0000,,y squared over 2 plus y\Nover 2 minus 5 over 8-- Dialogue: 0,1:32:29.85,1:32:41.93,Default,,0000,0000,0000,,oh my god-- 5 over 8, y\Ncubed over 3, between 2 up Dialogue: 0,1:32:41.93,1:32:43.81,Default,,0000,0000,0000,,and 0 down. Dialogue: 0,1:32:43.81,1:32:46.73,Default,,0000,0000,0000,,When I have 0 down,\NI plug y equals 0. Dialogue: 0,1:32:46.73,1:32:47.80,Default,,0000,0000,0000,,It's a piece of cake. Dialogue: 0,1:32:47.80,1:32:49.15,Default,,0000,0000,0000,,It's 0. Dialogue: 0,1:32:49.15,1:32:51.72,Default,,0000,0000,0000,,So what matters is\Nwhat I get when I plug Dialogue: 0,1:32:51.72,1:32:53.56,Default,,0000,0000,0000,,in the value 2 instead of y. Dialogue: 0,1:32:53.56,1:32:56.10,Default,,0000,0000,0000,,So what do I get? Dialogue: 0,1:32:56.10,1:33:07.02,Default,,0000,0000,0000,,4 over 2 is 2, plus 2 over 2\Nis 1, minus 2 cubed, thank god. Dialogue: 0,1:33:07.02,1:33:07.71,Default,,0000,0000,0000,,That's 8. Dialogue: 0,1:33:07.71,1:33:10.74,Default,,0000,0000,0000,,8 simplifies with 8 minus 5/3. Dialogue: 0,1:33:10.74,1:33:16.28,Default,,0000,0000,0000,, Dialogue: 0,1:33:16.28,1:33:23.65,Default,,0000,0000,0000,,So I got 9/3 minus 5/3,\Nand I did it carefully. Dialogue: 0,1:33:23.65,1:33:25.15,Default,,0000,0000,0000,,I did a good job. Dialogue: 0,1:33:25.15,1:33:27.74,Default,,0000,0000,0000,,I got the same thing, 4/3. Dialogue: 0,1:33:27.74,1:33:30.88,Default,,0000,0000,0000,,So no matter which\Nmethod, the vertical strip Dialogue: 0,1:33:30.88,1:33:34.39,Default,,0000,0000,0000,,or the horizontal strip\Nmethod, I get the same thing. Dialogue: 0,1:33:34.39,1:33:36.66,Default,,0000,0000,0000,,And of course, you'll\Nalways get the same answer Dialogue: 0,1:33:36.66,1:33:42.80,Default,,0000,0000,0000,,because this is what the Fubini\Ntheorem extended to this case Dialogue: 0,1:33:42.80,1:33:43.76,Default,,0000,0000,0000,,is telling you. Dialogue: 0,1:33:43.76,1:33:46.76,Default,,0000,0000,0000,,It doesn't matter the\Norder of integration. Dialogue: 0,1:33:46.76,1:33:51.23,Default,,0000,0000,0000,, Dialogue: 0,1:33:51.23,1:33:54.62,Default,,0000,0000,0000,,I would advise you to go\Nthrough the theory in the book. Dialogue: 0,1:33:54.62,1:33:57.95,Default,,0000,0000,0000,, Dialogue: 0,1:33:57.95,1:34:02.47,Default,,0000,0000,0000,,They teach you more about\Narea and volume on page 934. Dialogue: 0,1:34:02.47,1:34:07.91,Default,,0000,0000,0000,,I'd like you to read that. Dialogue: 0,1:34:07.91,1:34:11.58,Default,,0000,0000,0000,,And let's see what I want to do. Dialogue: 0,1:34:11.58,1:34:14.03,Default,,0000,0000,0000,,Which one shall I do? Dialogue: 0,1:34:14.03,1:34:17.95,Default,,0000,0000,0000,,There are a few examples\Nthat are worth it. Dialogue: 0,1:34:17.95,1:34:21.25,Default,,0000,0000,0000,, Dialogue: 0,1:34:21.25,1:34:29.01,Default,,0000,0000,0000,,I'll pick the one that gives\Npeople the most trouble. Dialogue: 0,1:34:29.01,1:34:29.67,Default,,0000,0000,0000,,How about that? Dialogue: 0,1:34:29.67,1:34:33.46,Default,,0000,0000,0000,,I take the few examples that\Ngive people the most trouble. Dialogue: 0,1:34:33.46,1:34:39.13,Default,,0000,0000,0000,,One example that popped up on\Nalmost each and every final Dialogue: 0,1:34:39.13,1:34:44.49,Default,,0000,0000,0000,,in the past 13 years\Nthat involves changing Dialogue: 0,1:34:44.49,1:34:46.46,Default,,0000,0000,0000,,the order of integration. Dialogue: 0,1:34:46.46,1:34:57.31,Default,,0000,0000,0000,, Dialogue: 0,1:34:57.31,1:35:10.35,Default,,0000,0000,0000,,So example problem on changing\Nthe order of integration. Dialogue: 0,1:35:10.35,1:35:14.65,Default,,0000,0000,0000,, Dialogue: 0,1:35:14.65,1:35:19.62,Default,,0000,0000,0000,,A very tricky, smart\Nproblem is the following. Dialogue: 0,1:35:19.62,1:35:30.97,Default,,0000,0000,0000,,Evaluate integral from 0\Nto 1, integral from x to 1, Dialogue: 0,1:35:30.97,1:35:34.21,Default,,0000,0000,0000,,e to the y squared dy dx. Dialogue: 0,1:35:34.21,1:35:41.82,Default,,0000,0000,0000,, Dialogue: 0,1:35:41.82,1:35:43.74,Default,,0000,0000,0000,,I don't know if you've\Nseen anything like that Dialogue: 0,1:35:43.74,1:35:46.14,Default,,0000,0000,0000,,in AP Calculus or Calc 2. Dialogue: 0,1:35:46.14,1:35:51.97,Default,,0000,0000,0000,,Maybe you have, in which case\Nyour professor probably told Dialogue: 0,1:35:51.97,1:35:54.20,Default,,0000,0000,0000,,you that this is nasty. Dialogue: 0,1:35:54.20,1:35:57.29,Default,,0000,0000,0000,, Dialogue: 0,1:35:57.29,1:35:59.52,Default,,0000,0000,0000,,You say, in what\Nsense is it nasty? Dialogue: 0,1:35:59.52,1:36:05.05,Default,,0000,0000,0000,,There is no expressible\Nanti-derivative. Dialogue: 0,1:36:05.05,1:36:21.60,Default,,0000,0000,0000,,So this cannot be expressed in\Nterms of elementary functions Dialogue: 0,1:36:21.60,1:36:22.10,Default,,0000,0000,0000,,explicitly. Dialogue: 0,1:36:22.10,1:36:28.59,Default,,0000,0000,0000,, Dialogue: 0,1:36:28.59,1:36:31.42,Default,,0000,0000,0000,,It's not that there\Nis no anti-derivative. Dialogue: 0,1:36:31.42,1:36:34.76,Default,,0000,0000,0000,,There is an anti-derivative--\Na whole family, actually-- Dialogue: 0,1:36:34.76,1:36:38.81,Default,,0000,0000,0000,,but you cannot express them in\Nterms of elementary functions. Dialogue: 0,1:36:38.81,1:36:42.88,Default,,0000,0000,0000,,And actually, most functions\Nare not so bad in real world, Dialogue: 0,1:36:42.88,1:36:44.39,Default,,0000,0000,0000,,in real life. Dialogue: 0,1:36:44.39,1:36:48.57,Default,,0000,0000,0000,,Now, could you compute, for\Nexample, integral from 1 to 3 Dialogue: 0,1:36:48.57,1:36:51.29,Default,,0000,0000,0000,,of e to the t squared dt? Dialogue: 0,1:36:51.29,1:36:52.10,Default,,0000,0000,0000,,Yes. Dialogue: 0,1:36:52.10,1:36:53.52,Default,,0000,0000,0000,,How do you do that? Dialogue: 0,1:36:53.52,1:36:55.50,Default,,0000,0000,0000,,With a calculator. Dialogue: 0,1:36:55.50,1:36:57.56,Default,,0000,0000,0000,,And what if you don't have one? Dialogue: 0,1:36:57.56,1:36:59.06,Default,,0000,0000,0000,,You go to the lab over there. Dialogue: 0,1:36:59.06,1:37:00.40,Default,,0000,0000,0000,,There is MATLAB. Dialogue: 0,1:37:00.40,1:37:03.10,Default,,0000,0000,0000,,MATLAB will compute it for you. Dialogue: 0,1:37:03.10,1:37:04.95,Default,,0000,0000,0000,,How does MATLAB know\Nhow to compute it Dialogue: 0,1:37:04.95,1:37:07.98,Default,,0000,0000,0000,,if there is no way to\Nexpress the anti-derivative Dialogue: 0,1:37:07.98,1:37:12.04,Default,,0000,0000,0000,,and take the value of the\Nanti-derivative between b Dialogue: 0,1:37:12.04,1:37:16.08,Default,,0000,0000,0000,,and a, like in the fundamental\Ntheorem of calculus? Dialogue: 0,1:37:16.08,1:37:20.70,Default,,0000,0000,0000,,Well, the calculator or the\Ncomputer program is smart. Dialogue: 0,1:37:20.70,1:37:24.70,Default,,0000,0000,0000,,He uses numerical analysis\Nto approximate this type Dialogue: 0,1:37:24.70,1:37:26.69,Default,,0000,0000,0000,,of integral. Dialogue: 0,1:37:26.69,1:37:27.90,Default,,0000,0000,0000,,So he's fooling you. Dialogue: 0,1:37:27.90,1:37:29.85,Default,,0000,0000,0000,,He's just playing smarty pants. Dialogue: 0,1:37:29.85,1:37:33.16,Default,,0000,0000,0000,,He's smarter than\Nyou at this point. Dialogue: 0,1:37:33.16,1:37:33.66,Default,,0000,0000,0000,,OK. Dialogue: 0,1:37:33.66,1:37:38.30,Default,,0000,0000,0000,,So you cannot do this by hand,\Nso this order of integration is Dialogue: 0,1:37:38.30,1:37:38.80,Default,,0000,0000,0000,,fruitless. Dialogue: 0,1:37:38.80,1:37:43.26,Default,,0000,0000,0000,, Dialogue: 0,1:37:43.26,1:37:47.37,Default,,0000,0000,0000,,And there are people who\Ntried to do this on the final. Dialogue: 0,1:37:47.37,1:37:48.83,Default,,0000,0000,0000,,Of course, they\Ndidn't get anywhere Dialogue: 0,1:37:48.83,1:37:51.06,Default,,0000,0000,0000,,because they couldn't\Nintegrate it. Dialogue: 0,1:37:51.06,1:37:55.92,Default,,0000,0000,0000,,The whole idea of this one\Nis to-- some professors Dialogue: 0,1:37:55.92,1:37:58.69,Default,,0000,0000,0000,,are so mean they\Ndon't even tell you, Dialogue: 0,1:37:58.69,1:38:00.57,Default,,0000,0000,0000,,hint, change the\Norder of integration Dialogue: 0,1:38:00.57,1:38:03.16,Default,,0000,0000,0000,,because it may work\Nthe other way around. Dialogue: 0,1:38:03.16,1:38:06.26,Default,,0000,0000,0000,,They just give it to you, and\Nthen people can spend an hour Dialogue: 0,1:38:06.26,1:38:08.42,Default,,0000,0000,0000,,and they don't get anywhere. Dialogue: 0,1:38:08.42,1:38:12.14,Default,,0000,0000,0000,,If you want to be mean to a\Nstudent, that's what you do. Dialogue: 0,1:38:12.14,1:38:17.30,Default,,0000,0000,0000,,So I will tell\Nyou that one needs Dialogue: 0,1:38:17.30,1:38:20.02,Default,,0000,0000,0000,,to change the order of\Nintegration for this. Dialogue: 0,1:38:20.02,1:38:21.10,Default,,0000,0000,0000,,This is the function. Dialogue: 0,1:38:21.10,1:38:26.04,Default,,0000,0000,0000,,We keep the function, but let's\Nsee what happens if you draw. Dialogue: 0,1:38:26.04,1:38:31.12,Default,,0000,0000,0000,,The domain will be\Nx between 0 and 1. Dialogue: 0,1:38:31.12,1:38:33.92,Default,,0000,0000,0000,,This is your x value. Dialogue: 0,1:38:33.92,1:38:37.35,Default,,0000,0000,0000,,y will be between x and 1. Dialogue: 0,1:38:37.35,1:38:39.99,Default,,0000,0000,0000,,So it's like you have a square. Dialogue: 0,1:38:39.99,1:38:43.69,Default,,0000,0000,0000,,y equals x is your\Ndiagonal of the square. Dialogue: 0,1:38:43.69,1:38:49.42,Default,,0000,0000,0000,,And you go from--\Nmore colors, please. Dialogue: 0,1:38:49.42,1:38:54.78,Default,,0000,0000,0000,,You go from y equals x on the\Nbottom and y equals 1 on top. Dialogue: 0,1:38:54.78,1:38:57.49,Default,,0000,0000,0000,,And so the domain is\Nthis beautiful triangle Dialogue: 0,1:38:57.49,1:39:02.71,Default,,0000,0000,0000,,that I make all in line\Nwith vertical strips. Dialogue: 0,1:39:02.71,1:39:06.32,Default,,0000,0000,0000,,This is what it means,\Nvertical strips. Dialogue: 0,1:39:06.32,1:39:11.92,Default,,0000,0000,0000,,But if I do horizontal strips, I\Nhave to change the color, blue. Dialogue: 0,1:39:11.92,1:39:14.54,Default,,0000,0000,0000,,And for horizontal\Nstrips, I'm going Dialogue: 0,1:39:14.54,1:39:16.70,Default,,0000,0000,0000,,to have a different problem. Dialogue: 0,1:39:16.70,1:39:20.22,Default,,0000,0000,0000,,Integral, integral dx dy. Dialogue: 0,1:39:20.22,1:39:23.11,Default,,0000,0000,0000,,And I just hope to god\Nthat what I'm going to get Dialogue: 0,1:39:23.11,1:39:27.00,Default,,0000,0000,0000,,is doable because if\Nnot, then I'm in trouble. Dialogue: 0,1:39:27.00,1:39:30.28,Default,,0000,0000,0000,,So help me on this one. Dialogue: 0,1:39:30.28,1:39:33.68,Default,,0000,0000,0000,,If y is between what and what? Dialogue: 0,1:39:33.68,1:39:35.18,Default,,0000,0000,0000,,It's a square. Dialogue: 0,1:39:35.18,1:39:38.61,Default,,0000,0000,0000,,It's a square, so this will\Nbe the same, 0 to 1, right? Dialogue: 0,1:39:38.61,1:39:39.42,Default,,0000,0000,0000,,STUDENT: Yep. Dialogue: 0,1:39:39.42,1:39:40.31,Default,,0000,0000,0000,,PROFESSOR: But Mr. X? Dialogue: 0,1:39:40.31,1:39:41.72,Default,,0000,0000,0000,,How about Mr. X? Dialogue: 0,1:39:41.72,1:39:45.93,Default,,0000,0000,0000,,STUDENT: And then it\Nwill be between 1 and y. Dialogue: 0,1:39:45.93,1:39:49.41,Default,,0000,0000,0000,,PROFESSOR: Between--\NMr. X is this guy. Dialogue: 0,1:39:49.41,1:39:51.88,Default,,0000,0000,0000,,And he doesn't go between 1. Dialogue: 0,1:39:51.88,1:39:54.84,Default,,0000,0000,0000,,He goes between the\Nsea level, which is Dialogue: 0,1:39:54.84,1:40:02.60,Default,,0000,0000,0000,,x equals 0, to x equals what? Dialogue: 0,1:40:02.60,1:40:03.49,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:40:03.49,1:40:04.77,Default,,0000,0000,0000,,PROFESSOR: Right? Dialogue: 0,1:40:04.77,1:40:10.16,Default,,0000,0000,0000,,So from x equals 0\Nthrough x equals y. Dialogue: 0,1:40:10.16,1:40:15.43,Default,,0000,0000,0000,,And you have the same individual\Ne to the y squared that before Dialogue: 0,1:40:15.43,1:40:17.22,Default,,0000,0000,0000,,went on your nerves. Dialogue: 0,1:40:17.22,1:40:20.13,Default,,0000,0000,0000,,Now he's not so bad, actually. Dialogue: 0,1:40:20.13,1:40:21.79,Default,,0000,0000,0000,,Why is he not so bad? Dialogue: 0,1:40:21.79,1:40:24.72,Default,,0000,0000,0000,,Look what happens in\Nthe first parentheses. Dialogue: 0,1:40:24.72,1:40:27.28,Default,,0000,0000,0000,,This is so beautiful\Nthat it's something Dialogue: 0,1:40:27.28,1:40:29.10,Default,,0000,0000,0000,,you didn't even hope for. Dialogue: 0,1:40:29.10,1:40:34.45,Default,,0000,0000,0000,,So we copy and paste\Nit from 0 to 1 dy. Dialogue: 0,1:40:34.45,1:40:38.33,Default,,0000,0000,0000,,These guys stay\Noutside and they wait. Dialogue: 0,1:40:38.33,1:40:40.37,Default,,0000,0000,0000,,Inside, it's our\Nbusiness what we do. Dialogue: 0,1:40:40.37,1:40:44.79,Default,,0000,0000,0000,,So Mr. X is independent\Nfrom e to the y squared. Dialogue: 0,1:40:44.79,1:40:46.83,Default,,0000,0000,0000,,So e to the y squared pulls out. Dialogue: 0,1:40:46.83,1:40:48.34,Default,,0000,0000,0000,,He's a constant. Dialogue: 0,1:40:48.34,1:40:53.39,Default,,0000,0000,0000,,And you have integral\Nof 1 dx between 0 and y. Dialogue: 0,1:40:53.39,1:40:56.67,Default,,0000,0000,0000,,How much is that? Dialogue: 0,1:40:56.67,1:40:57.92,Default,,0000,0000,0000,,1. Dialogue: 0,1:40:57.92,1:41:03.55,Default,,0000,0000,0000,,x between x equals\N0 and x equals y. Dialogue: 0,1:41:03.55,1:41:05.00,Default,,0000,0000,0000,,So it's y. Dialogue: 0,1:41:05.00,1:41:05.88,Default,,0000,0000,0000,,So I'm being serious. Dialogue: 0,1:41:05.88,1:41:07.60,Default,,0000,0000,0000,,So I should have said y. Dialogue: 0,1:41:07.60,1:41:11.83,Default,,0000,0000,0000,, Dialogue: 0,1:41:11.83,1:41:21.05,Default,,0000,0000,0000,,Now, if your professor would\Nhave given you, in Calc 2, Dialogue: 0,1:41:21.05,1:41:25.05,Default,,0000,0000,0000,,this, how would\Nyou have done it? Dialogue: 0,1:41:25.05,1:41:26.93,Default,,0000,0000,0000,,STUDENT: U-substitution. Dialogue: 0,1:41:26.93,1:41:28.01,Default,,0000,0000,0000,,PROFESSOR: U-substitution. Dialogue: 0,1:41:28.01,1:41:28.78,Default,,0000,0000,0000,,Excellent. Dialogue: 0,1:41:28.78,1:41:32.13,Default,,0000,0000,0000,,What kind of\Nu-substitution [INAUDIBLE]? Dialogue: 0,1:41:32.13,1:41:35.10,Default,,0000,0000,0000,,STUDENT: y squared equals u. Dialogue: 0,1:41:35.10,1:41:40.41,Default,,0000,0000,0000,,PROFESSOR: y squared\Nequals u, du equals 2y dy. Dialogue: 0,1:41:40.41,1:41:44.40,Default,,0000,0000,0000,,So y dy together. Dialogue: 0,1:41:44.40,1:41:45.97,Default,,0000,0000,0000,,They stick together. Dialogue: 0,1:41:45.97,1:41:47.33,Default,,0000,0000,0000,,They stick together. Dialogue: 0,1:41:47.33,1:41:49.56,Default,,0000,0000,0000,,They attract each\Nother as magnets. Dialogue: 0,1:41:49.56,1:41:56.93,Default,,0000,0000,0000,,So y dy is going to be\N1/2 du-- 1/2 pulls out-- Dialogue: 0,1:41:56.93,1:42:00.19,Default,,0000,0000,0000,,integral e to the u du. Dialogue: 0,1:42:00.19,1:42:00.69,Default,,0000,0000,0000,,Attention. Dialogue: 0,1:42:00.69,1:42:03.43,Default,,0000,0000,0000,,When y is moving\Nbetween 0 and 1, Dialogue: 0,1:42:03.43,1:42:06.21,Default,,0000,0000,0000,,u is moving also\Nbetween 0 and 1. Dialogue: 0,1:42:06.21,1:42:11.99,Default,,0000,0000,0000,,So it really should\Nbe a piece of cake. Dialogue: 0,1:42:11.99,1:42:13.50,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,1:42:13.50,1:42:15.88,Default,,0000,0000,0000,,Do you understand what I did? Dialogue: 0,1:42:15.88,1:42:18.97,Default,,0000,0000,0000,,Do you understand the words\Ncoming out of my mouth? Dialogue: 0,1:42:18.97,1:42:24.42,Default,,0000,0000,0000,, Dialogue: 0,1:42:24.42,1:42:25.40,Default,,0000,0000,0000,,It's easy. Dialogue: 0,1:42:25.40,1:42:29.49,Default,,0000,0000,0000,, Dialogue: 0,1:42:29.49,1:42:29.99,Default,,0000,0000,0000,,Good. Dialogue: 0,1:42:29.99,1:42:35.46,Default,,0000,0000,0000,,So what is integral\Nof e to the u du? Dialogue: 0,1:42:35.46,1:42:39.22,Default,,0000,0000,0000,,e to the u between\N1 up and 0 down. Dialogue: 0,1:42:39.22,1:42:43.98,Default,,0000,0000,0000,,So e to the u de to the 1\Nminus e to the 0 over 2. Dialogue: 0,1:42:43.98,1:42:47.93,Default,,0000,0000,0000,, Dialogue: 0,1:42:47.93,1:42:51.04,Default,,0000,0000,0000,,That is e minus 1 over 2. Dialogue: 0,1:42:51.04,1:42:54.32,Default,,0000,0000,0000,, Dialogue: 0,1:42:54.32,1:42:59.22,Default,,0000,0000,0000,,I could not have solved this\Nif I tried it by integration Dialogue: 0,1:42:59.22,1:43:02.67,Default,,0000,0000,0000,,with y first and then x. Dialogue: 0,1:43:02.67,1:43:04.60,Default,,0000,0000,0000,,The only way I\Ncould have done this Dialogue: 0,1:43:04.60,1:43:07.70,Default,,0000,0000,0000,,is by changing the\Norder of integration. Dialogue: 0,1:43:07.70,1:43:11.56,Default,,0000,0000,0000,,So how many times have I seen\Nthis in the past 12 years Dialogue: 0,1:43:11.56,1:43:12.51,Default,,0000,0000,0000,,on the final? Dialogue: 0,1:43:12.51,1:43:15.23,Default,,0000,0000,0000,,At least six times. Dialogue: 0,1:43:15.23,1:43:18.22,Default,,0000,0000,0000,,It's a problem that\Ncould be a little bit Dialogue: 0,1:43:18.22,1:43:21.15,Default,,0000,0000,0000,,hard if the student has\Nnever seen it before Dialogue: 0,1:43:21.15,1:43:23.81,Default,,0000,0000,0000,,and doesn't know what to\Ndo [? at that point. ?] Dialogue: 0,1:43:23.81,1:43:27.08,Default,,0000,0000,0000,,Let's do a few more\Nin the same category. Dialogue: 0,1:43:27.08,1:43:36.65,Default,,0000,0000,0000,, Dialogue: 0,1:43:36.65,1:43:37.61,Default,,0000,0000,0000,,STUDENT: Professor? Dialogue: 0,1:43:37.61,1:43:38.27,Default,,0000,0000,0000,,PROFESSOR: Yes? Dialogue: 0,1:43:38.27,1:43:40.89,Default,,0000,0000,0000,,STUDENT: Where did this shape--\Nwhere did this graph come from? Dialogue: 0,1:43:40.89,1:43:43.72,Default,,0000,0000,0000,,Were we just saying\Nit was with the same-- Dialogue: 0,1:43:43.72,1:43:44.64,Default,,0000,0000,0000,,PROFESSOR: OK. Dialogue: 0,1:43:44.64,1:43:46.75,Default,,0000,0000,0000,,I read it from here. Dialogue: 0,1:43:46.75,1:43:50.46,Default,,0000,0000,0000,,So this and that are the key. Dialogue: 0,1:43:50.46,1:43:55.16,Default,,0000,0000,0000,,This is telling me x is between\N0 and 1, and at the same, Dialogue: 0,1:43:55.16,1:43:59.01,Default,,0000,0000,0000,,time y is between x and 1. Dialogue: 0,1:43:59.01,1:44:02.51,Default,,0000,0000,0000,,And when I read this\Ninformation on the graph, Dialogue: 0,1:44:02.51,1:44:05.57,Default,,0000,0000,0000,,I say, well, x is\Nbetween 0 and 1. Dialogue: 0,1:44:05.57,1:44:09.07,Default,,0000,0000,0000,,Mr. Y has the freedom to go\Nbetween the first bisector, Dialogue: 0,1:44:09.07,1:44:14.07,Default,,0000,0000,0000,,which is that, and the\Ncap, his cap, y equals 1. Dialogue: 0,1:44:14.07,1:44:17.45,Default,,0000,0000,0000,,So that's how I got\Nto the line strips. Dialogue: 0,1:44:17.45,1:44:21.26,Default,,0000,0000,0000,,And from the line strips, I said\Nthat I need horizontal strips. Dialogue: 0,1:44:21.26,1:44:23.97,Default,,0000,0000,0000,,So I changed the\Ncolor and I said Dialogue: 0,1:44:23.97,1:44:27.69,Default,,0000,0000,0000,,the blue strips go between x. Dialogue: 0,1:44:27.69,1:44:31.86,Default,,0000,0000,0000,,x will be x equals\N0 and x equals y. Dialogue: 0,1:44:31.86,1:44:36.91,Default,,0000,0000,0000,,And then y between 0\Nand 1, just the same. Dialogue: 0,1:44:36.91,1:44:38.08,Default,,0000,0000,0000,,It's a little bit tricky. Dialogue: 0,1:44:38.08,1:44:42.36,Default,,0000,0000,0000,,That's why I want to do one or\Ntwo more problems like that, Dialogue: 0,1:44:42.36,1:44:46.84,Default,,0000,0000,0000,,because I know that I remember\N20-something years ago, Dialogue: 0,1:44:46.84,1:44:52.50,Default,,0000,0000,0000,,I myself needed a little\Nbit of time understanding Dialogue: 0,1:44:52.50,1:44:56.71,Default,,0000,0000,0000,,the meaning of reversing\Nthe order of integration. Dialogue: 0,1:44:56.71,1:44:58.97,Default,,0000,0000,0000,,STUDENT: Does it matter\Nwhich way you put it? Dialogue: 0,1:44:58.97,1:45:02.41,Default,,0000,0000,0000,,PROFESSOR: In this case, it's\Nimportant that you do reverse. Dialogue: 0,1:45:02.41,1:45:05.89,Default,,0000,0000,0000,,But in general, it's\Ndoable both ways. Dialogue: 0,1:45:05.89,1:45:10.02,Default,,0000,0000,0000,,I mean, in the other problems\NI'm going to give you today, Dialogue: 0,1:45:10.02,1:45:11.89,Default,,0000,0000,0000,,you should be able\Nto do either way. Dialogue: 0,1:45:11.89,1:45:19.13,Default,,0000,0000,0000,,So I'm looking for a problem\Nthat you could eventually Dialogue: 0,1:45:19.13,1:45:20.61,Default,,0000,0000,0000,,do another one. Dialogue: 0,1:45:20.61,1:45:25.53,Default,,0000,0000,0000,, Dialogue: 0,1:45:25.53,1:45:27.99,Default,,0000,0000,0000,,We don't have so many. Dialogue: 0,1:45:27.99,1:45:31.72,Default,,0000,0000,0000,,I'm going to go ahead and\Nlook into the homework. Dialogue: 0,1:45:31.72,1:45:32.22,Default,,0000,0000,0000,,Yeah. Dialogue: 0,1:45:32.22,1:45:34.84,Default,,0000,0000,0000,, Dialogue: 0,1:45:34.84,1:45:41.40,Default,,0000,0000,0000,,So it says, you\Nhave this integral, Dialogue: 0,1:45:41.40,1:45:44.17,Default,,0000,0000,0000,,the integral from 0\Nto 4 of the integral Dialogue: 0,1:45:44.17,1:45:49.88,Default,,0000,0000,0000,,from x squared to 4y dy dx. Dialogue: 0,1:45:49.88,1:45:55.99,Default,,0000,0000,0000,,Draw, compute, and also\Ncompute with reversing Dialogue: 0,1:45:55.99,1:45:58.99,Default,,0000,0000,0000,,the order of integration\Nto check your work. Dialogue: 0,1:45:58.99,1:46:01.16,Default,,0000,0000,0000,,When I say that,\Nit sounds horrible. Dialogue: 0,1:46:01.16,1:46:04.28,Default,,0000,0000,0000,,But in reality, the\Nmore you work on Dialogue: 0,1:46:04.28,1:46:08.12,Default,,0000,0000,0000,,that one, the more familiar\Nyou're going to feel. Dialogue: 0,1:46:08.12,1:46:10.50,Default,,0000,0000,0000,,So what did I just say? Dialogue: 0,1:46:10.50,1:46:12.74,Default,,0000,0000,0000,,Problem number 26. Dialogue: 0,1:46:12.74,1:46:18.20,Default,,0000,0000,0000,,You have integral\Nfrom 0 to 4, integral Dialogue: 0,1:46:18.20,1:46:24.56,Default,,0000,0000,0000,,from x squared to 4x dy dx. Dialogue: 0,1:46:24.56,1:46:27.40,Default,,0000,0000,0000,, Dialogue: 0,1:46:27.40,1:46:31.31,Default,,0000,0000,0000,,Interpret geometrically,\Nwhatever that means, Dialogue: 0,1:46:31.31,1:46:35.26,Default,,0000,0000,0000,,and then compute the\Nintegral in two ways, Dialogue: 0,1:46:35.26,1:46:37.87,Default,,0000,0000,0000,,with this given order\Nintegration, which Dialogue: 0,1:46:37.87,1:46:40.00,Default,,0000,0000,0000,,is what kind of strips, guys? Dialogue: 0,1:46:40.00,1:46:41.91,Default,,0000,0000,0000,,Vertical strips. Dialogue: 0,1:46:41.91,1:46:45.01,Default,,0000,0000,0000,,Or reversing the\Norder of integration. Dialogue: 0,1:46:45.01,1:46:50.22,Default,,0000,0000,0000,,And check that the answer is the\Nsame just to check your work. Dialogue: 0,1:46:50.22,1:46:51.96,Default,,0000,0000,0000,,STUDENT: So first-- Dialogue: 0,1:46:51.96,1:46:53.10,Default,,0000,0000,0000,,PROFESSOR: First you draw. Dialogue: 0,1:46:53.10,1:46:55.89,Default,,0000,0000,0000,,First you draw because\Nif you don't draw, Dialogue: 0,1:46:55.89,1:47:00.49,Default,,0000,0000,0000,,you don't understand what\Nthe problem is about. Dialogue: 0,1:47:00.49,1:47:01.73,Default,,0000,0000,0000,,And you say, wait a minute. Dialogue: 0,1:47:01.73,1:47:05.29,Default,,0000,0000,0000,,But couldn't I go ahead\Nand do it without drawing? Dialogue: 0,1:47:05.29,1:47:08.40,Default,,0000,0000,0000,,Yeah, but you're not\Ngoing to get too far. Dialogue: 0,1:47:08.40,1:47:11.87,Default,,0000,0000,0000,,So let's see what kind\Nof problem you have. Dialogue: 0,1:47:11.87,1:47:13.21,Default,,0000,0000,0000,,y and x. Dialogue: 0,1:47:13.21,1:47:16.80,Default,,0000,0000,0000,,y equals x squared is a what? Dialogue: 0,1:47:16.80,1:47:18.76,Default,,0000,0000,0000,,It's a pa-- Dialogue: 0,1:47:18.76,1:47:19.75,Default,,0000,0000,0000,,STUDENT: Parabola. Dialogue: 0,1:47:19.75,1:47:20.73,Default,,0000,0000,0000,,PROFESSOR: Parabola. Dialogue: 0,1:47:20.73,1:47:24.19,Default,,0000,0000,0000,,And this parabola should\Nbe nice and sassy. Dialogue: 0,1:47:24.19,1:47:25.68,Default,,0000,0000,0000,,Is it fat enough? Dialogue: 0,1:47:25.68,1:47:27.47,Default,,0000,0000,0000,,I think it is. Dialogue: 0,1:47:27.47,1:47:34.14,Default,,0000,0000,0000,,And the other one will\Nbe 4x, y equals 4x. Dialogue: 0,1:47:34.14,1:47:36.07,Default,,0000,0000,0000,,What does that look like? Dialogue: 0,1:47:36.07,1:47:39.34,Default,,0000,0000,0000,,It looks like a line passing\Nthrough the origin that Dialogue: 0,1:47:39.34,1:47:42.70,Default,,0000,0000,0000,,has slope 4, so the\Nslope is really high. Dialogue: 0,1:47:42.70,1:47:43.69,Default,,0000,0000,0000,,STUDENT: Just straight. Dialogue: 0,1:47:43.69,1:47:48.15,Default,,0000,0000,0000,, Dialogue: 0,1:47:48.15,1:47:51.87,Default,,0000,0000,0000,,PROFESSOR: y equals 4x\Nversus y equals x squared. Dialogue: 0,1:47:51.87,1:47:53.69,Default,,0000,0000,0000,,Now, do they meet? Dialogue: 0,1:47:53.69,1:47:57.34,Default,,0000,0000,0000,, Dialogue: 0,1:47:57.34,1:47:57.88,Default,,0000,0000,0000,,STUDENT: Yes. Dialogue: 0,1:47:57.88,1:47:58.51,Default,,0000,0000,0000,,PROFESSOR: Yes. Dialogue: 0,1:47:58.51,1:47:59.75,Default,,0000,0000,0000,,Exactly where do they meet? Dialogue: 0,1:47:59.75,1:48:00.30,Default,,0000,0000,0000,,Exactly here. Dialogue: 0,1:48:00.30,1:48:00.80,Default,,0000,0000,0000,,STUDENT: 4. Dialogue: 0,1:48:00.80,1:48:04.19,Default,,0000,0000,0000,,PROFESSOR: So 4x equals x\Nsquared, where do they meet? Dialogue: 0,1:48:04.19,1:48:06.93,Default,,0000,0000,0000,, Dialogue: 0,1:48:06.93,1:48:13.48,Default,,0000,0000,0000,,They meet at-- it has\Ntwo possible roots. Dialogue: 0,1:48:13.48,1:48:18.42,Default,,0000,0000,0000,,One is x equals\N0, which is here, Dialogue: 0,1:48:18.42,1:48:21.27,Default,,0000,0000,0000,,and one is x equals\N4, which is here. Dialogue: 0,1:48:21.27,1:48:26.72,Default,,0000,0000,0000,,So really, my graph looks\Njust the way it should look, Dialogue: 0,1:48:26.72,1:48:29.45,Default,,0000,0000,0000,,only my parabola is\Na little bit too fat. Dialogue: 0,1:48:29.45,1:48:33.84,Default,,0000,0000,0000,, Dialogue: 0,1:48:33.84,1:48:44.10,Default,,0000,0000,0000,,This is the point of\Ncoordinates 4 and 16. Dialogue: 0,1:48:44.10,1:48:46.41,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,1:48:46.41,1:48:52.35,Default,,0000,0000,0000,,And Mr. X is moving\Nbetween 0 and 4. Dialogue: 0,1:48:52.35,1:48:56.91,Default,,0000,0000,0000,,This is the maximum\Nlevel x can get. Dialogue: 0,1:48:56.91,1:49:01.73,Default,,0000,0000,0000,,And where he stops here\Nat 4, a miracle happens. Dialogue: 0,1:49:01.73,1:49:06.68,Default,,0000,0000,0000,,The two curves intersect each\Nother exactly at that point. Dialogue: 0,1:49:06.68,1:49:11.90,Default,,0000,0000,0000,,So this looks like a\Nleaf, a slice of orange. Dialogue: 0,1:49:11.90,1:49:12.40,Default,,0000,0000,0000,,Oh my god. Dialogue: 0,1:49:12.40,1:49:12.94,Default,,0000,0000,0000,,I don't know. Dialogue: 0,1:49:12.94,1:49:17.82,Default,,0000,0000,0000,,I'm already hungry so I cannot\Nwait to get out of here. Dialogue: 0,1:49:17.82,1:49:20.68,Default,,0000,0000,0000,,I bet you're hungry as well. Dialogue: 0,1:49:20.68,1:49:24.00,Default,,0000,0000,0000,,Let's do this problem\Nboth ways and then go Dialogue: 0,1:49:24.00,1:49:26.54,Default,,0000,0000,0000,,home or to have\Nsomething to eat. Dialogue: 0,1:49:26.54,1:49:31.96,Default,,0000,0000,0000,,How are you going to advise\Nme to solve it first? Dialogue: 0,1:49:31.96,1:49:34.22,Default,,0000,0000,0000,,It's already set\Nup to be solved. Dialogue: 0,1:49:34.22,1:49:35.48,Default,,0000,0000,0000,,So it's vertical strips. Dialogue: 0,1:49:35.48,1:49:37.91,Default,,0000,0000,0000,,And I will say\Nintegral from 0 to 4, Dialogue: 0,1:49:37.91,1:49:40.79,Default,,0000,0000,0000,,copy and paste the outer part. Dialogue: 0,1:49:40.79,1:49:46.09,Default,,0000,0000,0000,,Take the inner part, and do the\Ninner part because it's easy. Dialogue: 0,1:49:46.09,1:49:50.40,Default,,0000,0000,0000,,And if it's easy, you tell\Nme how I'm going to do it. Dialogue: 0,1:49:50.40,1:49:53.56,Default,,0000,0000,0000,,Integral of 1 dy is y. Dialogue: 0,1:49:53.56,1:49:58.82,Default,,0000,0000,0000,,y measured at 4x is 4x,\Nand y measured at x squared Dialogue: 0,1:49:58.82,1:50:01.12,Default,,0000,0000,0000,,is x squared. Dialogue: 0,1:50:01.12,1:50:01.79,Default,,0000,0000,0000,,Oh thank god. Dialogue: 0,1:50:01.79,1:50:05.73,Default,,0000,0000,0000,,This is so beautiful\Nand so easy. Dialogue: 0,1:50:05.73,1:50:08.65,Default,,0000,0000,0000,,Let's integrate again. Dialogue: 0,1:50:08.65,1:50:16.37,Default,,0000,0000,0000,,4 x squared over 2 times x cubed\Nover 3 between x equals 0 down Dialogue: 0,1:50:16.37,1:50:17.92,Default,,0000,0000,0000,,and x equals 4 up. Dialogue: 0,1:50:17.92,1:50:22.19,Default,,0000,0000,0000,, Dialogue: 0,1:50:22.19,1:50:23.81,Default,,0000,0000,0000,,What do I get? Dialogue: 0,1:50:23.81,1:50:30.05,Default,,0000,0000,0000,,I get 4 cubed over 2\Nminus 4 cubed over 3. Dialogue: 0,1:50:30.05,1:50:31.84,Default,,0000,0000,0000,,This 4 cubed is an obsession. Dialogue: 0,1:50:31.84,1:50:33.81,Default,,0000,0000,0000,,Kick him out. Dialogue: 0,1:50:33.81,1:50:35.74,Default,,0000,0000,0000,,1/2 minus 1/3. Dialogue: 0,1:50:35.74,1:50:39.61,Default,,0000,0000,0000,, Dialogue: 0,1:50:39.61,1:50:41.45,Default,,0000,0000,0000,,How much is 1/2 minus 1/3? Dialogue: 0,1:50:41.45,1:50:42.43,Default,,0000,0000,0000,,My son knows that. Dialogue: 0,1:50:42.43,1:50:43.58,Default,,0000,0000,0000,,STUDENT: 1/6. Dialogue: 0,1:50:43.58,1:50:44.16,Default,,0000,0000,0000,,PROFESSOR: OK. Dialogue: 0,1:50:44.16,1:50:46.06,Default,,0000,0000,0000,,1/6, yes. Dialogue: 0,1:50:46.06,1:50:48.71,Default,,0000,0000,0000,,So we simply take it. Dialogue: 0,1:50:48.71,1:50:49.96,Default,,0000,0000,0000,,We can leave it like that. Dialogue: 0,1:50:49.96,1:50:55.53,Default,,0000,0000,0000,,If you leave it like that on\Nthe exam, I don't mind at all. Dialogue: 0,1:50:55.53,1:50:58.68,Default,,0000,0000,0000,,But you could always put\N64 over 6 and simplify it. Dialogue: 0,1:50:58.68,1:51:01.51,Default,,0000,0000,0000,, Dialogue: 0,1:51:01.51,1:51:03.39,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,1:51:03.39,1:51:07.07,Default,,0000,0000,0000,,You can simplify\Nit and get what? Dialogue: 0,1:51:07.07,1:51:08.20,Default,,0000,0000,0000,,32 over 3. Dialogue: 0,1:51:08.20,1:51:10.98,Default,,0000,0000,0000,, Dialogue: 0,1:51:10.98,1:51:12.53,Default,,0000,0000,0000,,Don't give me decimals. Dialogue: 0,1:51:12.53,1:51:14.63,Default,,0000,0000,0000,,I'm not impressed. Dialogue: 0,1:51:14.63,1:51:16.38,Default,,0000,0000,0000,,You're not supposed\Nto use the calculator. Dialogue: 0,1:51:16.38,1:51:21.30,Default,,0000,0000,0000,,You are supposed to leave\Nthis is exact fraction Dialogue: 0,1:51:21.30,1:51:24.85,Default,,0000,0000,0000,,form like that, irreducible. Dialogue: 0,1:51:24.85,1:51:26.32,Default,,0000,0000,0000,,Let's do it the\Nother way around, Dialogue: 0,1:51:26.32,1:51:30.02,Default,,0000,0000,0000,,and that will be the\Nlast thing we do. Dialogue: 0,1:51:30.02,1:51:34.02,Default,,0000,0000,0000,,The other way around means\NI'll take another color. Dialogue: 0,1:51:34.02,1:51:36.98,Default,,0000,0000,0000,,I'll do the horizontal stripes. Dialogue: 0,1:51:36.98,1:51:40.01,Default,,0000,0000,0000,, Dialogue: 0,1:51:40.01,1:51:44.11,Default,,0000,0000,0000,,And I will have to rewrite\Nthe meaning of these two Dialogue: 0,1:51:44.11,1:51:49.53,Default,,0000,0000,0000,,branches of functions with\Nx expressed in terms of y. Dialogue: 0,1:51:49.53,1:51:51.71,Default,,0000,0000,0000,,That's the only thing\NI need to do, right? Dialogue: 0,1:51:51.71,1:51:55.91,Default,,0000,0000,0000,,So what is this? Dialogue: 0,1:51:55.91,1:51:59.21,Default,,0000,0000,0000,,If y is x squared, what is x? Dialogue: 0,1:51:59.21,1:52:00.15,Default,,0000,0000,0000,,STUDENT: Root y. Dialogue: 0,1:52:00.15,1:52:03.74,Default,,0000,0000,0000,,PROFESSOR: The inverse\Nfunction. x will be root of y. Dialogue: 0,1:52:03.74,1:52:06.00,Default,,0000,0000,0000,,You said very well. Dialogue: 0,1:52:06.00,1:52:07.36,Default,,0000,0000,0000,,So I have to write. Dialogue: 0,1:52:07.36,1:52:10.47,Default,,0000,0000,0000,,In [INAUDIBLE], I\Nhave what I need Dialogue: 0,1:52:10.47,1:52:13.08,Default,,0000,0000,0000,,to have for the line\Nhorizontal strip method. Dialogue: 0,1:52:13.08,1:52:16.06,Default,,0000,0000,0000,, Dialogue: 0,1:52:16.06,1:52:19.66,Default,,0000,0000,0000,,And then for the other one,\Nx is going to be y over 4. Dialogue: 0,1:52:19.66,1:52:22.61,Default,,0000,0000,0000,, Dialogue: 0,1:52:22.61,1:52:23.60,Default,,0000,0000,0000,,So what do I do? Dialogue: 0,1:52:23.60,1:52:32.38,Default,,0000,0000,0000,,So integral, integral, a\N1 that was here hidden, Dialogue: 0,1:52:32.38,1:52:35.83,Default,,0000,0000,0000,,but I'll put it because\Nthat's the integral. Dialogue: 0,1:52:35.83,1:52:38.56,Default,,0000,0000,0000,,And then I go dx dy. Dialogue: 0,1:52:38.56,1:52:44.99,Default,,0000,0000,0000,,All I have to care about is the\Nendpoints of the integration. Dialogue: 0,1:52:44.99,1:52:48.24,Default,,0000,0000,0000,,Now, pay attention a little\Nbit because Mr. Y is not Dialogue: 0,1:52:48.24,1:52:49.66,Default,,0000,0000,0000,,between 0 and 4. Dialogue: 0,1:52:49.66,1:52:53.21,Default,,0000,0000,0000,,I had very good\Nstudents under stress Dialogue: 0,1:52:53.21,1:52:55.62,Default,,0000,0000,0000,,in the final putting 0 and 4. Dialogue: 0,1:52:55.62,1:52:56.76,Default,,0000,0000,0000,,Don't do that. Dialogue: 0,1:52:56.76,1:52:59.24,Default,,0000,0000,0000,,So pay attention to the\Nlimits of integration. Dialogue: 0,1:52:59.24,1:53:01.03,Default,,0000,0000,0000,,What are the limits? Dialogue: 0,1:53:01.03,1:53:01.75,Default,,0000,0000,0000,,0 and-- Dialogue: 0,1:53:01.75,1:53:02.41,Default,,0000,0000,0000,,STUDENT: 16. Dialogue: 0,1:53:02.41,1:53:02.99,Default,,0000,0000,0000,,PROFESSOR: 16. Dialogue: 0,1:53:02.99,1:53:04.97,Default,,0000,0000,0000,,Very good. Dialogue: 0,1:53:04.97,1:53:09.61,Default,,0000,0000,0000,,And x will be between root\Ny-- well, which one is on top? Dialogue: 0,1:53:09.61,1:53:11.61,Default,,0000,0000,0000,,Which one is on the bottom? Dialogue: 0,1:53:11.61,1:53:17.11,Default,,0000,0000,0000,,Because if I move my head,\NI'll say that's on top Dialogue: 0,1:53:17.11,1:53:18.61,Default,,0000,0000,0000,,and that's on the bottom. Dialogue: 0,1:53:18.61,1:53:22.21,Default,,0000,0000,0000,,STUDENT: The right side\Nis always on the top. Dialogue: 0,1:53:22.21,1:53:25.73,Default,,0000,0000,0000,,PROFESSOR: So the one that\Nlooks higher is this one. Dialogue: 0,1:53:25.73,1:53:29.21,Default,,0000,0000,0000,,This is more than\Nthat in this frame. Dialogue: 0,1:53:29.21,1:53:36.58,Default,,0000,0000,0000,,So square of y is on top and\Ny over 4 is on the bottom. Dialogue: 0,1:53:36.58,1:53:38.98,Default,,0000,0000,0000,,I should get the same answer. Dialogue: 0,1:53:38.98,1:53:40.42,Default,,0000,0000,0000,,If I don't, then I'm in trouble. Dialogue: 0,1:53:40.42,1:53:43.22,Default,,0000,0000,0000,,So what do I get? Dialogue: 0,1:53:43.22,1:53:49.08,Default,,0000,0000,0000,,Integral from 0 to 16. Dialogue: 0,1:53:49.08,1:53:51.50,Default,,0000,0000,0000,,Tonight, when I\Ngo home, I'm going Dialogue: 0,1:53:51.50,1:53:57.02,Default,,0000,0000,0000,,to cook up the homework\Nfor 12.1 and 12.1 at least. Dialogue: 0,1:53:57.02,1:53:59.18,Default,,0000,0000,0000,,I'll put some problems\Nsimilar to that Dialogue: 0,1:53:59.18,1:54:02.59,Default,,0000,0000,0000,,because I want to emphasize\Nthe same type of problem Dialogue: 0,1:54:02.59,1:54:05.01,Default,,0000,0000,0000,,in at least two or three\Napplications for the homework Dialogue: 0,1:54:05.01,1:54:07.20,Default,,0000,0000,0000,,for the midterm. Dialogue: 0,1:54:07.20,1:54:10.58,Default,,0000,0000,0000,,And maybe one like that will\Nbe on the final as well. Dialogue: 0,1:54:10.58,1:54:13.28,Default,,0000,0000,0000,,It's very important for\Nyou to understand how, Dialogue: 0,1:54:13.28,1:54:15.27,Default,,0000,0000,0000,,with this kind of\Ndomain, you reverse Dialogue: 0,1:54:15.27,1:54:16.83,Default,,0000,0000,0000,,the order of integration. Dialogue: 0,1:54:16.83,1:54:19.76,Default,,0000,0000,0000,,Who's helping me here? Dialogue: 0,1:54:19.76,1:54:22.21,Default,,0000,0000,0000,,Root y. Dialogue: 0,1:54:22.21,1:54:26.07,Default,,0000,0000,0000,,What is root y\Nwhen-- y to the 1/2. Dialogue: 0,1:54:26.07,1:54:28.07,Default,,0000,0000,0000,,I need to integrate. Dialogue: 0,1:54:28.07,1:54:33.55,Default,,0000,0000,0000,,So I need minus y over 4 and dy. Dialogue: 0,1:54:33.55,1:54:39.03,Default,,0000,0000,0000,, Dialogue: 0,1:54:39.03,1:54:42.19,Default,,0000,0000,0000,,Can you help me integrate? Dialogue: 0,1:54:42.19,1:54:44.18,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:54:44.18,1:54:49.83,Default,,0000,0000,0000,,PROFESSOR: 2/3 y\Nto the 3/2 minus-- Dialogue: 0,1:54:49.83,1:54:51.12,Default,,0000,0000,0000,,STUDENT: y squared. Dialogue: 0,1:54:51.12,1:54:56.48,Default,,0000,0000,0000,,PROFESSOR: y squared\Nover 8, y equals 0 Dialogue: 0,1:54:56.48,1:54:58.49,Default,,0000,0000,0000,,on the bottom, piece of cake. Dialogue: 0,1:54:58.49,1:55:00.22,Default,,0000,0000,0000,,That will give me 0. Dialogue: 0,1:55:00.22,1:55:00.97,Default,,0000,0000,0000,,I'm so happy. Dialogue: 0,1:55:00.97,1:55:04.56,Default,,0000,0000,0000,,And y equals 16 on top. Dialogue: 0,1:55:04.56,1:55:09.83,Default,,0000,0000,0000,,So for 16, I have 2/3. Dialogue: 0,1:55:09.83,1:55:12.21,Default,,0000,0000,0000,,And who's telling me what else? Dialogue: 0,1:55:12.21,1:55:13.17,Default,,0000,0000,0000,,STUDENT: 64. Dialogue: 0,1:55:13.17,1:55:13.91,Default,,0000,0000,0000,,PROFESSOR: 64. Dialogue: 0,1:55:13.91,1:55:14.48,Default,,0000,0000,0000,,4 cubed. Dialogue: 0,1:55:14.48,1:55:22.70,Default,,0000,0000,0000,,I can leave it 4 cubed if I want\Nto minus another-- well here, Dialogue: 0,1:55:22.70,1:55:24.74,Default,,0000,0000,0000,,I have to pay attention. Dialogue: 0,1:55:24.74,1:55:27.35,Default,,0000,0000,0000,,So I have 16 here. Dialogue: 0,1:55:27.35,1:55:31.39,Default,,0000,0000,0000,,I got square root of\N16, which is 4, cubed. Dialogue: 0,1:55:31.39,1:55:38.72,Default,,0000,0000,0000,,Here, I put minus 4\Nsquared, which was there. Dialogue: 0,1:55:38.72,1:55:40.64,Default,,0000,0000,0000,,How do you want me to\Ndo this simplification? Dialogue: 0,1:55:40.64,1:55:41.91,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]. Dialogue: 0,1:55:41.91,1:55:44.84,Default,,0000,0000,0000,,PROFESSOR: I can\Ndo 4 to the fourth. Dialogue: 0,1:55:44.84,1:55:47.19,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,1:55:47.19,1:55:52.40,Default,,0000,0000,0000,,I can put, like you\Nprefer, 16 squared over 8. Dialogue: 0,1:55:52.40,1:55:57.90,Default,,0000,0000,0000,, Dialogue: 0,1:55:57.90,1:55:59.39,Default,,0000,0000,0000,,Is it the same answer? Dialogue: 0,1:55:59.39,1:56:00.15,Default,,0000,0000,0000,,I don't know. Dialogue: 0,1:56:00.15,1:56:02.41,Default,,0000,0000,0000,,Let's see. Dialogue: 0,1:56:02.41,1:56:09.05,Default,,0000,0000,0000,,This is really 4 to the 4,\Nso I have 4 times 4 cubed. Dialogue: 0,1:56:09.05,1:56:19.79,Default,,0000,0000,0000,,4 cubed gets out and\NI have 2/3 minus 1/2. Dialogue: 0,1:56:19.79,1:56:24.06,Default,,0000,0000,0000,, Dialogue: 0,1:56:24.06,1:56:27.64,Default,,0000,0000,0000,,And how much is that? Dialogue: 0,1:56:27.64,1:56:28.76,Default,,0000,0000,0000,,Again 1/6. Dialogue: 0,1:56:28.76,1:56:30.67,Default,,0000,0000,0000,,Are you guys with me? Dialogue: 0,1:56:30.67,1:56:31.62,Default,,0000,0000,0000,,1/6. Dialogue: 0,1:56:31.62,1:56:36.86,Default,,0000,0000,0000,,So again, I get 4 cubed\Nover 6, so I'm done. Dialogue: 0,1:56:36.86,1:56:40.42,Default,,0000,0000,0000,,4 cubed over 6 equals 32 over 3. Dialogue: 0,1:56:40.42,1:56:42.96,Default,,0000,0000,0000,,I am happy that\NI checked my work Dialogue: 0,1:56:42.96,1:56:44.42,Default,,0000,0000,0000,,through two different methods. Dialogue: 0,1:56:44.42,1:56:45.71,Default,,0000,0000,0000,,I got the same answer. Dialogue: 0,1:56:45.71,1:56:49.22,Default,,0000,0000,0000,, Dialogue: 0,1:56:49.22,1:56:51.50,Default,,0000,0000,0000,,Now, let me tell you something. Dialogue: 0,1:56:51.50,1:56:55.22,Default,,0000,0000,0000,,There were also times\Nwhen on the midterm Dialogue: 0,1:56:55.22,1:56:59.80,Default,,0000,0000,0000,,or on the final, due to\Nlack of time and everything, Dialogue: 0,1:56:59.80,1:57:02.93,Default,,0000,0000,0000,,we put the following\Nkind of problem. Dialogue: 0,1:57:02.93,1:57:11.29,Default,,0000,0000,0000,,Without solving this integral--\Nwithout solving-- indicate Dialogue: 0,1:57:11.29,1:57:16.29,Default,,0000,0000,0000,,the corresponding integral\Nwith the order reversed. Dialogue: 0,1:57:16.29,1:57:19.68,Default,,0000,0000,0000,,So all you have to\Ndo-- don't do that. Dialogue: 0,1:57:19.68,1:57:24.72,Default,,0000,0000,0000,,Just from here,\Nwrite this and stop. Dialogue: 0,1:57:24.72,1:57:27.58,Default,,0000,0000,0000,,Don't waste your time. Dialogue: 0,1:57:27.58,1:57:29.62,Default,,0000,0000,0000,,If you do the whole thing,\Nit's going to take you Dialogue: 0,1:57:29.62,1:57:30.58,Default,,0000,0000,0000,,10 minutes, 15 minutes. Dialogue: 0,1:57:30.58,1:57:33.90,Default,,0000,0000,0000,,If you do just reversing\Nthe order of integration, Dialogue: 0,1:57:33.90,1:57:38.26,Default,,0000,0000,0000,,I don't know what it takes, a\Nminute and a half, two minutes. Dialogue: 0,1:57:38.26,1:57:42.25,Default,,0000,0000,0000,,So in order to save\Ntime, at times, Dialogue: 0,1:57:42.25,1:57:46.17,Default,,0000,0000,0000,,we gave you just don't\Nsolve the problem. reverse Dialogue: 0,1:57:46.17,1:57:47.62,Default,,0000,0000,0000,,the order of integration. Dialogue: 0,1:57:47.62,1:57:54.40,Default,,0000,0000,0000,, Dialogue: 0,1:57:54.40,1:57:55.75,Default,,0000,0000,0000,,One last one. Dialogue: 0,1:57:55.75,1:57:58.23,Default,,0000,0000,0000,,One last one. Dialogue: 0,1:57:58.23,1:57:59.73,Default,,0000,0000,0000,,But I don't want to finish it. Dialogue: 0,1:57:59.73,1:58:03.23,Default,,0000,0000,0000,,I want to give you\Nthe answer at home, Dialogue: 0,1:58:03.23,1:58:05.68,Default,,0000,0000,0000,,or maybe you can finish it. Dialogue: 0,1:58:05.68,1:58:07.64,Default,,0000,0000,0000,,It should be shorter. Dialogue: 0,1:58:07.64,1:58:13.54,Default,,0000,0000,0000,,You have a circular parabola,\Nbut only the first quadrant. Dialogue: 0,1:58:13.54,1:58:16.49,Default,,0000,0000,0000,, Dialogue: 0,1:58:16.49,1:58:19.25,Default,,0000,0000,0000,,So x is positive. Dialogue: 0,1:58:19.25,1:58:20.26,Default,,0000,0000,0000,,STUDENT: Question. Dialogue: 0,1:58:20.26,1:58:21.26,Default,,0000,0000,0000,,PROFESSOR: I don't know. Dialogue: 0,1:58:21.26,1:58:22.48,Default,,0000,0000,0000,,I have to find it. Dialogue: 0,1:58:22.48,1:58:23.84,Default,,0000,0000,0000,,Find the volume. Dialogue: 0,1:58:23.84,1:58:25.47,Default,,0000,0000,0000,,Example 4, page 934. Dialogue: 0,1:58:25.47,1:58:28.56,Default,,0000,0000,0000,,Find the volume\Nof the solid bound Dialogue: 0,1:58:28.56,1:58:32.69,Default,,0000,0000,0000,,in the above-- this is a\Nlittle tricky-- by the plane z Dialogue: 0,1:58:32.69,1:58:38.01,Default,,0000,0000,0000,,equals y and below\Nin the xy plane Dialogue: 0,1:58:38.01,1:58:42.46,Default,,0000,0000,0000,,by the part of the disk\Nin the first quadrant. Dialogue: 0,1:58:42.46,1:58:47.94,Default,,0000,0000,0000,,So z equals y means this\Nis your f of x and y. Dialogue: 0,1:58:47.94,1:58:50.55,Default,,0000,0000,0000,,So they gave it to you. Dialogue: 0,1:58:50.55,1:58:54.43,Default,,0000,0000,0000,,But then they say, but\Nalso, in the xy plane, Dialogue: 0,1:58:54.43,1:59:00.12,Default,,0000,0000,0000,,you have to have the part of\Nthe disk in the first quadrant. Dialogue: 0,1:59:00.12,1:59:01.79,Default,,0000,0000,0000,,This is not so easy. Dialogue: 0,1:59:01.79,1:59:04.64,Default,,0000,0000,0000,,They draw it for you to\Nmake your life easier. Dialogue: 0,1:59:04.64,1:59:08.05,Default,,0000,0000,0000,,The first quadrant is that. Dialogue: 0,1:59:08.05,1:59:13.60,Default,,0000,0000,0000,,How do you write the unit\Ncircle, x squared equals 1, Dialogue: 0,1:59:13.60,1:59:16.72,Default,,0000,0000,0000,,x squared plus y squared\Nless than or equal to 1, Dialogue: 0,1:59:16.72,1:59:19.35,Default,,0000,0000,0000,,and x and y are both positive. Dialogue: 0,1:59:19.35,1:59:21.12,Default,,0000,0000,0000,,This is the first quadrant. Dialogue: 0,1:59:21.12,1:59:22.51,Default,,0000,0000,0000,,How do you compute? Dialogue: 0,1:59:22.51,1:59:26.68,Default,,0000,0000,0000,,So they say compute the\Nvolume, and I say just Dialogue: 0,1:59:26.68,1:59:27.85,Default,,0000,0000,0000,,set up the volume. Dialogue: 0,1:59:27.85,1:59:30.06,Default,,0000,0000,0000,,Forget about computing it. Dialogue: 0,1:59:30.06,1:59:33.47,Default,,0000,0000,0000,,I could put it in the\Nmidterm just like that. Dialogue: 0,1:59:33.47,1:59:36.38,Default,,0000,0000,0000,,Set up an integral\Nwithout solving it Dialogue: 0,1:59:36.38,1:59:46.11,Default,,0000,0000,0000,,that indicates the volume\Nunder z equals f of xy-- that's Dialogue: 0,1:59:46.11,1:59:50.98,Default,,0000,0000,0000,,the geography of z-- and above\Na certain domain in plane, Dialogue: 0,1:59:50.98,1:59:55.51,Default,,0000,0000,0000,,above D in plane. Dialogue: 0,1:59:55.51,1:59:58.09,Default,,0000,0000,0000,,So you have, OK, what\Nthis should teach you? Dialogue: 0,1:59:58.09,2:00:08.66,Default,,0000,0000,0000,,Should teach you that double\Nintegral over d f of xy da Dialogue: 0,2:00:08.66,2:00:10.99,Default,,0000,0000,0000,,can be solved. Dialogue: 0,2:00:10.99,2:00:12.55,Default,,0000,0000,0000,,Do I ask to be solved? Dialogue: 0,2:00:12.55,2:00:13.72,Default,,0000,0000,0000,,No. Dialogue: 0,2:00:13.72,2:00:14.44,Default,,0000,0000,0000,,Why? Dialogue: 0,2:00:14.44,2:00:18.10,Default,,0000,0000,0000,,Because you can finish\Nit later, finish at home. Dialogue: 0,2:00:18.10,2:00:27.23,Default,,0000,0000,0000,,Or maybe, I don't even want\Nyou to compute on the final. Dialogue: 0,2:00:27.23,2:00:29.28,Default,,0000,0000,0000,,So how do we do that? Dialogue: 0,2:00:29.28,2:00:32.97,Default,,0000,0000,0000,,f is y. Dialogue: 0,2:00:32.97,2:00:36.52,Default,,0000,0000,0000,,Would I be able to choose\Nwhichever order integration I Dialogue: 0,2:00:36.52,2:00:38.38,Default,,0000,0000,0000,,want? Dialogue: 0,2:00:38.38,2:00:40.00,Default,,0000,0000,0000,,It shouldn't matter which order. Dialogue: 0,2:00:40.00,2:00:43.02,Default,,0000,0000,0000,,It should be more\Nor less the same. Dialogue: 0,2:00:43.02,2:00:44.70,Default,,0000,0000,0000,,What if I do dy dx? Dialogue: 0,2:00:44.70,2:00:47.63,Default,,0000,0000,0000,, Dialogue: 0,2:00:47.63,2:00:52.25,Default,,0000,0000,0000,,Then I have to do the Fubini. Dialogue: 0,2:00:52.25,2:00:54.23,Default,,0000,0000,0000,,But it's not a\Nrectangular domain. Dialogue: 0,2:00:54.23,2:00:54.73,Default,,0000,0000,0000,,Aha. Dialogue: 0,2:00:54.73,2:00:56.63,Default,,0000,0000,0000,,So Magdalena, be a\Nlittle bit careful Dialogue: 0,2:00:56.63,2:01:00.41,Default,,0000,0000,0000,,because this is going to\Nbe two finite numbers, Dialogue: 0,2:01:00.41,2:01:01.86,Default,,0000,0000,0000,,but these are functions. Dialogue: 0,2:01:01.86,2:01:04.34,Default,,0000,0000,0000,,STUDENT: It will\Nbe an x function. Dialogue: 0,2:01:04.34,2:01:08.07,Default,,0000,0000,0000,,PROFESSOR: So the x\Nis between 0 and 1, Dialogue: 0,2:01:08.07,2:01:10.02,Default,,0000,0000,0000,,and that's going to be z. Dialogue: 0,2:01:10.02,2:01:11.48,Default,,0000,0000,0000,,You do vertical strips. Dialogue: 0,2:01:11.48,2:01:13.92,Default,,0000,0000,0000,,That's a piece of cake. Dialogue: 0,2:01:13.92,2:01:17.89,Default,,0000,0000,0000,,But if you do the\Nvertical strips, Dialogue: 0,2:01:17.89,2:01:21.98,Default,,0000,0000,0000,,you have to pay attention to\Nthe endpoints for x and y, Dialogue: 0,2:01:21.98,2:01:23.43,Default,,0000,0000,0000,,and one is easy. Dialogue: 0,2:01:23.43,2:01:24.49,Default,,0000,0000,0000,,Which one is trivial? Dialogue: 0,2:01:24.49,2:01:25.07,Default,,0000,0000,0000,,STUDENT: Zero. Dialogue: 0,2:01:25.07,2:01:26.40,Default,,0000,0000,0000,,PROFESSOR: The bottom one, zero. Dialogue: 0,2:01:26.40,2:01:29.35,Default,,0000,0000,0000,,The one that's nontrivial\Nis the upper one. Dialogue: 0,2:01:29.35,2:01:31.29,Default,,0000,0000,0000,,STUDENT: There will be 1 minus-- Dialogue: 0,2:01:31.29,2:01:33.60,Default,,0000,0000,0000,,STUDENT: Square root\Nof 1 minus y squared. Dialogue: 0,2:01:33.60,2:01:34.47,Default,,0000,0000,0000,,PROFESSOR: Very good. Dialogue: 0,2:01:34.47,2:01:36.21,Default,,0000,0000,0000,,Square root of 1\Nminus y squared. Dialogue: 0,2:01:36.21,2:01:41.11,Default,,0000,0000,0000,, Dialogue: 0,2:01:41.11,2:01:46.64,Default,,0000,0000,0000,,So if I were to go one more step\Nfurther without solving this, Dialogue: 0,2:01:46.64,2:01:51.40,Default,,0000,0000,0000,,I'm going to ask you, could\Nthis be solved by hand? Dialogue: 0,2:01:51.40,2:01:57.89,Default,,0000,0000,0000,,Well, so you have\Nit in the book-- Dialogue: 0,2:01:57.89,2:02:00.39,Default,,0000,0000,0000,,STUDENT: Professor, should be\Na [INAUDIBLE] minus x squared? Dialogue: 0,2:02:00.39,2:02:03.01,Default,,0000,0000,0000,, Dialogue: 0,2:02:03.01,2:02:03.80,Default,,0000,0000,0000,,PROFESSOR: Oh yeah. Dialogue: 0,2:02:03.80,2:02:04.86,Default,,0000,0000,0000,,1 minus x squared. Dialogue: 0,2:02:04.86,2:02:06.59,Default,,0000,0000,0000,,Excuse me. Dialogue: 0,2:02:06.59,2:02:08.19,Default,,0000,0000,0000,,Didn't I write it? Dialogue: 0,2:02:08.19,2:02:11.100,Default,,0000,0000,0000,,Yeah, here I should have written\Ny equals square root of 1 Dialogue: 0,2:02:11.100,2:02:14.30,Default,,0000,0000,0000,,minus x squared. Dialogue: 0,2:02:14.30,2:02:21.26,Default,,0000,0000,0000,,So when you do it-- thank you\Nso much-- you go integrate, Dialogue: 0,2:02:21.26,2:02:26.79,Default,,0000,0000,0000,,and you have y squared over 2. Dialogue: 0,2:02:26.79,2:02:29.60,Default,,0000,0000,0000,,And you evaluate\Nbetween y equals 0 Dialogue: 0,2:02:29.60,2:02:33.47,Default,,0000,0000,0000,,and y equals square\Nroot 1 minus x squared, Dialogue: 0,2:02:33.47,2:02:34.92,Default,,0000,0000,0000,,and then you do the [INAUDIBLE]. Dialogue: 0,2:02:34.92,2:02:41.22,Default,,0000,0000,0000,, Dialogue: 0,2:02:41.22,2:02:44.76,Default,,0000,0000,0000,,In the book, they\Ndo it differently. Dialogue: 0,2:02:44.76,2:02:50.33,Default,,0000,0000,0000,,They do it with respect to\Ndx and dy and integrate. Dialogue: 0,2:02:50.33,2:02:52.80,Default,,0000,0000,0000,,But it doesn't\Nmatter how you do it. Dialogue: 0,2:02:52.80,2:02:54.77,Default,,0000,0000,0000,,You should get the same answer. Dialogue: 0,2:02:54.77,2:02:58.33,Default,,0000,0000,0000,, Dialogue: 0,2:02:58.33,2:03:00.14,Default,,0000,0000,0000,,All right? Dialogue: 0,2:03:00.14,2:03:01.06,Default,,0000,0000,0000,,[INAUDIBLE]? Dialogue: 0,2:03:01.06,2:03:03.86,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nin that way, Dialogue: 0,2:03:03.86,2:03:06.82,Default,,0000,0000,0000,,doesn't the square root work out\Nbetter because there's already Dialogue: 0,2:03:06.82,2:03:07.69,Default,,0000,0000,0000,,a y there? Dialogue: 0,2:03:07.69,2:03:09.04,Default,,0000,0000,0000,,PROFESSOR: In the other case-- Dialogue: 0,2:03:09.04,2:03:10.90,Default,,0000,0000,0000,,STUDENT: Doing dy dx. Dialogue: 0,2:03:10.90,2:03:12.72,Default,,0000,0000,0000,,PROFESSOR: Yeah,\Nin the other way, Dialogue: 0,2:03:12.72,2:03:14.34,Default,,0000,0000,0000,,it works a little\Nbit differently. Dialogue: 0,2:03:14.34,2:03:17.42,Default,,0000,0000,0000,,You can do\Nu-substitution, I think. Dialogue: 0,2:03:17.42,2:03:20.46,Default,,0000,0000,0000,,So if you do it the other\Nway, it will be what? Dialogue: 0,2:03:20.46,2:03:24.43,Default,,0000,0000,0000,,Integral from 0 to\N1, integral form 0 Dialogue: 0,2:03:24.43,2:03:32.43,Default,,0000,0000,0000,,to square root of 1\Nminus y squared, y dx dy. Dialogue: 0,2:03:32.43,2:03:35.17,Default,,0000,0000,0000,,And what do you do in this case? Dialogue: 0,2:03:35.17,2:03:37.43,Default,,0000,0000,0000,,You have integral from 0 to 1. Dialogue: 0,2:03:37.43,2:03:42.77,Default,,0000,0000,0000,,Integral of y dx is going\Nto be y is a constant. Dialogue: 0,2:03:42.77,2:03:48.86,Default,,0000,0000,0000,,x between the two values will\Nbe simply 1 minus y squared dy. Dialogue: 0,2:03:48.86,2:03:49.92,Default,,0000,0000,0000,,So you're right. Dialogue: 0,2:03:49.92,2:03:52.58,Default,,0000,0000,0000,,Matthew saw that,\Nbecause he's a prophet, Dialogue: 0,2:03:52.58,2:03:56.09,Default,,0000,0000,0000,,and he could see\Ntwo steps ahead. Dialogue: 0,2:03:56.09,2:03:57.85,Default,,0000,0000,0000,,This is very nice\Nwhat you observed. Dialogue: 0,2:03:57.85,2:03:59.10,Default,,0000,0000,0000,,What do you do? Dialogue: 0,2:03:59.10,2:04:02.59,Default,,0000,0000,0000,,You take a u-substitution\Nwhen you go home. Dialogue: 0,2:04:02.59,2:04:06.01,Default,,0000,0000,0000,,You get u equals\N1 minus y squared. Dialogue: 0,2:04:06.01,2:04:12.80,Default,,0000,0000,0000,,du will be minus 2y\Ndy, and you go on. Dialogue: 0,2:04:12.80,2:04:17.17,Default,,0000,0000,0000,,So in the book, we got 1/3. Dialogue: 0,2:04:17.17,2:04:19.60,Default,,0000,0000,0000,,If you continue\Nwith this method, Dialogue: 0,2:04:19.60,2:04:20.91,Default,,0000,0000,0000,,I think it's the same answer. Dialogue: 0,2:04:20.91,2:04:21.49,Default,,0000,0000,0000,,STUDENT: Yeah. Dialogue: 0,2:04:21.49,2:04:21.99,Default,,0000,0000,0000,,I got 1/3. Dialogue: 0,2:04:21.99,2:04:23.06,Default,,0000,0000,0000,,PROFESSOR: You got 1/3. Dialogue: 0,2:04:23.06,2:04:26.07,Default,,0000,0000,0000,,So sounds good. Dialogue: 0,2:04:26.07,2:04:28.14,Default,,0000,0000,0000,,We will stop here. Dialogue: 0,2:04:28.14,2:04:29.73,Default,,0000,0000,0000,,You will get homework. Dialogue: 0,2:04:29.73,2:04:32.56,Default,,0000,0000,0000,,How long should I\Nleave that homework on? Dialogue: 0,2:04:32.56,2:04:35.81,Default,,0000,0000,0000,,Because I'm thinking maybe\Nanother month, but please Dialogue: 0,2:04:35.81,2:04:38.19,Default,,0000,0000,0000,,don't procrastinate. Dialogue: 0,2:04:38.19,2:04:41.43,Default,,0000,0000,0000,,So let's say until\Nthe end of March. Dialogue: 0,2:04:41.43,2:04:44.48,Default,,0000,0000,0000,,And keep in mind\Nthat we have included Dialogue: 0,2:04:44.48,2:04:48.08,Default,,0000,0000,0000,,one week of spring\Nbreak here, which you Dialogue: 0,2:04:48.08,2:04:51.36,Default,,0000,0000,0000,,can do whatever you want with. Dialogue: 0,2:04:51.36,2:04:57.74,Default,,0000,0000,0000,,Some of you may be in Florida\Nswimming and working on a tan, Dialogue: 0,2:04:57.74,2:04:59.24,Default,,0000,0000,0000,,and not working on homework. Dialogue: 0,2:04:59.24,2:05:01.94,Default,,0000,0000,0000,,So no matter how, plan ahead. Dialogue: 0,2:05:01.94,2:05:03.44,Default,,0000,0000,0000,,Plan ahead and you will do well. Dialogue: 0,2:05:03.44,2:05:10.39,Default,,0000,0000,0000,,31st of March for\Nthe whole chapter. Dialogue: 0,2:05:10.39,2:05:11.40,Default,,0000,0000,0000,,