WEBVTT 00:00:00.000 --> 00:00:04.392 PROFESSOR: I have some assignments 00:00:04.392 --> 00:00:06.344 that I want to give you back. 00:00:06.344 --> 00:00:09.510 And I'm just going to put them here, 00:00:09.510 --> 00:00:13.720 and I'll ask you to pick them up as soon as we take a break. 00:00:13.720 --> 00:00:16.850 00:00:16.850 --> 00:00:20.585 There are explanations there how they were computed in red. 00:00:20.585 --> 00:00:23.480 If you have questions, you can as me 00:00:23.480 --> 00:00:26.000 so I can ask my grader about it. 00:00:26.000 --> 00:00:28.950 00:00:28.950 --> 00:00:33.576 Now, I promised you that I would move on today, 00:00:33.576 --> 00:00:35.216 and that's what I'm going to do. 00:00:35.216 --> 00:00:39.480 I'm moving on to something that you're gong to love. 00:00:39.480 --> 00:00:47.844 [? Practically ?] chapter 12 is integration of functions 00:00:47.844 --> 00:00:49.320 of several variables. 00:00:49.320 --> 00:00:58.668 00:00:58.668 --> 00:01:01.320 And to warn you we're going to see 00:01:01.320 --> 00:01:08.880 how we introduce introduction to the double integral. 00:01:08.880 --> 00:01:15.521 00:01:15.521 --> 00:01:17.390 But you will say, wait a minute. 00:01:17.390 --> 00:01:22.480 I don't even know if I remember the simple integral. 00:01:22.480 --> 00:01:24.330 And that's why I'm here. 00:01:24.330 --> 00:01:31.520 I want to remind you what the definite integral was both 00:01:31.520 --> 00:01:35.420 as a formal definition-- let's do it as a formal definition 00:01:35.420 --> 00:01:39.930 first, then come up with a geometric interpretation based 00:01:39.930 --> 00:01:40.470 on that. 00:01:40.470 --> 00:01:45.000 And finally write down the definition 00:01:45.000 --> 00:01:49.380 and the fundamental theorem of calculus. 00:01:49.380 --> 00:01:52.390 So assume you have a function that's continuous. 00:01:52.390 --> 00:01:56.310 00:01:56.310 --> 00:02:05.090 Continuous over a certain integral of a, b interval in R. 00:02:05.090 --> 00:02:08.690 And you know that in that case, you 00:02:08.690 --> 00:02:23.040 can "define the definite integral of f 00:02:23.040 --> 00:02:29.476 of x from or between a and b." 00:02:29.476 --> 00:02:35.300 And as the notation is denoted, by integral from a to b f of x 00:02:35.300 --> 00:02:35.800 dx. 00:02:35.800 --> 00:02:42.490 00:02:42.490 --> 00:02:46.095 Well, how do we define this? 00:02:46.095 --> 00:02:47.340 This is just the notation. 00:02:47.340 --> 00:02:51.140 How do we define it? 00:02:51.140 --> 00:02:58.830 We have to have a set up, and we are thinking of a x, y frame. 00:02:58.830 --> 00:03:02.190 You have a function, f, that's continuous. 00:03:02.190 --> 00:03:05.934 00:03:05.934 --> 00:03:07.750 And you are thinking, oh, wait a minute. 00:03:07.750 --> 00:03:11.090 I would like to be able to evaluate 00:03:11.090 --> 00:03:12.466 the area under the integral. 00:03:12.466 --> 00:03:16.590 00:03:16.590 --> 00:03:19.675 And if you ask your teacher when you are in fourth grade, 00:03:19.675 --> 00:03:22.480 your teacher will say, well, I can give you some graphing 00:03:22.480 --> 00:03:23.550 paper. 00:03:23.550 --> 00:03:25.410 And with that graphing paper, you 00:03:25.410 --> 00:03:35.550 can eventually approximate your area like that. 00:03:35.550 --> 00:03:42.790 Sort of what you get here is like you draw a horizontal 00:03:42.790 --> 00:03:46.520 so that the little part above the horizontal 00:03:46.520 --> 00:03:49.100 cancels out with the little part below the horizontal. 00:03:49.100 --> 00:03:51.490 So more or less, the pink rectangle 00:03:51.490 --> 00:03:56.190 is a good approximation of the first slice. 00:03:56.190 --> 00:03:59.660 But you say yeah, but the first slice is a curvilinear slice. 00:03:59.660 --> 00:04:03.080 Yes, but we make it like a stop function. 00:04:03.080 --> 00:04:06.770 So then you say, OK, how about this fellow? 00:04:06.770 --> 00:04:11.230 I'm going to approximate it in a similar way, 00:04:11.230 --> 00:04:15.031 and I'm going to have a bunch of rectangles on this graphing 00:04:15.031 --> 00:04:15.530 paper. 00:04:15.530 --> 00:04:18.450 And I'm going to compute their areas, 00:04:18.450 --> 00:04:20.760 and I'm going to come up with an approximation, 00:04:20.760 --> 00:04:23.872 and I'll give it to my fourth grade teacher. 00:04:23.872 --> 00:04:26.830 And that's what we did in fourth grade, 00:04:26.830 --> 00:04:29.380 but this is not fourth grade. 00:04:29.380 --> 00:04:32.880 And actually, it's very relevant to us 00:04:32.880 --> 00:04:35.610 that this has applications to our life, 00:04:35.610 --> 00:04:38.630 to our digital world, that people did not 00:04:38.630 --> 00:04:44.490 understand when Riemann introduced the Riemann sum. 00:04:44.490 --> 00:04:49.270 They thought, OK, the idea makes sense that practically we 00:04:49.270 --> 00:04:54.220 have a huge picture here, and I'm 00:04:54.220 --> 00:04:59.430 taking a and b and a function that's continuous over a and b. 00:04:59.430 --> 00:05:02.180 And then I say I'm going to split this 00:05:02.180 --> 00:05:08.200 into a equidistant intervals. 00:05:08.200 --> 00:05:10.926 I don't know how many I want, but let 00:05:10.926 --> 00:05:12.089 me make them eight of them. 00:05:12.089 --> 00:05:12.630 I don't know. 00:05:12.630 --> 00:05:14.310 They have to have the same length. 00:05:14.310 --> 00:05:17.410 And I'll call this delta x. 00:05:17.410 --> 00:05:18.510 It has to be the same. 00:05:18.510 --> 00:05:21.782 And, you guys, please forgive me for the horrible picture. 00:05:21.782 --> 00:05:25.990 They don't look like the same step, delta x, 00:05:25.990 --> 00:05:28.630 but it should be the same. 00:05:28.630 --> 00:05:32.645 In each of them I arbitrarily, say it again, 00:05:32.645 --> 00:05:39.340 Magdalena, arbitrarily pick x1 star, and another point, 00:05:39.340 --> 00:05:44.920 x2 star wherever I want inside. 00:05:44.920 --> 00:05:47.580 I'm just getting [INAUDIBLE]. 00:05:47.580 --> 00:05:51.300 X4 star, and this is x8 star. 00:05:51.300 --> 00:05:54.130 But let's say that in general I don't know they are 8. 00:05:54.130 --> 00:05:56.270 They could be n. 00:05:56.270 --> 00:05:57.180 xn star. 00:05:57.180 --> 00:05:59.890 And passing to the limit with respect 00:05:59.890 --> 00:06:02.670 to n going to infinity, what am I going to get? 00:06:02.670 --> 00:06:06.800 Well, in the first cam I'm going up, 00:06:06.800 --> 00:06:08.860 and I'm hitting at what altitude? 00:06:08.860 --> 00:06:13.150 I'm hitting at the altitude called f of x1 star. 00:06:13.150 --> 00:06:17.240 And that's going to be the height of this-- what is this? 00:06:17.240 --> 00:06:17.820 Strip? 00:06:17.820 --> 00:06:18.350 Right? 00:06:18.350 --> 00:06:21.365 Or rectangle. 00:06:21.365 --> 00:06:21.865 OK. 00:06:21.865 --> 00:06:24.690 And I'm going to do the same with green 00:06:24.690 --> 00:06:26.990 for the second rectangle. 00:06:26.990 --> 00:06:32.166 I'll pick x2 star, and then that doesn't work. 00:06:32.166 --> 00:06:33.098 And I'll take this. 00:06:33.098 --> 00:06:34.962 Let's see if I can do the light green one, 00:06:34.962 --> 00:06:36.360 because spring is here. 00:06:36.360 --> 00:06:37.292 Let's see. 00:06:37.292 --> 00:06:38.880 That's beautiful. 00:06:38.880 --> 00:06:40.660 I go up. 00:06:40.660 --> 00:06:44.975 I hit here at x2 star. 00:06:44.975 --> 00:06:48.292 I get f of x2 star. 00:06:48.292 --> 00:06:50.310 And so on and so forth. 00:06:50.310 --> 00:06:53.476 00:06:53.476 --> 00:06:57.880 Until I get to, let's say, the last of the Mohicans. 00:06:57.880 --> 00:07:00.610 This will be xn minus 1, and this is going 00:07:00.610 --> 00:07:06.190 to be xn star, the purple guy. 00:07:06.190 --> 00:07:07.950 And this is going to be the height 00:07:07.950 --> 00:07:12.280 of that last of the Mohicans. 00:07:12.280 --> 00:07:19.440 So when I compute the sum, I call that approximating sum 00:07:19.440 --> 00:07:23.490 or Riemann approximating sum, because Riemann had nothing 00:07:23.490 --> 00:07:25.880 better to do than invent it. 00:07:25.880 --> 00:07:27.880 He didn't even know that we are going 00:07:27.880 --> 00:07:32.850 to get pixels that are in larger and larger quantities. 00:07:32.850 --> 00:07:36.040 Like, we get 3,000 by 900. 00:07:36.040 --> 00:07:41.440 He didn't know we are going to have all those digital gadgets. 00:07:41.440 --> 00:07:45.645 But passing to the limit practically should 00:07:45.645 --> 00:07:49.330 be easier to understand for teenagers now 00:07:49.330 --> 00:07:53.300 age, because it's like making the number of pixels 00:07:53.300 --> 00:07:57.880 larger and larger, and the pixels practically invisible. 00:07:57.880 --> 00:08:01.710 Remember, I mean, I don't know, those old TVs, 00:08:01.710 --> 00:08:04.240 color TVs where you could still see the squares? 00:08:04.240 --> 00:08:05.154 STUDENT: Mm-hm. 00:08:05.154 --> 00:08:06.070 PROFESSOR: Well, yeah. 00:08:06.070 --> 00:08:07.980 When you were little. 00:08:07.980 --> 00:08:10.850 But I remember them much better than you. 00:08:10.850 --> 00:08:13.970 And, yes, as the number of pixels will increase, 00:08:13.970 --> 00:08:18.190 that means I'm taking the limit and going larger and larger. 00:08:18.190 --> 00:08:20.610 That means practically limitless. 00:08:20.610 --> 00:08:22.720 Infinity will give me an ideal image. 00:08:22.720 --> 00:08:27.125 My eye will be as if I could see the image that's a curvilinear 00:08:27.125 --> 00:08:31.010 image as a real person. 00:08:31.010 --> 00:08:35.039 And, of course, the quality of our movies 00:08:35.039 --> 00:08:36.058 really increased a lot. 00:08:36.058 --> 00:08:41.210 And this is what I'm trying to emphasize here. 00:08:41.210 --> 00:08:47.120 So you have f of x1 star delta x plus the last rectangle 00:08:47.120 --> 00:08:51.580 area, f of xn star delta x. 00:08:51.580 --> 00:08:56.200 Well, as a mathematician, I don't write it like that. 00:08:56.200 --> 00:08:58.770 How do I write it as a mathematician? 00:08:58.770 --> 00:09:00.680 Well, we are funny people. 00:09:00.680 --> 00:09:02.150 We like Greek. 00:09:02.150 --> 00:09:03.050 It's all Greek to me. 00:09:03.050 --> 00:09:16.420 So we go sum and from-- no. k from 1 to n, f of x sub k star. 00:09:16.420 --> 00:09:23.742 So I have k from 1 to n exactly an rectangles area to add. 00:09:23.742 --> 00:09:25.685 And this is going to be [INAUDIBLE], which 00:09:25.685 --> 00:09:28.046 is the same everywhere. 00:09:28.046 --> 00:09:35.730 In that case, I made the partition is equal. 00:09:35.730 --> 00:09:39.190 So practically I have the same distance. 00:09:39.190 --> 00:09:41.100 And what is this limit? [? Lim ?] 00:09:41.100 --> 00:09:45.750 is going to be exactly integral from a to b of f of x dx. 00:09:45.750 --> 00:09:48.825 And I make a smile here, and I say I'm very happy. 00:09:48.825 --> 00:09:55.430 This is as a meaning is the area under the graph. 00:09:55.430 --> 00:09:57.510 If-- well, I didn't say something. 00:09:57.510 --> 00:10:01.390 If I want it to be positive, otherwise it's 00:10:01.390 --> 00:10:04.400 getting not to be the area under the graph. 00:10:04.400 --> 00:10:08.350 The integral will still be defined like that. 00:10:08.350 --> 00:10:12.110 But what's going to happen if I have, for example, half of it 00:10:12.110 --> 00:10:15.380 above and half of it below? 00:10:15.380 --> 00:10:18.180 I'm going to get this, and I'm going to get that. 00:10:18.180 --> 00:10:23.250 And when I add them, I'm going to get a negative answer, 00:10:23.250 --> 00:10:26.500 because this is a negative area, and that's a positive area 00:10:26.500 --> 00:10:28.680 and they try to annihilate each other. 00:10:28.680 --> 00:10:32.010 But this guy under the water is stronger, 00:10:32.010 --> 00:10:35.930 like an iceberg that's 20% on tip of the water, 00:10:35.930 --> 00:10:39.080 80% of the iceberg is under the water. 00:10:39.080 --> 00:10:39.830 So the same thing. 00:10:39.830 --> 00:10:45.400 I'm going to get a negative answer in volume [INAUDIBLE]. 00:10:45.400 --> 00:10:45.980 OK. 00:10:45.980 --> 00:10:49.080 Now, we remember that very well, but now we 00:10:49.080 --> 00:10:54.650 have to generalize this thingy to something else. 00:10:54.650 --> 00:10:57.470 00:10:57.470 --> 00:11:03.207 And I will give you a curvilinear domain. 00:11:03.207 --> 00:11:04.040 Where shall I erase? 00:11:04.040 --> 00:11:07.290 I don't know. 00:11:07.290 --> 00:11:09.390 Here. 00:11:09.390 --> 00:11:12.500 What if somebody gives you the image of a potatoe-- well, 00:11:12.500 --> 00:11:13.380 I don't know. 00:11:13.380 --> 00:11:14.650 Something. 00:11:14.650 --> 00:11:15.690 A blob. 00:11:15.690 --> 00:11:24.860 Some nice curvilinear domain-- and says, you know what? 00:11:24.860 --> 00:11:29.895 I want to approximate the area of this image, curvilinear 00:11:29.895 --> 00:11:35.140 image, to the best of my abilities. 00:11:35.140 --> 00:11:42.370 And compute it, and eventually I have some weighted sum of that. 00:11:42.370 --> 00:11:52.340 So if one would have to compute the area, 00:11:52.340 --> 00:11:55.970 it wouldn't be so hard, because we would say, 00:11:55.970 --> 00:12:05.116 OK, I have to "partition this domain 00:12:05.116 --> 00:12:19.766 into small sections using a rectangular partition 00:12:19.766 --> 00:12:31.450 or square partition." 00:12:31.450 --> 00:12:32.100 And how? 00:12:32.100 --> 00:12:34.805 Well, I'm going to-- you have to imagine 00:12:34.805 --> 00:12:41.192 that I have a bunch of a grid, and I'm 00:12:41.192 --> 00:12:43.180 partitioning the whole thing. 00:12:43.180 --> 00:12:53.630 00:12:53.630 --> 00:12:55.730 And you say, wait a minute. 00:12:55.730 --> 00:12:56.530 Wait a minute. 00:12:56.530 --> 00:12:57.690 It's not so easy. 00:12:57.690 --> 00:13:01.500 I mean, they are not all the same area, Magdalena. 00:13:01.500 --> 00:13:05.840 Even if you tried to make these equidistant in both directions, 00:13:05.840 --> 00:13:07.880 look at this guy. 00:13:07.880 --> 00:13:09.070 Look at that guy. 00:13:09.070 --> 00:13:10.810 He's much bigger than that. 00:13:10.810 --> 00:13:14.080 Look at this small guy, and so on. 00:13:14.080 --> 00:13:26.967 So we have to imagine that we look at the so-called normal 00:13:26.967 --> 00:13:27.550 the partition. 00:13:27.550 --> 00:13:34.070 00:13:34.070 --> 00:13:37.480 And let's say in the normal, or the length of the partition, 00:13:37.480 --> 00:13:38.990 is denoted like that. 00:13:38.990 --> 00:13:41.030 We have to give that a meaning. 00:13:41.030 --> 00:13:51.210 Well, let's say "this is the highest diameter 00:13:51.210 --> 00:14:04.146 for all subdomains in the picture." 00:14:04.146 --> 00:14:06.040 And you say, wait a minute. 00:14:06.040 --> 00:14:08.090 But these subdomains should have names. 00:14:08.090 --> 00:14:11.690 Well, they don't have names, but assume they have areas. 00:14:11.690 --> 00:14:16.385 This would be-- I have to find a way to denote them 00:14:16.385 --> 00:14:18.430 and be orderly. 00:14:18.430 --> 00:14:32.340 A1, A2, A3, A4, A5, AN, AM, AN, stuff like that. 00:14:32.340 --> 00:14:38.440 So practically I'm looking at the highest diameter. 00:14:38.440 --> 00:14:43.290 When I have a domain, I look at the largest instance 00:14:43.290 --> 00:14:44.900 inside that domain. 00:14:44.900 --> 00:14:47.050 So what would be the diameter? 00:14:47.050 --> 00:14:50.110 The largest distance between two points in that domain. 00:14:50.110 --> 00:14:52.265 I'll call that the diameter. 00:14:52.265 --> 00:14:52.765 OK. 00:14:52.765 --> 00:14:57.550 I want that diameter to go got 0 in the limit. 00:14:57.550 --> 00:15:03.360 So I want this partition to go to 0 in the limit. 00:15:03.360 --> 00:15:05.748 And that means I'm "shrinking" the pixels. 00:15:05.748 --> 00:15:08.712 00:15:08.712 --> 00:15:11.182 "Shrinking" in quotes, the pixels. 00:15:11.182 --> 00:15:17.120 00:15:17.120 --> 00:15:20.710 How would I mimic what I did here? 00:15:20.710 --> 00:15:23.435 Well, it would be easier to get the area. 00:15:23.435 --> 00:15:29.410 In this case, I would have some sort of A sum limit. 00:15:29.410 --> 00:15:30.100 I'm sorry. 00:15:30.100 --> 00:15:36.340 The curvilinear area of the domain. 00:15:36.340 --> 00:15:40.570 Let's call it-- what do you want to call it? 00:15:40.570 --> 00:15:45.300 D for domain-- inside the domain. 00:15:45.300 --> 00:15:46.208 OK? 00:15:46.208 --> 00:15:48.476 This whole thing would be what? 00:15:48.476 --> 00:16:01.240 Would be limit of summation of, let's say, limit of what kind? 00:16:01.240 --> 00:16:04.000 k from 1 to n. 00:16:04.000 --> 00:16:06.270 Limit n goes to infinity. 00:16:06.270 --> 00:16:16.270 K from 1 to n of these tiny A sub k's, 00:16:16.270 --> 00:16:17.400 areas of the subdomain. 00:16:17.400 --> 00:16:24.520 00:16:24.520 --> 00:16:25.293 Wait a minute. 00:16:25.293 --> 00:16:29.970 But you say, but what if I want something else? 00:16:29.970 --> 00:16:34.080 Like, I'm going to build some geography. 00:16:34.080 --> 00:16:35.110 This is the domain. 00:16:35.110 --> 00:16:38.560 That's something like on a map, and I'm 00:16:38.560 --> 00:16:40.670 going to build a mountain on top of it. 00:16:40.670 --> 00:16:43.400 I'll take some Play-Do, I'll take some Play-Do, 00:16:43.400 --> 00:16:46.266 and I'm going to model some geography. 00:16:46.266 --> 00:16:47.390 And you say, wait a minute. 00:16:47.390 --> 00:16:49.390 Do you make mountains? 00:16:49.390 --> 00:16:52.150 I'm afraid to make Rocky Mountains, because they 00:16:52.150 --> 00:16:55.850 may have points where the function is not smooth. 00:16:55.850 --> 00:16:58.310 If I don't have derivative at the peak, 00:16:58.310 --> 00:17:01.260 them I'm in trouble, in general. 00:17:01.260 --> 00:17:03.020 Although you say, well, but the function 00:17:03.020 --> 00:17:04.740 has to be only continuous. 00:17:04.740 --> 00:17:05.260 I know. 00:17:05.260 --> 00:17:05.760 I know. 00:17:05.760 --> 00:17:09.980 But I don't want any kind of really nasty singularity 00:17:09.980 --> 00:17:12.300 where I can have a crack in the mountain 00:17:12.300 --> 00:17:15.829 or a well or something like that. 00:17:15.829 --> 00:17:18.589 So I assume the geography to be smooth, 00:17:18.589 --> 00:17:21.420 the function of [INAUDIBLE] is continuous, 00:17:21.420 --> 00:17:23.310 and the picture should look something 00:17:23.310 --> 00:17:27.654 like-- let's see if I can do that. 00:17:27.654 --> 00:17:33.960 00:17:33.960 --> 00:17:38.170 The projection, the shadow of this geography, 00:17:38.170 --> 00:17:43.395 would be the domain, [? D. ?] And this is equal, f of x what? 00:17:43.395 --> 00:17:44.550 You say, what? 00:17:44.550 --> 00:17:46.380 Magdalena, I don't understand. 00:17:46.380 --> 00:17:51.720 The exact shadow of this fellow where I have the sun on top 00:17:51.720 --> 00:17:54.360 here-- that's the sun. 00:17:54.360 --> 00:17:59.330 Spring is coming-- the shade is the plain, or domain, x, y. 00:17:59.330 --> 00:18:03.280 I take all my points in x, y. 00:18:03.280 --> 00:18:05.630 I mean, I take really all my points in x, y, 00:18:05.630 --> 00:18:10.355 and the value of the altitude on this geography at the point 00:18:10.355 --> 00:18:13.676 x, y would be z equals f of x, y. 00:18:13.676 --> 00:18:20.775 And somebody's asking me, OK, if this would be a can of Coke, 00:18:20.775 --> 00:18:23.900 it would be easy to compute the volume, right? 00:18:23.900 --> 00:18:27.680 Practically you have a constant altitude everywhere, 00:18:27.680 --> 00:18:30.150 and you have the area of the base times the height, 00:18:30.150 --> 00:18:32.750 and that's your volume. 00:18:32.750 --> 00:18:39.460 But what if somebody asks you to find the volume under the hat? 00:18:39.460 --> 00:18:47.115 "Find the volume undo this graph." 00:18:47.115 --> 00:18:51.027 STUDENT: I would take it more as two functions. 00:18:51.027 --> 00:18:53.472 So the top line would be the one function, 00:18:53.472 --> 00:18:55.428 and the bottom line would be another function. 00:18:55.428 --> 00:18:58.777 So if you take the volume of the top function minus the volume 00:18:58.777 --> 00:19:00.318 of the bottom function, it'd give you 00:19:00.318 --> 00:19:02.780 the total volume of the object. 00:19:02.780 --> 00:19:05.570 PROFESSOR: And actually, I want the total volume 00:19:05.570 --> 00:19:07.660 above the sea level. 00:19:07.660 --> 00:19:12.750 So I'm going to-- sometimes I can take it up 00:19:12.750 --> 00:19:16.115 to a certain level where-- let's say the mountain is up to here, 00:19:16.115 --> 00:19:18.490 and I want it only up to here. 00:19:18.490 --> 00:19:22.220 So I want everything, including the-- the walls 00:19:22.220 --> 00:19:24.137 would be cylindrical. 00:19:24.137 --> 00:19:24.720 STUDENT: Yeah. 00:19:24.720 --> 00:19:26.140 PROFESSOR: If I want all the volume, 00:19:26.140 --> 00:19:27.764 that's going to be a little bit easier. 00:19:27.764 --> 00:19:29.470 Let's see why. 00:19:29.470 --> 00:19:31.703 I will have limit. 00:19:31.703 --> 00:19:35.631 The idea is, as you said very well, limit. 00:19:35.631 --> 00:19:37.104 n goes to infinity. 00:19:37.104 --> 00:19:42.150 A sum k from 1 to n. 00:19:42.150 --> 00:19:44.880 And what kind of partition can I build? 00:19:44.880 --> 00:19:47.640 I'll take the line, and I'll say, 00:19:47.640 --> 00:19:53.030 I'll build myself a partition with a, 00:19:53.030 --> 00:19:57.510 let's say, the typical domain, AK. 00:19:57.510 --> 00:20:00.960 I have A1, A2 A3, A4, AK, AN. 00:20:00.960 --> 00:20:03.473 How may of those little domains? 00:20:03.473 --> 00:20:04.455 AN. 00:20:04.455 --> 00:20:07.940 That will be all the little subdomains 00:20:07.940 --> 00:20:12.410 inside the green curve. 00:20:12.410 --> 00:20:14.510 The green loop. 00:20:14.510 --> 00:20:16.660 In that case, what do I do? 00:20:16.660 --> 00:20:24.260 For each of these guys, I go up, and I go, oh, my god, this 00:20:24.260 --> 00:20:27.380 looks like a skyscraper, but the corners, 00:20:27.380 --> 00:20:29.085 when I go through this surface, are 00:20:29.085 --> 00:20:30.900 in the different dimensions. 00:20:30.900 --> 00:20:32.430 What am I going to do? 00:20:32.430 --> 00:20:34.760 That forces me to build a skyscraper 00:20:34.760 --> 00:20:38.570 by thinking I take a point in the domain, 00:20:38.570 --> 00:20:45.110 I go up until that hits the surface, pinches the surface, 00:20:45.110 --> 00:20:47.620 and this is the altitude that I'm going 00:20:47.620 --> 00:20:50.280 to select for my skyscraper. 00:20:50.280 --> 00:20:54.490 And here I'm going to have another skyscraper, and here 00:20:54.490 --> 00:20:57.800 another one and another one, so practically it's dense. 00:20:57.800 --> 00:21:02.450 I have a skyscraper next to the other or a less like 00:21:02.450 --> 00:21:02.960 [INAUDIBLE]. 00:21:02.960 --> 00:21:06.550 Not so many gaps in certain areas. 00:21:06.550 --> 00:21:12.640 So I'm going to say f of x kappa star. 00:21:12.640 --> 00:21:18.680 Now those would be the altitudes of the buildings. 00:21:18.680 --> 00:21:20.920 Magdalena, you don't know how to spell. 00:21:20.920 --> 00:21:27.972 Altitudes of the buildings. 00:21:27.972 --> 00:21:31.860 00:21:31.860 --> 00:21:32.695 What are they? 00:21:32.695 --> 00:21:34.055 Parallel [INAUDIBLE] by P's. 00:21:34.055 --> 00:21:36.230 Can you say parallel by P? 00:21:36.230 --> 00:21:37.040 OK. 00:21:37.040 --> 00:21:40.722 [INAUDIBLE] what. 00:21:40.722 --> 00:21:49.720 Ak where Ak will be the basis of the area of the basis. 00:21:49.720 --> 00:21:51.670 is of my building. 00:21:51.670 --> 00:21:54.860 00:21:54.860 --> 00:21:55.490 OK. 00:21:55.490 --> 00:21:59.290 The green part will be the flat area 00:21:59.290 --> 00:22:03.580 of the floor of the skyscraper. 00:22:03.580 --> 00:22:05.600 Is this hard? 00:22:05.600 --> 00:22:06.830 Gosh, yes. 00:22:06.830 --> 00:22:12.713 If you want to do it by hand and take the limit 00:22:12.713 --> 00:22:15.620 you would really kill yourself in the process. 00:22:15.620 --> 00:22:17.420 This is how you introduce it. 00:22:17.420 --> 00:22:21.900 You can prove this limit exists, and you can prove that limits 00:22:21.900 --> 00:22:32.805 exist and will be the volume of the region under the geography 00:22:32.805 --> 00:22:38.760 z equals f of x,y and above the sea level. 00:22:38.760 --> 00:22:43.233 00:22:43.233 --> 00:22:46.779 The seal level meaning z equals z. 00:22:46.779 --> 00:22:49.717 STUDENT: What's under a of k? 00:22:49.717 --> 00:22:50.300 PROFESSOR: Ak. 00:22:50.300 --> 00:22:51.466 STUDENT: What is [INAUDIBLE] 00:22:51.466 --> 00:22:54.078 PROFESSOR: Volume of the region. 00:22:54.078 --> 00:22:55.745 STUDENT: Oh, I know, like what under it? 00:22:55.745 --> 00:22:56.411 PROFESSOR: Here? 00:22:56.411 --> 00:22:57.259 STUDENT: No, up. 00:22:57.259 --> 00:22:58.047 PROFESSOR: Here? 00:22:58.047 --> 00:22:58.588 STUDENT: Yes. 00:22:58.588 --> 00:23:00.605 PROFESSOR: Area of the basis of a building. 00:23:00.605 --> 00:23:01.563 STUDENT: Oh, the basis. 00:23:01.563 --> 00:23:04.070 PROFESSOR: So practically this green thingy 00:23:04.070 --> 00:23:10.730 is a basis like the base rate. 00:23:10.730 --> 00:23:13.400 How large is the basement of that building. 00:23:13.400 --> 00:23:16.240 Ak. 00:23:16.240 --> 00:23:18.470 Now how am I going to write this? 00:23:18.470 --> 00:23:19.400 This is something new. 00:23:19.400 --> 00:23:27.280 We have to invent a notion for it, and since it's Ak, 00:23:27.280 --> 00:23:31.390 looks more or less like a square or a rectangle. 00:23:31.390 --> 00:23:35.120 You think, well, wouldn't-- OK, if it's a rectangle, 00:23:35.120 --> 00:23:38.410 I know I'm going to get delta x and delta y right? 00:23:38.410 --> 00:23:41.535 The width times the height, whatever those two dimensions. 00:23:41.535 --> 00:23:42.400 It makes sense. 00:23:42.400 --> 00:23:44.730 But what if I have this domain that's 00:23:44.730 --> 00:23:47.270 curvilinear or that domain or that domain. 00:23:47.270 --> 00:23:49.690 Of course, the diameter of such a domain 00:23:49.690 --> 00:23:54.000 is less than the diameter of the partition, so I'm very happy. 00:23:54.000 --> 00:23:55.870 The highest diameter, say I can get it here, 00:23:55.870 --> 00:23:59.416 and this is shrinking to zero, and pixels 00:23:59.416 --> 00:24:01.580 are shrinking to zero. 00:24:01.580 --> 00:24:05.410 But what am I going to do about those guys? 00:24:05.410 --> 00:24:09.720 Well, you can assume that I am still approximating 00:24:09.720 --> 00:24:14.750 with some squares and as the pixels are getting 00:24:14.750 --> 00:24:17.315 to be many, many, many more, it doesn't 00:24:17.315 --> 00:24:19.520 matter that I'm doing this. 00:24:19.520 --> 00:24:21.797 Let me show you what I'm doing. 00:24:21.797 --> 00:24:29.070 So on the floor, on the-- this is the city floor, whatever. 00:24:29.070 --> 00:24:32.180 What we do in practice, we approximate that 00:24:32.180 --> 00:24:42.270 like on the graphing paper with tiny square domains, 00:24:42.270 --> 00:24:48.800 and we call them delta Ak will be delta Sk times delta Yk, 00:24:48.800 --> 00:24:53.640 and I tried to make it a uniform partition as much as I can. 00:24:53.640 --> 00:24:56.230 Now as the number of pixels goes to infinity 00:24:56.230 --> 00:24:59.360 and those pixels will become smaller and smaller, 00:24:59.360 --> 00:25:04.240 it doesn't there that the actual contour of your Riemann sum 00:25:04.240 --> 00:25:07.090 will look like graphing paper. 00:25:07.090 --> 00:25:10.410 It will get refined, more refined, more refined, smoother 00:25:10.410 --> 00:25:12.710 and smoother, and it's going to be 00:25:12.710 --> 00:25:17.890 really close to the ideal image, which is a curve. 00:25:17.890 --> 00:25:20.342 So as that end goes to infinity, you're 00:25:20.342 --> 00:25:24.940 not going to see this-- what is this called-- zig zag thingy. 00:25:24.940 --> 00:25:25.810 Not anymore. 00:25:25.810 --> 00:25:31.950 The zig zag thingy will go into the limit to the green curve. 00:25:31.950 --> 00:25:34.566 This is what the pixels are about. 00:25:34.566 --> 00:25:38.300 This is how our life changed a lot. 00:25:38.300 --> 00:25:39.070 OK? 00:25:39.070 --> 00:25:39.840 All right. 00:25:39.840 --> 00:25:41.850 Now good. 00:25:41.850 --> 00:25:45.020 How am I going compute this thing? 00:25:45.020 --> 00:25:47.730 00:25:47.730 --> 00:25:52.430 Well, I don't know, but let me give it a name first. 00:25:52.430 --> 00:25:56.130 It's going to be double integral over-- what 00:25:56.130 --> 00:25:58.960 do want the floor to be called? 00:25:58.960 --> 00:26:02.271 00:26:02.271 --> 00:26:04.440 We called d domain before. 00:26:04.440 --> 00:26:06.460 What should I call this? 00:26:06.460 --> 00:26:09.870 Big D. Not round. 00:26:09.870 --> 00:26:13.870 Over D. That's the floor, the foundation 00:26:13.870 --> 00:26:16.655 of the whole city-- of the whole area of the city 00:26:16.655 --> 00:26:18.050 that I'm looking at. 00:26:18.050 --> 00:26:28.190 Then I have f of xy, da, and what is this? 00:26:28.190 --> 00:26:29.640 This is exactly that. 00:26:29.640 --> 00:26:35.370 It's the limit of sum of the-- what is the difference here? 00:26:35.370 --> 00:26:37.040 You say, wait a minute, Magdalena, 00:26:37.040 --> 00:26:40.160 but I think I don't understand what you did. 00:26:40.160 --> 00:26:43.930 You tried to copy the concept from here, 00:26:43.930 --> 00:26:47.630 but you forgot you have a function of two variables. 00:26:47.630 --> 00:26:52.560 In that case, this mister, whoever it is that goes up 00:26:52.560 --> 00:26:57.820 is not xk, it's XkYk. 00:26:57.820 --> 00:27:02.060 So I have two variables-- doesn't change anything 00:27:02.060 --> 00:27:03.880 for the couple. 00:27:03.880 --> 00:27:08.190 This couple represents a point on the skyscraper 00:27:08.190 --> 00:27:16.320 so that when I go up, I hit the roof with this exact altitude. 00:27:16.320 --> 00:27:19.670 So what is the double integral of a continuous function 00:27:19.670 --> 00:27:26.480 f of x and y, two variables, with respect to area level. 00:27:26.480 --> 00:27:32.780 Well, it's going to be just the limit of this huge thing. 00:27:32.780 --> 00:27:37.670 In fact, it's how do we compute it? 00:27:37.670 --> 00:27:40.580 Let's see how we compute it in practice. 00:27:40.580 --> 00:27:42.821 It shouldn't be a big deal. 00:27:42.821 --> 00:27:56.078 00:27:56.078 --> 00:27:57.970 What if I have a rectangular domain, 00:27:57.970 --> 00:28:00.855 and that's going to make my life easier. 00:28:00.855 --> 00:28:05.840 I'm going to have a rectangular domain in plane, 00:28:05.840 --> 00:28:07.880 and which one is the x-axis? 00:28:07.880 --> 00:28:09.550 This one. 00:28:09.550 --> 00:28:14.750 From A to B, I have the x moving between a and Mr. y 00:28:14.750 --> 00:28:19.550 says, I'm going to be between c and d. 00:28:19.550 --> 00:28:23.180 C is here, and d is here. 00:28:23.180 --> 00:28:28.480 So this is going to be the so-called rectangle 00:28:28.480 --> 00:28:37.150 a, b cross c, d meaning the set of all the pairs-- 00:28:37.150 --> 00:28:42.020 or the couples xy-- inside it, what does it mean? 00:28:42.020 --> 00:28:45.560 x, y you playing with the property there. 00:28:45.560 --> 00:28:48.510 X is between a and b, thank god. 00:28:48.510 --> 00:28:50.210 It's easy. 00:28:50.210 --> 00:28:53.530 And y must be between c and d, also easy. 00:28:53.530 --> 00:28:57.885 A, b, c, d are fixed real numbers in this order. 00:28:57.885 --> 00:29:02.054 A is less than b, and c is less. 00:29:02.054 --> 00:29:05.030 And we have this geography on top, 00:29:05.030 --> 00:29:09.080 and I will tell you what it looks like. 00:29:09.080 --> 00:29:13.720 I'm going to try and draw some beautiful geography. 00:29:13.720 --> 00:29:19.620 And now I'm thinking of my son, who is 10. 00:29:19.620 --> 00:29:23.590 He played with this kind of toy that was exactly this color, 00:29:23.590 --> 00:29:25.795 lime, and it had needles. 00:29:25.795 --> 00:29:27.965 Do you guys remember that toy? 00:29:27.965 --> 00:29:30.910 I am sure you're young enough to remember that. 00:29:30.910 --> 00:29:35.330 You have your palm like that, and you see this square thingy, 00:29:35.330 --> 00:29:37.430 and it's all made of needles that 00:29:37.430 --> 00:29:40.810 look like thin, tiny skyscrapers, 00:29:40.810 --> 00:29:47.400 and you push through and all those needles go up and take 00:29:47.400 --> 00:29:49.660 the shape of your hand. 00:29:49.660 --> 00:29:51.925 And of course, he would put it on his face, 00:29:51.925 --> 00:29:54.350 and you could see his face and so on. 00:29:54.350 --> 00:29:56.100 But what is that? 00:29:56.100 --> 00:29:59.890 That's exactly the Riemann sum, the Riemann approximation, 00:29:59.890 --> 00:30:02.200 because if you think of all those needles or tiny-- 00:30:02.200 --> 00:30:07.300 what are they, like the tiny skyscrapers-- 00:30:07.300 --> 00:30:11.610 the sum of the them approximates the curvilinear shape. 00:30:11.610 --> 00:30:15.890 If you put that over your face, your face is nice and smooth, 00:30:15.890 --> 00:30:19.190 curvilinear except for a few single areas, 00:30:19.190 --> 00:30:24.600 but if you actually look at that needle 00:30:24.600 --> 00:30:27.640 thingy that is giving the figure, 00:30:27.640 --> 00:30:30.060 you recognize the figure. 00:30:30.060 --> 00:30:33.560 It's like a pattern recognition, but it's not your face. 00:30:33.560 --> 00:30:34.910 I mean it is and it's not. 00:30:34.910 --> 00:30:39.220 It's an approximation of your face, a very rough face. 00:30:39.220 --> 00:30:42.140 You have to take that rough model of your face 00:30:42.140 --> 00:30:43.850 and smooth it out. 00:30:43.850 --> 00:30:44.450 How? 00:30:44.450 --> 00:30:49.930 By passing to the limit, and this is what 00:30:49.930 --> 00:30:52.240 animation is doing actually. 00:30:52.240 --> 00:30:55.915 On top of that you want this to have some other properties-- 00:30:55.915 --> 00:31:01.250 illumination of some sort-- light coming from what angle. 00:31:01.250 --> 00:31:04.910 That is all rendering techniques are actually 00:31:04.910 --> 00:31:06.810 applied mathematics. 00:31:06.810 --> 00:31:09.550 In animation, the people who programmed 00:31:09.550 --> 00:31:12.940 Toy Story-- that was a long time ago, 00:31:12.940 --> 00:31:16.760 but everything that came after Toy Story 2 00:31:16.760 --> 00:31:20.550 was based on mathematical rendering techniques. 00:31:20.550 --> 00:31:23.825 Everything based on the notion of length. 00:31:23.825 --> 00:31:25.078 All right. 00:31:25.078 --> 00:31:28.012 So the way we compute this in practice 00:31:28.012 --> 00:31:31.081 is going to be very simple, because you're going to think, 00:31:31.081 --> 00:31:33.330 how am I going to do the rectangle for the rectangles? 00:31:33.330 --> 00:31:35.800 That'll be very easy. 00:31:35.800 --> 00:31:43.570 I split the rectangle perfectly into other tiny rectangle. 00:31:43.570 --> 00:31:46.890 Every rectangle will have the same dimension. 00:31:46.890 --> 00:31:48.670 Delta x and delta y. 00:31:48.670 --> 00:31:51.515 00:31:51.515 --> 00:31:53.040 Does it makes sense? 00:31:53.040 --> 00:31:55.920 So practically when I go to the limit, 00:31:55.920 --> 00:32:02.220 I have summation f of xk star, yk star 00:32:02.220 --> 00:32:06.840 inside the delta x delta y delta Magdalena, 00:32:06.840 --> 00:32:11.530 the same kind of displacement when I take k from 1 to n, 00:32:11.530 --> 00:32:16.170 and I pass to the limit according to the partition, 00:32:16.170 --> 00:32:17.840 what's going to happen? 00:32:17.840 --> 00:32:21.240 These guys, according to Mr. Linux, 00:32:21.240 --> 00:32:26.120 will go to be infinitesimal elements, dx, dy. 00:32:26.120 --> 00:32:29.530 This whole thing will go to double integral 00:32:29.530 --> 00:32:36.660 of f of x and y, and Mr. y says, OK 00:32:36.660 --> 00:32:39.330 it's like you want him to integrate him one at a time. 00:32:39.330 --> 00:32:43.612 This is actually something that we are going to see in a second 00:32:43.612 --> 00:32:45.030 and verify it. 00:32:45.030 --> 00:32:50.420 X goes between a and b, and y goes between c and d, 00:32:50.420 --> 00:32:55.008 and this is an application of a big theorem called 00:32:55.008 --> 00:33:01.450 Fubini's Theorem that says, wait a minute, 00:33:01.450 --> 00:33:06.830 if you do it like this over a rectangle a,b cross c,d, 00:33:06.830 --> 00:33:12.360 you're double integral can be written as three things. 00:33:12.360 --> 00:33:18.210 Double integral over your square domain f of x,y dA, 00:33:18.210 --> 00:33:22.140 or you integral from c to d, integral from a to b, 00:33:22.140 --> 00:33:28.170 f of x,y dx dy, or you can also swap the order, 00:33:28.170 --> 00:33:32.142 because you say, well, you can do the integration with respect 00:33:32.142 --> 00:33:34.770 to y first. 00:33:34.770 --> 00:33:36.800 Nobody stops you from doing that, 00:33:36.800 --> 00:33:40.832 and y has to be between what and what? 00:33:40.832 --> 00:33:41.540 STUDENT: C and d. 00:33:41.540 --> 00:33:42.915 PROFESSOR: C and d, thank you. 00:33:42.915 --> 00:33:47.190 And then whatever you get, you get to integrate that 00:33:47.190 --> 00:33:52.170 with respect to x from a to b. 00:33:52.170 --> 00:33:56.280 So no matter in what order you do it, 00:33:56.280 --> 00:33:59.200 you'll get the same thing. 00:33:59.200 --> 00:34:02.990 Let's see an easy example, and you'll say, well, 00:34:02.990 --> 00:34:06.320 start with some [INAUDIBLE] example, Magdalena, 00:34:06.320 --> 00:34:08.610 because we are just starting, and that's 00:34:08.610 --> 00:34:10.090 exactly what I'm going to. 00:34:10.090 --> 00:34:11.639 I will just misbehave. 00:34:11.639 --> 00:34:14.630 I'm not going to go by the book. 00:34:14.630 --> 00:34:19.409 And I will say I'm going by whatever I want to go. 00:34:19.409 --> 00:34:27.590 X is between 0, 2, and y is between 0 and 2 00:34:27.590 --> 00:34:32.969 and 3-- this is 2, this is 3-- and my domain 00:34:32.969 --> 00:34:38.630 will be the rectangle 0, 2 times 0, 3. 00:34:38.630 --> 00:34:42.161 This is neat on the floor. 00:34:42.161 --> 00:34:59.940 Compute the volume of the box of basis d and height 5. 00:34:59.940 --> 00:35:02.070 Can I draw that? 00:35:02.070 --> 00:35:03.610 It gets out of the picture. 00:35:03.610 --> 00:35:04.570 I'm just kidding. 00:35:04.570 --> 00:35:07.830 This is 5, and that's sort of the box. 00:35:07.830 --> 00:35:10.690 00:35:10.690 --> 00:35:13.590 And you say, wait a minute, I know that from third grade-- 00:35:13.590 --> 00:35:16.040 I mean, first grade, whenever. 00:35:16.040 --> 00:35:17.230 How do we do that? 00:35:17.230 --> 00:35:21.340 We go 2 units times 3 units that's 00:35:21.340 --> 00:35:25.460 going to be 6 square inches on the bottom of the box, 00:35:25.460 --> 00:35:27.586 and then times 5. 00:35:27.586 --> 00:35:31.437 So the volume has to be 2 times 3 times 5, which 00:35:31.437 --> 00:35:35.490 is 30 square inches. 00:35:35.490 --> 00:35:37.090 I don't care what it is. 00:35:37.090 --> 00:35:39.370 I'm a mathematician, right? 00:35:39.370 --> 00:35:39.870 OK. 00:35:39.870 --> 00:35:44.000 How does somebody who just learned Tonelli's-- Fubini 00:35:44.000 --> 00:35:46.964 Tonelli's Theorem do the problem. 00:35:46.964 --> 00:35:49.180 That person will say, wait a minute, 00:35:49.180 --> 00:35:54.800 now I know that the function is going to be z 00:35:54.800 --> 00:36:00.430 equals f of xy, which in this case happens to be cost. 00:36:00.430 --> 00:36:05.020 According to what you told us, the theorem you claim Magdalena 00:36:05.020 --> 00:36:07.130 proved to this theorem, but there is a sketch 00:36:07.130 --> 00:36:08.954 of the proof in the book. 00:36:08.954 --> 00:36:13.360 According to this, the double integral 00:36:13.360 --> 00:36:21.400 that you have over the domain d, and this is dA. 00:36:21.400 --> 00:36:30.530 DA will be called element of area, which is also dx dy. 00:36:30.530 --> 00:36:34.510 This can be solved in two different ways. 00:36:34.510 --> 00:36:38.285 You take integral from-- where is x going? 00:36:38.285 --> 00:36:41.970 Do we want to do it first in x or in y? 00:36:41.970 --> 00:36:44.925 If we put dy dx, that means we integrate with respect 00:36:44.925 --> 00:36:49.250 to y first, and y goes between 0 and 3, 00:36:49.250 --> 00:36:53.200 so I have to pay attention to the limits of integration. 00:36:53.200 --> 00:36:56.290 And then x between 0 and 2 and again 00:36:56.290 --> 00:36:58.850 I have to pay attention to the limits of integration 00:36:58.850 --> 00:37:03.121 all the time and, here, who is my f? 00:37:03.121 --> 00:37:06.670 Is the altitude 5 that's constant in my case? 00:37:06.670 --> 00:37:08.500 I'm not worried about it. 00:37:08.500 --> 00:37:10.640 Let me see if I get 30? 00:37:10.640 --> 00:37:16.490 I'm just checking if this theorem was true or is just 00:37:16.490 --> 00:37:20.981 something that you cannot apply. 00:37:20.981 --> 00:37:25.618 How do you integrate 5 with respect to y? 00:37:25.618 --> 00:37:26.479 STUDENT: 5y. 00:37:26.479 --> 00:37:27.520 PROFESSOR: 5y, very good. 00:37:27.520 --> 00:37:34.120 So it's going to be 5y between y equals 0 down and y equals 3 00:37:34.120 --> 00:37:38.745 up, and how much is that 5y, we're 00:37:38.745 --> 00:37:41.440 doing y equals 0 down and y equals 3 up, 00:37:41.440 --> 00:37:42.910 what number is that? 00:37:42.910 --> 00:37:44.870 STUDENT: 25. 00:37:44.870 --> 00:37:45.850 PROFESSOR: What? 00:37:45.850 --> 00:37:46.830 STUDENT: 25. 00:37:46.830 --> 00:37:47.810 PROFESSOR: 25? 00:37:47.810 --> 00:37:49.770 STUDENT: One [INAUDIBLE] 15. 00:37:49.770 --> 00:37:54.200 PROFESSOR: No, you did-- you are thinking ahead. 00:37:54.200 --> 00:38:00.010 So I go 5 times 3 minus 5 times 0 equals 15. 00:38:00.010 --> 00:38:04.110 So when I compute this variation of 5y between y 00:38:04.110 --> 00:38:06.816 equals 3 and y equals 0, I just block in 00:38:06.816 --> 00:38:08.260 and make the difference. 00:38:08.260 --> 00:38:09.730 Why do I do that? 00:38:09.730 --> 00:38:15.790 It's the simplest application of that FT, fundamental theorem. 00:38:15.790 --> 00:38:19.480 The one that I did not specify in [INAUDIBLE]. 00:38:19.480 --> 00:38:23.490 I should have specified when I have a g function that 00:38:23.490 --> 00:38:28.190 is continuous between alpha and beta, how do we 00:38:28.190 --> 00:38:30.300 integrate with respect to x? 00:38:30.300 --> 00:38:33.485 I get the antiderivative of rule G. Let's call 00:38:33.485 --> 00:38:37.140 that big G. Compute it at the end points, 00:38:37.140 --> 00:38:39.190 and I make the difference. 00:38:39.190 --> 00:38:41.789 So I compute the antiderivative at an endpoint-- 00:38:41.789 --> 00:38:44.372 at the other endpoint-- then I'm going to make the difference. 00:38:44.372 --> 00:38:49.700 That's the same thing I do here, so 5 times 3 is 15, 00:38:49.700 --> 00:38:54.410 5 times 0 is 0, 15 minus 0 is 15. 00:38:54.410 --> 00:38:56.030 I can keep moving. 00:38:56.030 --> 00:38:59.100 Everything in the parentheses is the number 15. 00:38:59.100 --> 00:39:03.518 I copy and paste, and that should be a piece of cake. 00:39:03.518 --> 00:39:07.011 What do I get? 00:39:07.011 --> 00:39:09.510 STUDENT: 15. 00:39:09.510 --> 00:39:15.560 PROFESSOR: I get 15 times x between 0 and 2. 00:39:15.560 --> 00:39:16.910 Integral of 1 is x. 00:39:16.910 --> 00:39:19.780 Integral of 1 is x with respect to x, 00:39:19.780 --> 00:39:24.200 so I get 15 times 2, which is 30, and you go, duh, 00:39:24.200 --> 00:39:26.830 [INAUDIBLE]. 00:39:26.830 --> 00:39:28.780 That was elementary mathematics. 00:39:28.780 --> 00:39:32.050 Yes, you were lucky you knew that volume of the box, 00:39:32.050 --> 00:39:35.730 but what if somebody gave you a curvilinear area? 00:39:35.730 --> 00:39:39.360 What if somebody gave you something quite complicated? 00:39:39.360 --> 00:39:40.700 What would you do? 00:39:40.700 --> 00:39:43.453 You have know calculus. 00:39:43.453 --> 00:39:45.918 That's your only chance. 00:39:45.918 --> 00:39:51.720 If you don't calculus, you are dead meat. 00:39:51.720 --> 00:40:00.680 So I'm saying, how about another problem. 00:40:00.680 --> 00:40:04.500 That look like it's complicated, but calculus 00:40:04.500 --> 00:40:08.460 is something [INAUDIBLE] with that. 00:40:08.460 --> 00:40:15.755 Suppose that I have a square in the plane between-- this 00:40:15.755 --> 00:40:19.910 is x and y-- do you want square 0,1 0,1 00:40:19.910 --> 00:40:22.630 or you want minus 1 to 1 minus 1 to 1. 00:40:22.630 --> 00:40:26.160 00:40:26.160 --> 00:40:28.080 It doesn't matter. 00:40:28.080 --> 00:40:32.970 Well, let's take minus 1 to 1 and minus 1 to 1, 00:40:32.970 --> 00:40:35.990 and I'll try to draw as well as I can, 00:40:35.990 --> 00:40:38.182 which I cannot but it's OK. 00:40:38.182 --> 00:40:41.560 You will forgive me. 00:40:41.560 --> 00:40:42.430 This is the floor. 00:40:42.430 --> 00:40:45.615 00:40:45.615 --> 00:40:48.210 If I were just a little tiny square 00:40:48.210 --> 00:40:52.300 in this room plus the equivalent square in that room 00:40:52.300 --> 00:40:53.790 and that room and that room. 00:40:53.790 --> 00:40:56.340 This is the origin. 00:40:56.340 --> 00:40:57.650 Are you guys with me? 00:40:57.650 --> 00:41:00.030 So what you're looking at right now 00:41:00.030 --> 00:41:05.418 is this square foot of carpet that I have, 00:41:05.418 --> 00:41:11.905 but I have another one here and another one behind the wall, 00:41:11.905 --> 00:41:15.590 and so do I everything in mind? 00:41:15.590 --> 00:41:21.130 X is between minus 1 and 1, y is between minus 1 and 1. 00:41:21.130 --> 00:41:24.518 00:41:24.518 --> 00:41:30.350 And somebody gives you z to be a positive function, 00:41:30.350 --> 00:41:36.076 continuous function, which is x squared plus y squared. 00:41:36.076 --> 00:41:37.415 And you go, already. 00:41:37.415 --> 00:41:39.290 Oh, my god. 00:41:39.290 --> 00:41:41.950 I already have this kind of hard function. 00:41:41.950 --> 00:41:44.350 It's not a hard thing to do. 00:41:44.350 --> 00:41:45.380 Let's draw that. 00:41:45.380 --> 00:41:47.990 What are we going to get? 00:41:47.990 --> 00:41:55.980 Your favorite [INAUDIBLE] that goes like this. 00:41:55.980 --> 00:41:59.780 And imagine what's going to happen 00:41:59.780 --> 00:42:03.400 with this is like a vase. 00:42:03.400 --> 00:42:06.850 Inside, it has this circular paraboloid. 00:42:06.850 --> 00:42:17.760 But the walls of this vase are-- I cannot draw better than that. 00:42:17.760 --> 00:42:25.310 So the walls of this vase are squares. 00:42:25.310 --> 00:42:30.040 And what you have inside is the carved circular paraboloid. 00:42:30.040 --> 00:42:32.580 00:42:32.580 --> 00:42:45.330 Now I'm asking you, how do I find 00:42:45.330 --> 00:43:02.460 volume of the body under and above D, which is minus 1, 00:43:02.460 --> 00:43:03.510 1, minus 1, 1. 00:43:03.510 --> 00:43:05.410 It's hard to draw that, right? 00:43:05.410 --> 00:43:06.970 It's hard to draw. 00:43:06.970 --> 00:43:09.810 So what do we do? 00:43:09.810 --> 00:43:15.507 00:43:15.507 --> 00:43:16.590 We start imagining things. 00:43:16.590 --> 00:43:20.060 00:43:20.060 --> 00:43:24.110 Actually, when you cut with a plane that is y equals 1, 00:43:24.110 --> 00:43:27.870 you would get a parabola. 00:43:27.870 --> 00:43:35.880 And so when you look at what the picture is going to look like, 00:43:35.880 --> 00:43:39.580 you're going to have a parabola like this, 00:43:39.580 --> 00:43:41.790 a parabola like that, exactly the same, 00:43:41.790 --> 00:43:45.990 a parallel parabola like this and a parabola like that. 00:43:45.990 --> 00:43:49.270 Now I started drawing better. 00:43:49.270 --> 00:43:51.581 And you say, how did you start drawing better? 00:43:51.581 --> 00:43:53.370 Well, with a little bit of practice. 00:43:53.370 --> 00:43:59.560 Where are the maxima of this thing? 00:43:59.560 --> 00:44:00.510 At the corners. 00:44:00.510 --> 00:44:01.200 Why is that? 00:44:01.200 --> 00:44:05.650 Because at the corners, you get 1, 1 for both. 00:44:05.650 --> 00:44:10.270 Of course, to do the absolute extrema, minimum, maximum, 00:44:10.270 --> 00:44:14.710 we would have to go back to section 11.7 and do the thing. 00:44:14.710 --> 00:44:19.130 But practically, it's easy to see that at the corners, 00:44:19.130 --> 00:44:23.050 you have the height 2 because this is the point 1, 1. 00:44:23.050 --> 00:44:28.930 And the same height, 2 and 2 and 2, are at every corner. 00:44:28.930 --> 00:44:31.870 That would be the maximum that you have. 00:44:31.870 --> 00:44:39.340 So you have 1 minus 1 and so on-- minus 1, 1, and minus 1, 00:44:39.340 --> 00:44:42.389 minus 1, who is behind me, minus 1, minus 1. 00:44:42.389 --> 00:44:46.650 That goes all the way to 2. 00:44:46.650 --> 00:44:51.240 So it's hard to do an approximation with a three 00:44:51.240 --> 00:44:53.090 dimensional model. 00:44:53.090 --> 00:44:54.560 Thank god there is calculus. 00:44:54.560 --> 00:44:59.110 So you say integral of x squared plus y squared, 00:44:59.110 --> 00:45:05.930 as simple as that, da over the domain, D, which is minus 1, 00:45:05.930 --> 00:45:07.820 1, minus 1, 1. 00:45:07.820 --> 00:45:10.370 How do you write it according to the theorem 00:45:10.370 --> 00:45:13.390 that I told you about, Fubini-Tonelli? 00:45:13.390 --> 00:45:19.640 Then you have integral integral x squared plus y squared dy dx. 00:45:19.640 --> 00:45:22.430 00:45:22.430 --> 00:45:25.210 Doesn't matter which one I'm taking. 00:45:25.210 --> 00:45:26.620 I can do dy dx. 00:45:26.620 --> 00:45:27.860 I can do dx dy. 00:45:27.860 --> 00:45:31.220 I just have to pay attention to the endpoints. 00:45:31.220 --> 00:45:33.360 Lucky for you the endpoints are the same. 00:45:33.360 --> 00:45:35.460 y is between minus 1 and 1. 00:45:35.460 --> 00:45:37.443 x is between minus 1 and 1. 00:45:37.443 --> 00:45:40.754 00:45:40.754 --> 00:45:44.720 I wouldn't known how to compute the volume of this vase made 00:45:44.720 --> 00:45:46.094 of marble or made of whatever you 00:45:46.094 --> 00:45:53.835 want to make it unless I knew to compute this integral. 00:45:53.835 --> 00:45:58.582 Now you have to help me because it's not hard 00:45:58.582 --> 00:46:03.902 but it's not easy either, so we need a little bit of attention. 00:46:03.902 --> 00:46:05.860 We always start from the inside to the outside. 00:46:05.860 --> 00:46:10.560 The outer person has to be just neglected for the time being 00:46:10.560 --> 00:46:14.580 and I focus all my attention to this integration. 00:46:14.580 --> 00:46:18.055 And when I integrate with respect to y, 00:46:18.055 --> 00:46:20.450 y is the variable for me. 00:46:20.450 --> 00:46:22.800 Nothing else exists for the time being, 00:46:22.800 --> 00:46:27.540 but y being a variable, x being like a constant. 00:46:27.540 --> 00:46:29.990 So when you integrate x squared plus y squared 00:46:29.990 --> 00:46:34.510 with respect to y, you have to pay attention a little bit. 00:46:34.510 --> 00:46:39.790 It's about the same if you had 7 squared plus y squared. 00:46:39.790 --> 00:46:43.410 So this x squared is like a constant. 00:46:43.410 --> 00:46:45.363 So what do you get inside? 00:46:45.363 --> 00:46:47.362 Let's apply the fundamental theorem of calculus. 00:46:47.362 --> 00:46:48.292 STUDENT: x squared y. 00:46:48.292 --> 00:46:49.250 PROFESSOR: x squared y. 00:46:49.250 --> 00:46:49.960 Excellent. 00:46:49.960 --> 00:46:51.960 I'm very proud of you. 00:46:51.960 --> 00:46:52.670 Plus? 00:46:52.670 --> 00:46:53.800 STUDENT: y cubed over 3. 00:46:53.800 --> 00:46:55.080 PROFESSOR: y cubed over three. 00:46:55.080 --> 00:46:57.460 Again, I'm proud of you. 00:46:57.460 --> 00:47:03.310 Evaluated between y equals minus 1 down, y equals 1 up. 00:47:03.310 --> 00:47:07.020 And I will do the math later because I'm getting tired. 00:47:07.020 --> 00:47:09.730 00:47:09.730 --> 00:47:11.790 Now let's do the math. 00:47:11.790 --> 00:47:13.275 I don't know what I'm going to get. 00:47:13.275 --> 00:47:18.930 I get minus 1 to 1, a big bracket, and dx. 00:47:18.930 --> 00:47:21.930 And in this big bracket, I have to do the difference 00:47:21.930 --> 00:47:23.270 between two values. 00:47:23.270 --> 00:47:26.920 So I put two parentheses. 00:47:26.920 --> 00:47:29.960 When y equals 1, I get x squared 1-- 00:47:29.960 --> 00:47:33.800 I'm not going to write that down-- plus 1 cubed 00:47:33.800 --> 00:47:36.510 over 3, 1/3. 00:47:36.510 --> 00:47:41.920 I'm done with evaluating this sausage thingy at 1. 00:47:41.920 --> 00:47:44.310 It's an expression that I evaluate. 00:47:44.310 --> 00:47:46.580 It could be a lot longer. 00:47:46.580 --> 00:47:49.130 I'm not planning to give you long expressions in the midterm 00:47:49.130 --> 00:47:51.936 because you're going to make algebra mistakes, 00:47:51.936 --> 00:47:55.270 and that's not what I want. 00:47:55.270 --> 00:48:01.150 For minus 1, what do we have Minus x squared. 00:48:01.150 --> 00:48:04.500 What is y equals minus 1 plugged in here? 00:48:04.500 --> 00:48:05.455 Minus 1/3. 00:48:05.455 --> 00:48:09.092 00:48:09.092 --> 00:48:10.520 I have to pay attention. 00:48:10.520 --> 00:48:15.650 You realize that if I mess up a sign, it's all done. 00:48:15.650 --> 00:48:21.220 So in this case, I say, but this I have minus, minus. 00:48:21.220 --> 00:48:24.182 A minus in front of a minus is a plus, 00:48:24.182 --> 00:48:30.870 so I'm practically doubling the x squared plus 1/3 00:48:30.870 --> 00:48:33.930 and taking it between minus 1 and 1 00:48:33.930 --> 00:48:36.680 and just with respect to x. 00:48:36.680 --> 00:48:38.160 So you say, wait a minute. 00:48:38.160 --> 00:48:38.970 But that's easy. 00:48:38.970 --> 00:48:41.120 I've done that when I was in Calc 1. 00:48:41.120 --> 00:48:41.860 Of course. 00:48:41.860 --> 00:48:47.035 This is the nice part that you get, a simple integral 00:48:47.035 --> 00:48:51.770 from the ones in Calc 1. 00:48:51.770 --> 00:48:56.530 Let's solve this one and find out what the area will be. 00:48:56.530 --> 00:48:59.168 What do we get? 00:48:59.168 --> 00:48:59.980 Is it hard? 00:48:59.980 --> 00:49:00.920 No. 00:49:00.920 --> 00:49:02.150 Kick Mr. 2 out. 00:49:02.150 --> 00:49:04.970 He's just messing up with your life. 00:49:04.970 --> 00:49:06.150 Kick him out. 00:49:06.150 --> 00:49:08.500 2, out. 00:49:08.500 --> 00:49:11.900 And then integral of x squared plus 1/3 00:49:11.900 --> 00:49:15.720 is going to be x cubed over 3 plus-- 00:49:15.720 --> 00:49:16.686 STUDENT: x over 3. 00:49:16.686 --> 00:49:18.618 PROFESSOR: x over 3, very good. 00:49:18.618 --> 00:49:22.965 Evaluated between x equals minus 1 down, x equals 1 up. 00:49:22.965 --> 00:49:26.360 00:49:26.360 --> 00:49:27.729 Let's see what we get. 00:49:27.729 --> 00:49:30.942 2 times bracket. 00:49:30.942 --> 00:49:33.565 I'll put a parentheses for the first fractions, 00:49:33.565 --> 00:49:36.980 and another minus, and another parentheses. 00:49:36.980 --> 00:49:42.070 What's the first edition of fractions that I get? 00:49:42.070 --> 00:49:44.440 1/3 plus 1/3. 00:49:44.440 --> 00:49:47.360 I'll put 2/3 because I'm lazy. 00:49:47.360 --> 00:49:49.218 Then minus what? 00:49:49.218 --> 00:49:51.310 STUDENT: Minus 1/3. 00:49:51.310 --> 00:49:56.370 PROFESSOR: Minus 1/3 minus 1/3, minus 2/3. 00:49:56.370 --> 00:50:01.120 And now I should be able to not beat around the bush. 00:50:01.120 --> 00:50:04.444 Tell me what the answer will be in the end. 00:50:04.444 --> 00:50:06.270 STUDENT: 8/3. 00:50:06.270 --> 00:50:08.251 PROFESSOR: 8/3. 00:50:08.251 --> 00:50:09.790 Does that make sense? 00:50:09.790 --> 00:50:12.430 When you do that in math, you should always 00:50:12.430 --> 00:50:16.890 think-- one of the famous professors at Harvard 00:50:16.890 --> 00:50:21.650 was saying one time she asked the students, 00:50:21.650 --> 00:50:23.805 how many hours of life do we have have 00:50:23.805 --> 00:50:25.770 in one day, blah, blah, blah? 00:50:25.770 --> 00:50:30.280 And many students came up with 36, 37. 00:50:30.280 --> 00:50:35.560 So always make sure that the answer you get makes sense. 00:50:35.560 --> 00:50:37.890 This is part of a cube, right? 00:50:37.890 --> 00:50:42.920 It's like carved in a cube or a rectangle. 00:50:42.920 --> 00:50:46.490 00:50:46.490 --> 00:50:48.570 Now, what's the height? 00:50:48.570 --> 00:50:53.236 If this were to go up all the way to 2, 00:50:53.236 --> 00:50:58.610 it would be 2, 2, and 2. 00:50:58.610 --> 00:51:04.430 2 times 2 times 2 equals 8, and what we got is 8 over 3. 00:51:04.430 --> 00:51:08.830 Now, using our imagination, it makes sense. 00:51:08.830 --> 00:51:11.380 If I got a 16, I would say, oh my god. 00:51:11.380 --> 00:51:12.220 No, no, no, no. 00:51:12.220 --> 00:51:14.190 What is that? 00:51:14.190 --> 00:51:17.972 So a little bit, I would think, does this make sense or not? 00:51:17.972 --> 00:51:21.710 00:51:21.710 --> 00:51:24.400 Let's do one more, a similar one. 00:51:24.400 --> 00:51:28.180 Now I'm going to count on you a little bit more. 00:51:28.180 --> 00:51:39.246 00:51:39.246 --> 00:51:41.230 STUDENT: Professor, did you calculate that 00:51:41.230 --> 00:51:45.555 by just doing a quarter, and then just multiplying it by 4? 00:51:45.555 --> 00:51:47.096 Because then that would just leave us 00:51:47.096 --> 00:51:48.794 with zeroes [INAUDIBLE]. 00:51:48.794 --> 00:51:50.710 PROFESSOR: You mean in that particular figure? 00:51:50.710 --> 00:51:51.209 Yeah. 00:51:51.209 --> 00:51:54.170 STUDENT: Yeah, because it was perfectly [INAUDIBLE]. 00:51:54.170 --> 00:51:54.880 PROFESSOR: Yeah. 00:51:54.880 --> 00:51:56.660 It's nice. 00:51:56.660 --> 00:52:02.859 It's a little bit related to some other problems that 00:52:02.859 --> 00:52:03.650 come from pyramids. 00:52:03.650 --> 00:52:06.510 00:52:06.510 --> 00:52:16.246 By the way, how can you compute the volume of a square pyramid? 00:52:16.246 --> 00:52:21.470 00:52:21.470 --> 00:52:26.360 Suppose that you have the same problem. 00:52:26.360 --> 00:52:30.610 Minus 1 to 1 for x and y. 00:52:30.610 --> 00:52:34.760 Minus 1 to 1, minus 1 to 1. 00:52:34.760 --> 00:53:04.560 Let's say the pyramid would have the something like that. 00:53:04.560 --> 00:53:06.646 What would be the volume of such a pyramid? 00:53:06.646 --> 00:53:10.470 00:53:10.470 --> 00:53:12.860 STUDENT: [INAUDIBLE]. 00:53:12.860 --> 00:53:17.960 PROFESSOR: The height is h for extra credit. 00:53:17.960 --> 00:53:32.636 Can you compute the volume of this pyramid 00:53:32.636 --> 00:53:33.980 using double integrals? 00:53:33.980 --> 00:53:40.920 00:53:40.920 --> 00:53:48.620 Say the height is h and the bases is the square minus 1, 00:53:48.620 --> 00:53:51.740 1, minus 1, 1. 00:53:51.740 --> 00:53:54.420 I'm sure it can be done, but you know-- 00:53:54.420 --> 00:53:58.062 now I'm testing what you remember in terms of geometry 00:53:58.062 --> 00:54:00.691 because we will deal with geometry a lot 00:54:00.691 --> 00:54:03.100 in volumes and areas. 00:54:03.100 --> 00:54:07.475 So how do you do that in general, guys? 00:54:07.475 --> 00:54:09.950 STUDENT: 1/3 [INAUDIBLE]. 00:54:09.950 --> 00:54:14.260 PROFESSOR: 1/3 the height times the area 00:54:14.260 --> 00:54:18.536 of the bases, which is what? 00:54:18.536 --> 00:54:20.490 2 times 2. 00:54:20.490 --> 00:54:28.290 2 times 2, 3, over 3, 4/3 h. 00:54:28.290 --> 00:54:30.070 Can you prove that with calculus? 00:54:30.070 --> 00:54:31.110 That's all I'm saying. 00:54:31.110 --> 00:54:33.660 One point extra credit. 00:54:33.660 --> 00:54:36.295 Can you prove that with calculus? 00:54:36.295 --> 00:54:40.610 Actually, you would have to use what you learned. 00:54:40.610 --> 00:54:44.528 You can use Calc 2 as well. 00:54:44.528 --> 00:54:46.710 Do you guys remember that there were 00:54:46.710 --> 00:54:53.270 some cross-sectional areas, like this would be made of cheese, 00:54:53.270 --> 00:54:56.750 and you come with a vertical knife and cut cross sections. 00:54:56.750 --> 00:54:57.910 They go like that. 00:54:57.910 --> 00:54:59.300 But that's awfully hard. 00:54:59.300 --> 00:55:02.995 Maybe you can do it differently with Calc 3 instead of Calc 2. 00:55:02.995 --> 00:55:07.650 00:55:07.650 --> 00:55:10.118 Let's pick one from the book as well. 00:55:10.118 --> 00:55:31.100 00:55:31.100 --> 00:55:33.180 OK. 00:55:33.180 --> 00:55:38.930 So the same idea of using the Fubini-Tonelli argument 00:55:38.930 --> 00:55:45.500 and have an iterative-- evaluate the following double integral 00:55:45.500 --> 00:55:48.820 over the rectangle of vertices 0, 0-- 00:55:48.820 --> 00:55:52.120 write it down-- 3, 0, 3, 2, and 0, 2. 00:55:52.120 --> 00:56:01.552 So on the bases, you have a rectangle of vertices 3, 0, 0, 00:56:01.552 --> 00:56:14.370 0, 3, 2, and 0, 2. 00:56:14.370 --> 00:56:18.650 And then somebody tells you, find us 00:56:18.650 --> 00:56:29.350 the double integral of 2 minus y da 00:56:29.350 --> 00:56:35.850 over r where r represents the rectangle that we talked about. 00:56:35.850 --> 00:56:37.730 This is exactly [INAUDIBLE]. 00:56:37.730 --> 00:56:42.440 00:56:42.440 --> 00:56:45.300 And the answer we should get is 6. 00:56:45.300 --> 00:56:48.920 And I'm saying on top of what we said in the book, 00:56:48.920 --> 00:56:52.980 can you give a geometric interpretation? 00:56:52.980 --> 00:56:55.190 Does this have a geometric interpretation 00:56:55.190 --> 00:56:57.162 you can think of or not? 00:56:57.162 --> 00:57:01.374 00:57:01.374 --> 00:57:04.190 Well, first of all, what is this animal? 00:57:04.190 --> 00:57:07.050 According to the Fubini theorem, this animal 00:57:07.050 --> 00:57:14.160 will have to be-- I have it over a rectangle, 00:57:14.160 --> 00:57:18.110 so assume x will be between a and b, y 00:57:18.110 --> 00:57:22.047 will be between c and d. 00:57:22.047 --> 00:57:24.873 I have to figure out who those are. 00:57:24.873 --> 00:57:31.660 2 minus y and dy dx. 00:57:31.660 --> 00:57:35.675 00:57:35.675 --> 00:57:37.749 Where is y between? 00:57:37.749 --> 00:57:39.540 I should draw the picture for the rectangle 00:57:39.540 --> 00:57:42.760 because otherwise, it's not so easy to see. 00:57:42.760 --> 00:57:50.902 I have 0, 0 here, 3, 0 here, 3, 2 over here, shouldn't be hard. 00:57:50.902 --> 00:57:53.220 So this is going to be 0, 2. 00:57:53.220 --> 00:57:57.460 That's the y-axis and that's the x-axis. 00:57:57.460 --> 00:58:00.930 Let's see if we can see it. 00:58:00.930 --> 00:58:05.240 And what is the meaning of the 6, I'm asking you? 00:58:05.240 --> 00:58:07.150 I don't know. 00:58:07.150 --> 00:58:11.380 x should be between 0 and 3, right? 00:58:11.380 --> 00:58:15.330 y should be between 0 and 2, right? 00:58:15.330 --> 00:58:16.890 Now you are experts in this. 00:58:16.890 --> 00:58:20.916 We've done this twice, and you already know how to do it. 00:58:20.916 --> 00:58:23.160 Integral from 0 to 3. 00:58:23.160 --> 00:58:27.040 Then I take that, and that's going 00:58:27.040 --> 00:58:39.270 to be 2y minus y squared over 2 between y 00:58:39.270 --> 00:58:43.465 equals 0 down and y equals 2 up dx. 00:58:43.465 --> 00:58:48.044 00:58:48.044 --> 00:58:54.690 That means integral from 0 to 3, bracket minus bracket 00:58:54.690 --> 00:58:58.690 to make my life easier, dx. 00:58:58.690 --> 00:59:02.450 Now, there is no x, thank god. 00:59:02.450 --> 00:59:04.615 So that means I'm going to have a constant 00:59:04.615 --> 00:59:09.920 minus another constant, which means I go 4 minus 4 over 2. 00:59:09.920 --> 00:59:12.950 2, right? 00:59:12.950 --> 00:59:18.430 The other one, for 0, I get 0. 00:59:18.430 --> 00:59:20.600 I'm very happy I get 0 because in that case, 00:59:20.600 --> 00:59:25.170 it's obvious that I get 2 times 3, which is 6. 00:59:25.170 --> 00:59:29.410 So I got what the book said I'm going to get. 00:59:29.410 --> 00:59:32.130 But do I have a geometric interpretation of that? 00:59:32.130 --> 00:59:37.160 I would like to see if anybody can-- 00:59:37.160 --> 00:59:41.140 I'm going to give you a break in a few minues-- 00:59:41.140 --> 00:59:45.970 if anybody can think of a geometric interpretation. 00:59:45.970 --> 00:59:52.630 What is this f of xy if I were to interpret this as a graph? 00:59:52.630 --> 00:59:55.100 x equals f of x and y. 00:59:55.100 --> 00:59:55.760 Is this-- 00:59:55.760 --> 00:59:57.520 STUDENT: 2 minus y. 00:59:57.520 --> 01:00:04.550 PROFESSOR: So z equals 2 minus y is a plane, right? 01:00:04.550 --> 01:00:08.144 STUDENT: Yes, but then you have the parabola is going down. 01:00:08.144 --> 01:00:11.210 PROFESSOR: And how do I get to draw this plane the best? 01:00:11.210 --> 01:00:13.881 Because there are many ways to do it. 01:00:13.881 --> 01:00:16.700 I look at this wall. 01:00:16.700 --> 01:00:19.120 The y-axis is this. 01:00:19.120 --> 01:00:21.093 The z-axis is the vertical line. 01:00:21.093 --> 01:00:23.406 So I'm looking at this plane. 01:00:23.406 --> 01:00:27.630 y plus z must be equal to 2. 01:00:27.630 --> 01:00:29.850 So when is y plus z equal to 2? 01:00:29.850 --> 01:00:34.150 When I am on a line in the plane. 01:00:34.150 --> 01:00:39.150 I'm going to draw that line with pink because I like pink. 01:00:39.150 --> 01:00:41.240 This is y plus z equals 2. 01:00:41.240 --> 01:00:44.480 01:00:44.480 --> 01:00:50.480 And imagine this line will be shifted by parallelism as it 01:00:50.480 --> 01:00:54.940 comes towards you on all these other parallel vertical planes 01:00:54.940 --> 01:00:57.720 that are parallel to the board. 01:00:57.720 --> 01:01:04.770 So I'm going to have an entire plane like that, 01:01:04.770 --> 01:01:09.300 and I'm going to stop here. 01:01:09.300 --> 01:01:13.105 When I'm in the plane that's called x equals 3-- 01:01:13.105 --> 01:01:15.390 this is the plane called x equals 01:01:15.390 --> 01:01:20.160 3-- I have exactly this triangle, this [INAUDIBLE]. 01:01:20.160 --> 01:01:23.691 It's in the plane that faces me here. 01:01:23.691 --> 01:01:26.060 I don't know if you realize that. 01:01:26.060 --> 01:01:30.719 I'll help you make a house or something nice. 01:01:30.719 --> 01:01:32.651 I think I'm getting hungry. 01:01:32.651 --> 01:01:35.650 I imagine this again as being a piece of cheese, 01:01:35.650 --> 01:01:39.750 or it looks even like a piece of cake would be with layers. 01:01:39.750 --> 01:01:42.850 01:01:42.850 --> 01:01:47.560 So our question is, if we didn't know calculus 01:01:47.560 --> 01:01:51.210 but we knew how to draw this, and somebody gave you 01:01:51.210 --> 01:01:53.787 this at the GRE or whatever exam, 01:01:53.787 --> 01:01:55.620 how could you have done it without calculus? 01:01:55.620 --> 01:02:00.482 Just by cheating and pretending, I know how to do it, 01:02:00.482 --> 01:02:02.920 but you've never done a double integral in your life. 01:02:02.920 --> 01:02:06.178 So I know it's a volume. 01:02:06.178 --> 01:02:08.655 How do I get the volume? 01:02:08.655 --> 01:02:10.154 What kind of geometric body is that? 01:02:10.154 --> 01:02:11.645 STUDENT: A triangle. 01:02:11.645 --> 01:02:13.633 STUDENT: It's a triangular prism. 01:02:13.633 --> 01:02:15.618 PROFESSOR: It's a triangular prism. 01:02:15.618 --> 01:02:16.118 Good. 01:02:16.118 --> 01:02:19.625 And a triangular prism has what volume formula? 01:02:19.625 --> 01:02:20.750 STUDENT: Base times height. 01:02:20.750 --> 01:02:22.446 PROFESSOR: Base times the height. 01:02:22.446 --> 01:02:25.932 And the height has what area? 01:02:25.932 --> 01:02:27.440 Let's see. 01:02:27.440 --> 01:02:30.218 The base would be that, right? 01:02:30.218 --> 01:02:34.170 And the height would be 3. 01:02:34.170 --> 01:02:36.146 Am I right or not? 01:02:36.146 --> 01:02:37.581 The height would be 3. 01:02:37.581 --> 01:02:38.122 This is not-- 01:02:38.122 --> 01:02:38.990 STUDENT: It's 2. 01:02:38.990 --> 01:02:39.490 Yeah. 01:02:39.490 --> 01:02:40.324 STUDENT: No, it's 3. 01:02:40.324 --> 01:02:41.906 DR. MAGDALENA TODA: From here to here? 01:02:41.906 --> 01:02:42.450 STUDENT: 3. 01:02:42.450 --> 01:02:43.575 DR. MAGDALENA TODA: It's 3. 01:02:43.575 --> 01:02:46.860 So how much is that? 01:02:46.860 --> 01:02:47.690 How much-- OK. 01:02:47.690 --> 01:02:50.010 From here to here is 2. 01:02:50.010 --> 01:02:54.200 From here to here, it's how much? 01:02:54.200 --> 01:02:56.119 STUDENT: The height is only-- I see-- 01:02:56.119 --> 01:02:57.065 STUDENT: It's also 2. 01:02:57.065 --> 01:02:59.440 DR. MAGDALENA TODA: It's also 2 because look at that. 01:02:59.440 --> 01:03:01.825 It's an isosceles triangle. 01:03:01.825 --> 01:03:03.730 This is 45 to 45. 01:03:03.730 --> 01:03:05.250 So this is also 2. 01:03:05.250 --> 01:03:08.860 2 to-- that's 90 degrees, 45, 45. 01:03:08.860 --> 01:03:09.360 OK. 01:03:09.360 --> 01:03:13.220 So the area of the shaded purple triangle-- how much is that? 01:03:13.220 --> 01:03:13.964 STUDENT: 2. 01:03:13.964 --> 01:03:14.880 DR. MAGDALENA TODA: 2. 01:03:14.880 --> 01:03:17.120 2 times 2 over 2. 01:03:17.120 --> 01:03:19.710 2 times 3 equals 6. 01:03:19.710 --> 01:03:22.220 I don't need calculus. 01:03:22.220 --> 01:03:24.170 In this case, I don't need calculus. 01:03:24.170 --> 01:03:27.170 But when I have those nasty curvilinear 01:03:27.170 --> 01:03:31.958 z equals f of x, y, complicated expressions, I have no choice. 01:03:31.958 --> 01:03:34.946 I have to do the double integral. 01:03:34.946 --> 01:03:37.950 But in this case, even if I didn't know how to do it, 01:03:37.950 --> 01:03:39.200 I would still get the 6. 01:03:39.200 --> 01:03:39.976 Yes, sir? 01:03:39.976 --> 01:03:42.738 STUDENT: What if we did that on the exam? 01:03:42.738 --> 01:03:44.321 DR. MAGDALENA TODA: Well, that's good. 01:03:44.321 --> 01:03:45.736 I will then keep it in mind. 01:03:45.736 --> 01:03:46.235 Yes. 01:03:46.235 --> 01:03:48.560 It doesn't matter to me. 01:03:48.560 --> 01:03:50.900 I have other colleagues who really care about the method 01:03:50.900 --> 01:03:52.350 and start complaining. 01:03:52.350 --> 01:03:55.560 I don't care how you get to the answer 01:03:55.560 --> 01:03:57.450 as long as you got the right answer. 01:03:57.450 --> 01:03:59.400 Let me tell you my logic. 01:03:59.400 --> 01:04:03.830 Suppose somebody hired you thinking you're a good worker, 01:04:03.830 --> 01:04:05.327 and you're smart and so on. 01:04:05.327 --> 01:04:09.818 Would they care how you got to the solution of the problem? 01:04:09.818 --> 01:04:14.220 As long as the problem was solved correctly, no. 01:04:14.220 --> 01:04:18.250 And actually, the elementary way is the fastest 01:04:18.250 --> 01:04:20.120 because it's just 10 seconds. 01:04:20.120 --> 01:04:20.923 You draw. 01:04:20.923 --> 01:04:21.690 You imagine. 01:04:21.690 --> 01:04:23.180 You know what it is. 01:04:23.180 --> 01:04:28.310 So your boss will want you to find the fastest way to provide 01:04:28.310 --> 01:04:29.230 the correct solution. 01:04:29.230 --> 01:04:33.480 He's not going to care how you got that. 01:04:33.480 --> 01:04:35.580 So no matter how you do it, as long 01:04:35.580 --> 01:04:39.990 as you've got the right answer, I'm going to be happy. 01:04:39.990 --> 01:04:49.230 I want to ask you to please go to page 927 in the book 01:04:49.230 --> 01:04:50.380 and read. 01:04:50.380 --> 01:04:52.580 It's only one page. 01:04:52.580 --> 01:04:55.340 That whole end section, 12.1. 01:04:55.340 --> 01:04:59.928 It's called an informal argument for Fubini's theorem. 01:04:59.928 --> 01:05:05.736 Practically, it's a proof of Fubini's theorem, page 927. 01:05:05.736 --> 01:05:08.640 And then I'm going to go ahead and start the homework 01:05:08.640 --> 01:05:11.820 four, if you don't mind. 01:05:11.820 --> 01:05:15.920 I'm going to go into WeBWork and give you homework four. 01:05:15.920 --> 01:05:18.580 And the first few problems that you 01:05:18.580 --> 01:05:21.100 are going to be expected to solve 01:05:21.100 --> 01:05:27.313 will be out of 12.1, which is really easy. 01:05:27.313 --> 01:05:28.792 I'll give you a few minutes back. 01:05:28.792 --> 01:05:32.736 And we go on with 12.2, and it's very similar. 01:05:32.736 --> 01:05:34.708 You're going to like that. 01:05:34.708 --> 01:05:39.638 And then we'll go home or wherever we need to go. 01:05:39.638 --> 01:05:42.103 So you have a few minutes of a break. 01:05:42.103 --> 01:05:45.560 Pick up your extra credits. 01:05:45.560 --> 01:05:47.129 I'll call the names. 01:05:47.129 --> 01:05:48.596 Lily. 01:05:48.596 --> 01:05:52.019 You got a lot of points. 01:05:52.019 --> 01:05:54.953 And [INAUDIBLE]. 01:05:54.953 --> 01:05:56.909 And you have two separate ones. 01:05:56.909 --> 01:05:58.365 Nathan. 01:05:58.365 --> 01:05:58.865 Nathan? 01:05:58.865 --> 01:06:02.310 01:06:02.310 --> 01:06:03.040 Rachel Smith. 01:06:03.040 --> 01:06:05.730 01:06:05.730 --> 01:06:06.230 Austin. 01:06:06.230 --> 01:06:09.278 01:06:09.278 --> 01:06:09.778 Thank you. 01:06:09.778 --> 01:06:12.736 01:06:12.736 --> 01:06:13.722 Edgar. 01:06:13.722 --> 01:06:16.180 [INAUDIBLE] 01:06:16.180 --> 01:06:16.680 Aaron. 01:06:16.680 --> 01:06:24.068 01:06:24.068 --> 01:06:24.568 Andre. 01:06:24.568 --> 01:06:32.456 01:06:32.456 --> 01:06:35.407 Aaron. 01:06:35.407 --> 01:06:35.907 Kasey. 01:06:35.907 --> 01:06:39.851 01:06:39.851 --> 01:06:43.490 Kasey came up with a very good idea 01:06:43.490 --> 01:06:47.530 that I will write a review sample. 01:06:47.530 --> 01:06:48.615 Did I promise that? 01:06:48.615 --> 01:06:52.200 A review sample for the midterm. 01:06:52.200 --> 01:06:53.720 And so I said yes. 01:06:53.720 --> 01:06:56.696 01:06:56.696 --> 01:07:01.160 Karen and Matthew. 01:07:01.160 --> 01:07:07.604 01:07:07.604 --> 01:07:08.104 Reagan. 01:07:08.104 --> 01:07:16.040 01:07:16.040 --> 01:07:17.910 Aaron. 01:07:17.910 --> 01:07:20.510 When you submitted, you submitted. 01:07:20.510 --> 01:07:21.194 Yeah. 01:07:21.194 --> 01:07:21.860 And [INAUDIBLE]. 01:07:21.860 --> 01:07:25.860 01:07:25.860 --> 01:07:26.860 here. 01:07:26.860 --> 01:07:27.860 And I'm done. 01:07:27.860 --> 01:07:46.360 01:07:46.360 --> 01:07:48.360 STUDENT: Did we turn in [INAUDIBLE]? 01:07:48.360 --> 01:07:49.860 DR. MAGDALENA TODA: Yes, absolutely. 01:07:49.860 --> 01:08:08.860 01:08:08.860 --> 01:08:12.438 Now once we go over 12.2, you will say, oh, 01:08:12.438 --> 01:08:14.434 but I understand the Fubini theorem. 01:08:14.434 --> 01:08:21.439 01:08:21.439 --> 01:08:23.926 I didn't know whether there's room for Fubini, 01:08:23.926 --> 01:08:29.250 because once I cover the more general case, which is in 12.2, 01:08:29.250 --> 01:08:33.580 you are going to understand Why Fubini-Tonelli 01:08:33.580 --> 01:08:36.970 works for rectangles. 01:08:36.970 --> 01:08:47.220 So if I think of a domain that is of the following form, 01:08:47.220 --> 01:08:54.207 in the x, y plane, I go x is between and and b, right? 01:08:54.207 --> 01:08:59.685 That's my favorite x. 01:08:59.685 --> 01:09:02.340 So I take the pink segment, and I 01:09:02.340 --> 01:09:05.050 say, everything that happens-- it's going 01:09:05.050 --> 01:09:08.450 to happen on top of this world. 01:09:08.450 --> 01:09:11.359 I have, let's say, two functions. 01:09:11.359 --> 01:09:14.100 To make my life easier, I'll assume both of them [INAUDIBLE] 01:09:14.100 --> 01:09:15.830 one bigger than the other. 01:09:15.830 --> 01:09:23.880 But in case they are not both positive, 01:09:23.880 --> 01:09:28.080 I just need f to be bigger than g for every point. 01:09:28.080 --> 01:09:32.742 And the same argument will function. 01:09:32.742 --> 01:09:38.890 This is f, continuous positive. 01:09:38.890 --> 01:09:41.996 Then g, continuous positive but smaller 01:09:41.996 --> 01:09:44.912 in values than this one. 01:09:44.912 --> 01:09:47.828 01:09:47.828 --> 01:09:48.800 Yes, sir? 01:09:48.800 --> 01:09:51.189 STUDENT: [INAUDIBLE] 12.2 that we're starting? 01:09:51.189 --> 01:09:52.229 DR. MAGDALENA TODA: 12.2. 01:09:52.229 --> 01:09:56.005 And you are more organized than I am, and I appreciate it. 01:09:56.005 --> 01:10:02.453 So integration over a non-rectangular domain. 01:10:02.453 --> 01:10:06.930 01:10:06.930 --> 01:10:10.361 And we call this a type one because this 01:10:10.361 --> 01:10:12.325 is what many books are using. 01:10:12.325 --> 01:10:17.030 And this is that x is between two fixed end points. 01:10:17.030 --> 01:10:21.270 But y is between two variable end points. 01:10:21.270 --> 01:10:24.300 So what's going to happen to y? 01:10:24.300 --> 01:10:29.480 y is going to take values between the lower, 01:10:29.480 --> 01:10:34.752 the bottom one, which is g of x, and the upper one, 01:10:34.752 --> 01:10:37.062 which is f of x. 01:10:37.062 --> 01:10:39.730 So this is how we define the domain that's 01:10:39.730 --> 01:10:45.230 shaded by me with black shades, vertical strips here. 01:10:45.230 --> 01:10:47.980 This is the domain. 01:10:47.980 --> 01:10:56.240 Now you really do not need to prove 01:10:56.240 --> 01:11:09.250 that double integral over 1 dA over-- let's call 01:11:09.250 --> 01:11:15.420 the domain D-- is what? 01:11:15.420 --> 01:11:17.980 01:11:17.980 --> 01:11:27.776 Integral between f of x minus g of x from a to b dx. 01:11:27.776 --> 01:11:31.080 01:11:31.080 --> 01:11:32.087 And you say, what? 01:11:32.087 --> 01:11:33.670 Magdalena, what are you trying to say? 01:11:33.670 --> 01:11:34.760 OK. 01:11:34.760 --> 01:11:37.330 Let's go back and say, what if somebody 01:11:37.330 --> 01:11:41.370 would have asked you the same question in calculus 2? 01:11:41.370 --> 01:11:44.640 Saying, guys I have a question about the area 01:11:44.640 --> 01:11:49.080 in the shaded strip, vertical strip thing. 01:11:49.080 --> 01:11:50.869 How are we going to compute that? 01:11:50.869 --> 01:11:53.590 And you would say, oh, I have an idea. 01:11:53.590 --> 01:12:04.020 I take the area under the graph f, and I shade that in orange. 01:12:04.020 --> 01:12:05.917 And I know what that is. 01:12:05.917 --> 01:12:07.500 So you would say, I know what that is. 01:12:07.500 --> 01:12:08.990 That's going to be what? 01:12:08.990 --> 01:12:13.276 Integral from a to be f of x dx. 01:12:13.276 --> 01:12:16.890 Let's call that A1, right? 01:12:16.890 --> 01:12:19.631 A1. 01:12:19.631 --> 01:12:27.920 Then you go, minus the area with-- I'm just going to shade 01:12:27.920 --> 01:12:32.128 that, brown strips under g. 01:12:32.128 --> 01:12:35.050 01:12:35.050 --> 01:12:37.980 g of x dx. 01:12:37.980 --> 01:12:39.320 And call that A2. 01:12:39.320 --> 01:12:42.310 01:12:42.310 --> 01:12:45.300 A1 minus A2. 01:12:45.300 --> 01:12:49.330 We know both of these formulas from where? 01:12:49.330 --> 01:12:52.900 Calc 1 because that's where you learned about the area 01:12:52.900 --> 01:12:55.040 under the graph of a curve. 01:12:55.040 --> 01:12:57.950 This is the area under the graph of a curve f. 01:12:57.950 --> 01:13:00.880 This is the area under the graph of the curve g. 01:13:00.880 --> 01:13:04.550 The black striped area is their difference. 01:13:04.550 --> 01:13:05.160 All right. 01:13:05.160 --> 01:13:07.190 And so how much is that? 01:13:07.190 --> 01:13:09.130 I'm sorry I put the wrong thing. 01:13:09.130 --> 01:13:11.560 a, b. 01:13:11.560 --> 01:13:13.530 That's going to be integral from a to b. 01:13:13.530 --> 01:13:15.890 Now you say, wait, wait, wait a minute. 01:13:15.890 --> 01:13:17.126 Based on what? 01:13:17.126 --> 01:13:20.260 Based on some sort of additivity property 01:13:20.260 --> 01:13:23.640 of the integral of one variable, which says integral 01:13:23.640 --> 01:13:27.025 from a to b of f plus g. 01:13:27.025 --> 01:13:29.270 You can have f plus, minus g. 01:13:29.270 --> 01:13:30.540 It doesn't matter. 01:13:30.540 --> 01:13:31.940 dx. 01:13:31.940 --> 01:13:37.882 You have integral from a to b f dx plus integral from a to b g 01:13:37.882 --> 01:13:39.176 dx. 01:13:39.176 --> 01:13:42.390 It doesn't matter what. 01:13:42.390 --> 01:13:46.050 You can have a linear combination of f and g. 01:13:46.050 --> 01:13:46.910 Yes, Matthew? 01:13:46.910 --> 01:13:49.180 MATTHEW: So this is just for the domain? 01:13:49.180 --> 01:13:52.790 So if you put it, that would be down. 01:13:52.790 --> 01:13:55.530 So there might be another formula up here 01:13:55.530 --> 01:13:57.350 that would be curved surface. 01:13:57.350 --> 01:13:59.560 And this is the bottom, so you're using integral 01:13:59.560 --> 01:14:01.260 to find the base, and then you're 01:14:01.260 --> 01:14:03.706 going to plug that integral into the other integral. 01:14:03.706 --> 01:14:06.080 DR. MAGDALENA TODA: So I'm just using the property that's 01:14:06.080 --> 01:14:10.660 called linearity of the simple integral, 01:14:10.660 --> 01:14:14.660 meaning that if I have even a linear combination like af 01:14:14.660 --> 01:14:22.246 plus bg, then a-- I have not a. 01:14:22.246 --> 01:14:26.758 Let me call it big A and big B. Big A Af integral 01:14:26.758 --> 01:14:29.360 of f plus big B integral of g. 01:14:29.360 --> 01:14:30.610 You've learned that in Calc 2. 01:14:30.610 --> 01:14:34.390 I'm doing this to apply it for these areas that are subtracted 01:14:34.390 --> 01:14:36.250 from one another. 01:14:36.250 --> 01:14:39.120 If I were to add, as you said, I would put something 01:14:39.120 --> 01:14:39.990 on top of that. 01:14:39.990 --> 01:14:44.650 And then it would be like a superimposition onto it. 01:14:44.650 --> 01:14:54.190 So I have integral from a to b of f of x minus g of x dx. 01:14:54.190 --> 01:14:56.840 And I claim that this is the same 01:14:56.840 --> 01:15:06.900 as double integral of the 1dA over the domain D. 01:15:06.900 --> 01:15:10.166 How can you write that differently? 01:15:10.166 --> 01:15:12.040 I'll tell you how you write that differently. 01:15:12.040 --> 01:15:19.150 Integral from a to b of integral from-- what's 01:15:19.150 --> 01:15:21.340 the bottom value of Mr. Y? 01:15:21.340 --> 01:15:23.880 01:15:23.880 --> 01:15:26.620 So Mr. X knows what he's doing. 01:15:26.620 --> 01:15:28.620 He goes all the way from a to b. 01:15:28.620 --> 01:15:31.330 The bottom value of y is g of x. 01:15:31.330 --> 01:15:36.250 You go from the bottom value of y g of x to the upper value 01:15:36.250 --> 01:15:38.080 f of x. 01:15:38.080 --> 01:15:42.320 And then you here put 1 and dy. 01:15:42.320 --> 01:15:44.660 Is this the same thing? 01:15:44.660 --> 01:15:46.440 You say, OK, I know this one. 01:15:46.440 --> 01:15:49.290 I know this one from calc 2. 01:15:49.290 --> 01:15:53.890 But Magdalena, the one you gave us is new. 01:15:53.890 --> 01:15:55.450 It's new and not new, guys. 01:15:55.450 --> 01:15:59.360 This is Fubini's theorem but generalized 01:15:59.360 --> 01:16:01.460 to something that depends on x. 01:16:01.460 --> 01:16:02.890 So how do I do that? 01:16:02.890 --> 01:16:05.080 Integral of 1dy. 01:16:05.080 --> 01:16:07.228 That's what? 01:16:07.228 --> 01:16:11.988 That's y measured between two values that don't depend on y. 01:16:11.988 --> 01:16:16.670 They depend only on x, g of x on the bottom, f of x on the top. 01:16:16.670 --> 01:16:20.155 So this is exactly the integral from a to b. 01:16:20.155 --> 01:16:22.360 In terms of the round parentheses, 01:16:22.360 --> 01:16:25.830 I put-- what is y between f of x and g of x? 01:16:25.830 --> 01:16:30.000 f of x minus g of x dx. 01:16:30.000 --> 01:16:34.474 So it is exactly the same thing from Calc 2 01:16:34.474 --> 01:16:36.410 expressed as a double integral. 01:16:36.410 --> 01:16:42.220 01:16:42.220 --> 01:16:42.920 All right. 01:16:42.920 --> 01:16:54.156 Now This is a type one region that we talked about. 01:16:54.156 --> 01:16:59.600 A type two region is a similar region, practically. 01:16:59.600 --> 01:17:03.130 What you have to keep in mind is they're both 01:17:03.130 --> 01:17:05.770 given here as examples. 01:17:05.770 --> 01:17:09.255 But the technique is absolutely the same. 01:17:09.255 --> 01:17:13.100 If instead of taking this picture, 01:17:13.100 --> 01:17:20.012 I would take y to move between fixed values, 01:17:20.012 --> 01:17:26.278 like y has to be between c and d-- this is my y. 01:17:26.278 --> 01:17:28.770 These are the fixed values. 01:17:28.770 --> 01:17:33.240 And then give me some nice colors. 01:17:33.240 --> 01:17:42.180 This curve and that curve-- OK, I 01:17:42.180 --> 01:17:49.860 have to rotate my head because then this is going to be x. 01:17:49.860 --> 01:17:51.780 This is going to be y. 01:17:51.780 --> 01:17:57.000 And the blue thingy has to be a function of y. 01:17:57.000 --> 01:17:58.810 x is a function of y. 01:17:58.810 --> 01:18:01.338 So how do I call that? 01:18:01.338 --> 01:18:09.950 I have x or whatever equals big F of y. 01:18:09.950 --> 01:18:16.930 And here in the red one, I have x equals big G of y. 01:18:16.930 --> 01:18:23.280 And how am I going to evaluate the striped area? 01:18:23.280 --> 01:18:30.550 Of course striped because I have again y is between c and d. 01:18:30.550 --> 01:18:33.770 And what's moving is Mr. X. 01:18:33.770 --> 01:18:37.480 And Mr. X refuses to have fixed variables. 01:18:37.480 --> 01:18:41.961 Now he goes, I move from the bottom, which is G of y, 01:18:41.961 --> 01:18:46.871 to the top, which is F of y. 01:18:46.871 --> 01:18:50.800 How am I going to write the double integral 01:18:50.800 --> 01:18:58.236 over this domain of 1dA, where dA is dxdy. 01:18:58.236 --> 01:19:00.380 Who's going to tell me? 01:19:00.380 --> 01:19:05.140 Similarly, the same reasoning as for this one. 01:19:05.140 --> 01:19:10.310 I'm going to have the integral from what to what 01:19:10.310 --> 01:19:12.190 of integral from what to what? 01:19:12.190 --> 01:19:14.840 Who comes first, dx or dy? 01:19:14.840 --> 01:19:15.524 STUDENT: dx. 01:19:15.524 --> 01:19:16.940 DR. MAGDALENA TODA: dx, very good. 01:19:16.940 --> 01:19:18.610 And dy at the end. 01:19:18.610 --> 01:19:23.130 So y will be between c and d, and x 01:19:23.130 --> 01:19:31.530 is going to be between G of y and F of y. 01:19:31.530 --> 01:19:32.440 And here is y. 01:19:32.440 --> 01:19:35.310 01:19:35.310 --> 01:19:38.510 How can I rewrite this integral? 01:19:38.510 --> 01:19:40.010 Very easily. 01:19:40.010 --> 01:19:46.070 The integral from c to d of the guy on top, 01:19:46.070 --> 01:19:54.090 the blue guy, F of y, minus the guy on the bottom, G of y, dy. 01:19:54.090 --> 01:20:00.170 Some people call the vertical stip method 01:20:00.170 --> 01:20:02.910 compared to the horizontal strip method, where 01:20:02.910 --> 01:20:05.150 in this kind of horizontal strip method, 01:20:05.150 --> 01:20:08.020 you just have to view x as a function of y 01:20:08.020 --> 01:20:11.730 and rotate your head and apply the same reasoning as before. 01:20:11.730 --> 01:20:13.140 It's not a big deal. 01:20:13.140 --> 01:20:15.900 You just need a little bit of imagination, 01:20:15.900 --> 01:20:20.464 and the result is the same. 01:20:20.464 --> 01:20:24.622 An example that's not too hard-- I 01:20:24.622 --> 01:20:26.452 want to give you several examples. 01:20:26.452 --> 01:20:29.450 01:20:29.450 --> 01:20:31.326 We have plenty of time. 01:20:31.326 --> 01:20:36.050 Now it says, we have a triangular region. 01:20:36.050 --> 01:20:40.980 And that is enclosed by lines y equals 0, y equals 2x, 01:20:40.980 --> 01:20:43.730 and x equals 1. 01:20:43.730 --> 01:20:47.520 Let's see what that means and be able to draw it. 01:20:47.520 --> 01:20:51.390 It's very important to be able to draw in this chapter. 01:20:51.390 --> 01:20:54.770 If you're not, just learn how to draw, 01:20:54.770 --> 01:20:56.690 and that will give you lots of ideas 01:20:56.690 --> 01:20:58.392 on how to solve the problems. 01:20:58.392 --> 01:21:18.270 01:21:18.270 --> 01:21:22.878 Chapter 12 is included completely on the midterm. 01:21:22.878 --> 01:21:25.368 So the midterm is on the 2nd of April. 01:21:25.368 --> 01:21:29.850 For the midterm, we have chapter 10, those three sections. 01:21:29.850 --> 01:21:32.340 Then we have chapter 11 completely, 01:21:32.340 --> 01:21:40.190 and then we have chapter 12 not completely, up to 12.6. 01:21:40.190 --> 01:21:40.760 All right. 01:21:40.760 --> 01:21:44.400 So what did I say? 01:21:44.400 --> 01:21:49.140 I have a triangular region that is obtained by intersecting 01:21:49.140 --> 01:21:50.900 the following lines. 01:21:50.900 --> 01:21:58.730 y equals 0, x equals 1, and y equals 2x. 01:21:58.730 --> 01:22:01.938 Can I draw them and see how they intersect? 01:22:01.938 --> 01:22:03.340 It shouldn't be a big problem. 01:22:03.340 --> 01:22:05.850 This is a line that passes through the origin 01:22:05.850 --> 01:22:07.990 and has slope 2. 01:22:07.990 --> 01:22:10.980 So it should be very easy to draw. 01:22:10.980 --> 01:22:18.270 At 1, x equals 1, the y will be 2 for this line of slope 2. 01:22:18.270 --> 01:22:20.850 So I'll try to draw. 01:22:20.850 --> 01:22:23.720 Does this look double to you? 01:22:23.720 --> 01:22:29.260 So this is 2. 01:22:29.260 --> 01:22:32.152 This is the point 1, 2. 01:22:32.152 --> 01:22:35.220 And that's the line y equals 2x. 01:22:35.220 --> 01:22:38.410 And that's the line y equals 0. 01:22:38.410 --> 01:22:40.220 And that's the line x equals 1. 01:22:40.220 --> 01:22:43.450 So can I shade this triangle? 01:22:43.450 --> 01:22:47.560 Yeah, I can eventually, depending on what they ask me. 01:22:47.560 --> 01:22:49.175 What do they ask me? 01:22:49.175 --> 01:22:58.100 Find the double integral of x plus y dA 01:22:58.100 --> 01:23:05.600 with respect to the area element over T, T being the triangle. 01:23:05.600 --> 01:23:09.508 So now I'm going to ask, did they say by what method? 01:23:09.508 --> 01:23:12.650 Unfortunately, they say, do it by both methods. 01:23:12.650 --> 01:23:17.190 That means both by x intregration first and then 01:23:17.190 --> 01:23:20.090 y integration and the other way around. 01:23:20.090 --> 01:23:23.310 So they ask you to change the order of the integration 01:23:23.310 --> 01:23:24.720 or do what? 01:23:24.720 --> 01:23:27.270 Switch from vertical strip method 01:23:27.270 --> 01:23:29.600 to horizontal strip method. 01:23:29.600 --> 01:23:31.150 You should get the same answer. 01:23:31.150 --> 01:23:34.160 That's a typical final exam problem. 01:23:34.160 --> 01:23:40.410 When we test you, if you are able to do this 01:23:40.410 --> 01:23:43.314 through the vertical strip or horizontal 01:23:43.314 --> 01:23:45.250 strip and change the order of integration. 01:23:45.250 --> 01:23:47.560 If I do it with the vertical strip method, 01:23:47.560 --> 01:23:52.010 who comes first, the dy or the dx? 01:23:52.010 --> 01:23:53.310 Think a little bit. 01:23:53.310 --> 01:23:55.610 Where do I put d-- Fubini [INAUDIBLE] 01:23:55.610 --> 01:23:58.652 comes dy dx or dx dy? 01:23:58.652 --> 01:23:59.600 STUDENT: dy. 01:23:59.600 --> 01:24:01.550 PROFESSOR: dy dx. 01:24:01.550 --> 01:24:04.360 So VSM. 01:24:04.360 --> 01:24:06.560 You're going to laugh. 01:24:06.560 --> 01:24:07.890 It's not written in the book. 01:24:07.890 --> 01:24:10.810 It's like a childish name, Vertical Strip Method, 01:24:10.810 --> 01:24:12.615 meeting integration with respect to y 01:24:12.615 --> 01:24:14.924 and then with respect to x. 01:24:14.924 --> 01:24:17.726 It helped my students through the last decade 01:24:17.726 --> 01:24:19.594 to remember about the vertical strips. 01:24:19.594 --> 01:24:25.490 And that's why I say something that's not using the book, VSM. 01:24:25.490 --> 01:24:35.810 Now, I have integral from-- so who is Mr. X going from 0 to 1? 01:24:35.810 --> 01:24:36.345 He's stable. 01:24:36.345 --> 01:24:37.750 He's happy. 01:24:37.750 --> 01:24:39.700 He's going between two fixed values. 01:24:39.700 --> 01:24:43.770 y goes between the bottom line, which is 0. 01:24:43.770 --> 01:24:44.650 We are lucky. 01:24:44.650 --> 01:24:47.600 It's a really nice problem. 01:24:47.600 --> 01:24:51.396 Going to y equals 2x. 01:24:51.396 --> 01:24:54.020 So it's not hard at all. 01:24:54.020 --> 01:24:59.230 And we have to integrate the function x plus y. 01:24:59.230 --> 01:25:01.768 It should be a piece of cake. 01:25:01.768 --> 01:25:06.600 Let's do this together because you've accumulated seniority 01:25:06.600 --> 01:25:07.675 in this type of problem. 01:25:07.675 --> 01:25:10.520 01:25:10.520 --> 01:25:12.440 What do I put inside? 01:25:12.440 --> 01:25:14.580 What's integral of x plus y with respect to y? 01:25:14.580 --> 01:25:15.987 Is it hard? 01:25:15.987 --> 01:25:19.270 01:25:19.270 --> 01:25:23.446 xy plus-- somebody tell me. 01:25:23.446 --> 01:25:25.110 STUDENT: y squared. 01:25:25.110 --> 01:25:29.160 PROFESSOR: y squared over 2, between y 01:25:29.160 --> 01:25:33.450 equals 0 on the bottom, y equals 2x on top. 01:25:33.450 --> 01:25:36.860 I have to be smart and plug in the values y. 01:25:36.860 --> 01:25:39.009 Otherwise, I'll never make it. 01:25:39.009 --> 01:25:39.800 STUDENT: Professor? 01:25:39.800 --> 01:25:40.900 PROFESSOR: Yes, sir? 01:25:40.900 --> 01:25:43.310 STUDENT: Why did you take 2x as the final value 01:25:43.310 --> 01:25:45.310 because you have a specified triangle. 01:25:45.310 --> 01:25:48.970 PROFESSOR: Because y equals 2x is the expression 01:25:48.970 --> 01:25:52.100 of the upper function. 01:25:52.100 --> 01:25:54.775 The upper function is the line y equals 2x. 01:25:54.775 --> 01:25:56.190 They provided that. 01:25:56.190 --> 01:25:59.660 So from the bottom function to the upper function, 01:25:59.660 --> 01:26:02.376 the vertical strips go between two functions. 01:26:02.376 --> 01:26:05.280 01:26:05.280 --> 01:26:07.720 So when I plug in here y equals 2x, 01:26:07.720 --> 01:26:09.723 I have to pay attention to my algebra. 01:26:09.723 --> 01:26:13.820 If I forget the 2, it's all over for me, zero points. 01:26:13.820 --> 01:26:16.260 Well, not zero points, but 10% credit. 01:26:16.260 --> 01:26:20.090 I have no idea what I would get, so I have to pay attention. 01:26:20.090 --> 01:26:26.616 2x times x is 2x squared plus 2x all squared-- guys, 01:26:26.616 --> 01:26:30.520 keep an eye on me-- 4x squared over 2. 01:26:30.520 --> 01:26:36.430 I put the first value in a pink parentheses, 01:26:36.430 --> 01:26:40.804 and then I move on to the line parentheses. 01:26:40.804 --> 01:26:42.840 Evaluate it at 0. 01:26:42.840 --> 01:26:44.506 That line is very lucky. 01:26:44.506 --> 01:26:50.910 I get a 0 because y equals 0 will give me 0. 01:26:50.910 --> 01:26:54.180 What am I going to get here? 01:26:54.180 --> 01:26:56.521 2x squared plus 2x squared. 01:26:56.521 --> 01:26:57.021 Good. 01:26:57.021 --> 01:26:59.406 What's 2x squared plus 2x squared? 01:26:59.406 --> 01:27:00.260 4x squared. 01:27:00.260 --> 01:27:01.916 So a 4 goes out. 01:27:01.916 --> 01:27:03.320 Kick him out. 01:27:03.320 --> 01:27:06.190 Integral from 0 to 1 x squared dx. 01:27:06.190 --> 01:27:08.050 Integral of x squared is? 01:27:08.050 --> 01:27:11.650 01:27:11.650 --> 01:27:14.100 Integral of x squared is? 01:27:14.100 --> 01:27:15.100 STUDENT: x cubed over 3. 01:27:15.100 --> 01:27:16.183 PROFESSOR: x cubed over 3. 01:27:16.183 --> 01:27:19.332 And if you take it between 1 and 0, you get? 01:27:19.332 --> 01:27:20.660 STUDENT: 1. 01:27:20.660 --> 01:27:21.310 PROFESSOR: 1/3. 01:27:21.310 --> 01:27:23.700 1/3 times 4 is 4/3. 01:27:23.700 --> 01:27:26.820 01:27:26.820 --> 01:27:29.140 Suppose this is going to happen on the midterm, 01:27:29.140 --> 01:27:32.400 and I'm asking you to do it reversing the integration 01:27:32.400 --> 01:27:33.930 order. 01:27:33.930 --> 01:27:37.520 Then you are going to check your own work very beautifully 01:27:37.520 --> 01:27:41.720 in the sense that you say, well, now 01:27:41.720 --> 01:27:45.880 I'm going to see if I made a mistake in this one. 01:27:45.880 --> 01:27:46.690 What do I do? 01:27:46.690 --> 01:27:50.480 I erase the whole thing, and instead of vertical strips, 01:27:50.480 --> 01:27:55.840 I'm going to put horizontal strips. 01:27:55.840 --> 01:28:01.020 And you say, well, life is a little bit harder in this case 01:28:01.020 --> 01:28:04.526 because in this case, I have to look 01:28:04.526 --> 01:28:10.850 at y between fixed values, y between 0 and 1. 01:28:10.850 --> 01:28:17.785 So y is between 0 and 1-- 0 and 2, fixed values. 01:28:17.785 --> 01:28:22.840 And Mr. X says, I'm going between two functions of y. 01:28:22.840 --> 01:28:26.230 I don't know what those functions of y are. 01:28:26.230 --> 01:28:28.380 I'm puzzled. 01:28:28.380 --> 01:28:30.600 You have to help Mr. X know where 01:28:30.600 --> 01:28:34.520 he's going because his life right now is a little bit hard. 01:28:34.520 --> 01:28:39.295 So what is the function for the blue? 01:28:39.295 --> 01:28:42.205 01:28:42.205 --> 01:28:44.160 Now he's not blue anymore. 01:28:44.160 --> 01:28:45.140 He's brown. 01:28:45.140 --> 01:28:47.840 x equals 1. 01:28:47.840 --> 01:28:50.100 So he knows what he's going to be. 01:28:50.100 --> 01:28:52.812 What is the x function for the red line 01:28:52.812 --> 01:28:54.580 that [INAUDIBLE] asked about? 01:28:54.580 --> 01:28:55.464 STUDENT: y over 2. 01:28:55.464 --> 01:28:57.550 PROFESSOR: x must be y over 2. 01:28:57.550 --> 01:29:01.010 It's the same thing, but I have to express x in terms of y. 01:29:01.010 --> 01:29:05.282 So I erase and I say x equals y over 2. 01:29:05.282 --> 01:29:06.700 Same thing. 01:29:06.700 --> 01:29:11.140 So x has to be between what and what, the bottom and the top? 01:29:11.140 --> 01:29:13.680 Well, I turn my head. 01:29:13.680 --> 01:29:19.391 The top must be x equals 1, and the bottom one is y over 2. 01:29:19.391 --> 01:29:24.980 That's the bottom one, the bottom value for x. 01:29:24.980 --> 01:29:27.450 Now wish me luck because I have to get the same thing. 01:29:27.450 --> 01:29:35.520 So integral from 0 to 2 of integral from y over 2 to 1. 01:29:35.520 --> 01:29:37.512 Changing the order of integration 01:29:37.512 --> 01:29:40.650 doesn't change the integrand, which is exactly 01:29:40.650 --> 01:29:43.414 the same function, f of xy. 01:29:43.414 --> 01:29:46.810 This is the f function. 01:29:46.810 --> 01:29:47.769 Then what changes? 01:29:47.769 --> 01:29:48.810 The order of integration. 01:29:48.810 --> 01:29:52.168 So I go dx first, dy next and stop. 01:29:52.168 --> 01:29:55.030 01:29:55.030 --> 01:30:00.230 I copy and paste the outer ones, and I focus my attention 01:30:00.230 --> 01:30:05.600 to the red parentheses inside, which I'm 01:30:05.600 --> 01:30:07.890 going to copy and paste here. 01:30:07.890 --> 01:30:12.480 I'll have to do some math very carefully. 01:30:12.480 --> 01:30:13.500 So what do I have? 01:30:13.500 --> 01:30:17.140 I have x plus y integrated with respect to x. 01:30:17.140 --> 01:30:19.232 If I rush, it's a bad thing. 01:30:19.232 --> 01:30:20.800 STUDENT: So that would be x squared. 01:30:20.800 --> 01:30:21.674 PROFESSOR: x squared. 01:30:21.674 --> 01:30:22.780 STUDENT: Over 2. 01:30:22.780 --> 01:30:23.742 PROFESSOR: Over 2. 01:30:23.742 --> 01:30:25.350 STUDENT: Plus xy. 01:30:25.350 --> 01:30:29.610 PROFESSOR: Plus xy taken between the following. 01:30:29.610 --> 01:30:32.760 When x equals 1, I have it on top. 01:30:32.760 --> 01:30:38.350 When x equals y over 2, I have it on the bottom. 01:30:38.350 --> 01:30:39.690 OK. 01:30:39.690 --> 01:30:42.585 This red thing, I'm a little bit too lazy. 01:30:42.585 --> 01:30:47.854 I'll copy and paste it separately. 01:30:47.854 --> 01:30:52.480 For the upper part, it's really easy to compute. 01:30:52.480 --> 01:30:53.480 What do I get? 01:30:53.480 --> 01:31:01.890 When x is 1, 1/2, 1/2 plus when x is 1, y. 01:31:01.890 --> 01:31:07.120 Minus integral of-- when x is y over 2, 01:31:07.120 --> 01:31:12.700 I get y squared over 4 up here over 2. 01:31:12.700 --> 01:31:19.360 So I should get y squared over 8 plus-- 01:31:19.360 --> 01:31:21.780 I've got an x equals y over 2. 01:31:21.780 --> 01:31:23.590 What do I get? 01:31:23.590 --> 01:31:26.500 y squared over 2. 01:31:26.500 --> 01:31:28.720 Is this hard? 01:31:28.720 --> 01:31:31.631 It's very easy to make an algebra mistake on such 01:31:31.631 --> 01:31:32.672 a problem, unfortunately. 01:31:32.672 --> 01:31:37.024 I have y plus 1/2 plus what? 01:31:37.024 --> 01:31:40.856 What is 1/2 plus 1/8? 01:31:40.856 --> 01:31:41.820 STUDENT: 5/8. 01:31:41.820 --> 01:31:48.520 PROFESSOR: 5 over 8 with a minus y squared. 01:31:48.520 --> 01:31:52.400 01:31:52.400 --> 01:31:54.270 So hopefully I did this right. 01:31:54.270 --> 01:32:00.970 Now I'll go, OK, integral from 0 to 2 of all of this animal, y 01:32:00.970 --> 01:32:06.105 plus 1/2 minus 5 over 8, y squared. 01:32:06.105 --> 01:32:10.660 What happens if I don't get the right answer? 01:32:10.660 --> 01:32:12.940 Then I go back and check my work because I 01:32:12.940 --> 01:32:14.980 know I'm supposed to get 4/3. 01:32:14.980 --> 01:32:16.150 That was easy. 01:32:16.150 --> 01:32:23.310 So what is integral of this sausage, whatever it is? 01:32:23.310 --> 01:32:29.850 y squared over 2 plus y over 2 minus 5 over 8-- 01:32:29.850 --> 01:32:41.930 oh my god-- 5 over 8, y cubed over 3, between 2 up 01:32:41.930 --> 01:32:43.810 and 0 down. 01:32:43.810 --> 01:32:46.730 When I have 0 down, I plug y equals 0. 01:32:46.730 --> 01:32:47.800 It's a piece of cake. 01:32:47.800 --> 01:32:49.146 It's 0. 01:32:49.146 --> 01:32:51.720 So what matters is what I get when I plug 01:32:51.720 --> 01:32:53.560 in the value 2 instead of y. 01:32:53.560 --> 01:32:56.100 So what do I get? 01:32:56.100 --> 01:33:07.020 4 over 2 is 2, plus 2 over 2 is 1, minus 2 cubed, thank god. 01:33:07.020 --> 01:33:07.706 That's 8. 01:33:07.706 --> 01:33:10.740 8 simplifies with 8 minus 5/3. 01:33:10.740 --> 01:33:16.280 01:33:16.280 --> 01:33:23.650 So I got 9/3 minus 5/3, and I did it carefully. 01:33:23.650 --> 01:33:25.150 I did a good job. 01:33:25.150 --> 01:33:27.735 I got the same thing, 4/3. 01:33:27.735 --> 01:33:30.880 So no matter which method, the vertical strip 01:33:30.880 --> 01:33:34.390 or the horizontal strip method, I get the same thing. 01:33:34.390 --> 01:33:36.660 And of course, you'll always get the same answer 01:33:36.660 --> 01:33:42.800 because this is what the Fubini theorem extended to this case 01:33:42.800 --> 01:33:43.760 is telling you. 01:33:43.760 --> 01:33:46.762 It doesn't matter the order of integration. 01:33:46.762 --> 01:33:51.230 01:33:51.230 --> 01:33:54.616 I would advise you to go through the theory in the book. 01:33:54.616 --> 01:33:57.950 01:33:57.950 --> 01:34:02.474 They teach you more about area and volume on page 934. 01:34:02.474 --> 01:34:07.910 I'd like you to read that. 01:34:07.910 --> 01:34:11.580 And let's see what I want to do. 01:34:11.580 --> 01:34:14.030 Which one shall I do? 01:34:14.030 --> 01:34:17.950 There are a few examples that are worth it. 01:34:17.950 --> 01:34:21.250 01:34:21.250 --> 01:34:29.010 I'll pick the one that gives people the most trouble. 01:34:29.010 --> 01:34:29.670 How about that? 01:34:29.670 --> 01:34:33.460 I take the few examples that give people the most trouble. 01:34:33.460 --> 01:34:39.130 One example that popped up on almost each and every final 01:34:39.130 --> 01:34:44.490 in the past 13 years that involves changing 01:34:44.490 --> 01:34:46.462 the order of integration. 01:34:46.462 --> 01:34:57.308 01:34:57.308 --> 01:35:10.350 So example problem on changing the order of integration. 01:35:10.350 --> 01:35:14.650 01:35:14.650 --> 01:35:19.615 A very tricky, smart problem is the following. 01:35:19.615 --> 01:35:30.970 Evaluate integral from 0 to 1, integral from x to 1, 01:35:30.970 --> 01:35:34.208 e to the y squared dy dx. 01:35:34.208 --> 01:35:41.824 01:35:41.824 --> 01:35:43.740 I don't know if you've seen anything like that 01:35:43.740 --> 01:35:46.135 in AP Calculus or Calc 2. 01:35:46.135 --> 01:35:51.970 Maybe you have, in which case your professor probably told 01:35:51.970 --> 01:35:54.200 you that this is nasty. 01:35:54.200 --> 01:35:57.290 01:35:57.290 --> 01:35:59.515 You say, in what sense is it nasty? 01:35:59.515 --> 01:36:05.050 There is no expressible anti-derivative. 01:36:05.050 --> 01:36:21.600 So this cannot be expressed in terms of elementary functions 01:36:21.600 --> 01:36:22.100 explicitly. 01:36:22.100 --> 01:36:28.587 01:36:28.587 --> 01:36:31.420 It's not that there is no anti-derivative. 01:36:31.420 --> 01:36:34.760 There is an anti-derivative-- a whole family, actually-- 01:36:34.760 --> 01:36:38.810 but you cannot express them in terms of elementary functions. 01:36:38.810 --> 01:36:42.880 And actually, most functions are not so bad in real world, 01:36:42.880 --> 01:36:44.390 in real life. 01:36:44.390 --> 01:36:48.570 Now, could you compute, for example, integral from 1 to 3 01:36:48.570 --> 01:36:51.290 of e to the t squared dt? 01:36:51.290 --> 01:36:52.100 Yes. 01:36:52.100 --> 01:36:53.524 How do you do that? 01:36:53.524 --> 01:36:55.500 With a calculator. 01:36:55.500 --> 01:36:57.560 And what if you don't have one? 01:36:57.560 --> 01:36:59.060 You go to the lab over there. 01:36:59.060 --> 01:37:00.400 There is MATLAB. 01:37:00.400 --> 01:37:03.105 MATLAB will compute it for you. 01:37:03.105 --> 01:37:04.950 How does MATLAB know how to compute it 01:37:04.950 --> 01:37:07.976 if there is no way to express the anti-derivative 01:37:07.976 --> 01:37:12.035 and take the value of the anti-derivative between b 01:37:12.035 --> 01:37:16.080 and a, like in the fundamental theorem of calculus? 01:37:16.080 --> 01:37:20.705 Well, the calculator or the computer program is smart. 01:37:20.705 --> 01:37:24.700 He uses numerical analysis to approximate this type 01:37:24.700 --> 01:37:26.690 of integral. 01:37:26.690 --> 01:37:27.900 So he's fooling you. 01:37:27.900 --> 01:37:29.850 He's just playing smarty pants. 01:37:29.850 --> 01:37:33.160 He's smarter than you at this point. 01:37:33.160 --> 01:37:33.660 OK. 01:37:33.660 --> 01:37:38.305 So you cannot do this by hand, so this order of integration is 01:37:38.305 --> 01:37:38.805 fruitless. 01:37:38.805 --> 01:37:43.260 01:37:43.260 --> 01:37:47.373 And there are people who tried to do this on the final. 01:37:47.373 --> 01:37:48.831 Of course, they didn't get anywhere 01:37:48.831 --> 01:37:51.062 because they couldn't integrate it. 01:37:51.062 --> 01:37:55.920 The whole idea of this one is to-- some professors 01:37:55.920 --> 01:37:58.690 are so mean they don't even tell you, 01:37:58.690 --> 01:38:00.566 hint, change the order of integration 01:38:00.566 --> 01:38:03.160 because it may work the other way around. 01:38:03.160 --> 01:38:06.260 They just give it to you, and then people can spend an hour 01:38:06.260 --> 01:38:08.420 and they don't get anywhere. 01:38:08.420 --> 01:38:12.145 If you want to be mean to a student, that's what you do. 01:38:12.145 --> 01:38:17.305 So I will tell you that one needs 01:38:17.305 --> 01:38:20.020 to change the order of integration for this. 01:38:20.020 --> 01:38:21.100 This is the function. 01:38:21.100 --> 01:38:26.040 We keep the function, but let's see what happens if you draw. 01:38:26.040 --> 01:38:31.120 The domain will be x between 0 and 1. 01:38:31.120 --> 01:38:33.920 This is your x value. 01:38:33.920 --> 01:38:37.350 y will be between x and 1. 01:38:37.350 --> 01:38:39.990 So it's like you have a square. 01:38:39.990 --> 01:38:43.690 y equals x is your diagonal of the square. 01:38:43.690 --> 01:38:49.420 And you go from-- more colors, please. 01:38:49.420 --> 01:38:54.780 You go from y equals x on the bottom and y equals 1 on top. 01:38:54.780 --> 01:38:57.490 And so the domain is this beautiful triangle 01:38:57.490 --> 01:39:02.710 that I make all in line with vertical strips. 01:39:02.710 --> 01:39:06.320 This is what it means, vertical strips. 01:39:06.320 --> 01:39:11.916 But if I do horizontal strips, I have to change the color, blue. 01:39:11.916 --> 01:39:14.545 And for horizontal strips, I'm going 01:39:14.545 --> 01:39:16.696 to have a different problem. 01:39:16.696 --> 01:39:20.220 Integral, integral dx dy. 01:39:20.220 --> 01:39:23.110 And I just hope to god that what I'm going to get 01:39:23.110 --> 01:39:27.000 is doable because if not, then I'm in trouble. 01:39:27.000 --> 01:39:30.280 So help me on this one. 01:39:30.280 --> 01:39:33.680 If y is between what and what? 01:39:33.680 --> 01:39:35.175 It's a square. 01:39:35.175 --> 01:39:38.612 It's a square, so this will be the same, 0 to 1, right? 01:39:38.612 --> 01:39:39.416 STUDENT: Yep. 01:39:39.416 --> 01:39:40.310 PROFESSOR: But Mr. X? 01:39:40.310 --> 01:39:41.715 How about Mr. X? 01:39:41.715 --> 01:39:45.930 STUDENT: And then it will be between 1 and y. 01:39:45.930 --> 01:39:49.410 PROFESSOR: Between-- Mr. X is this guy. 01:39:49.410 --> 01:39:51.880 And he doesn't go between 1. 01:39:51.880 --> 01:39:54.844 He goes between the sea level, which is 01:39:54.844 --> 01:40:02.600 x equals 0, to x equals what? 01:40:02.600 --> 01:40:03.490 STUDENT: [INAUDIBLE]. 01:40:03.490 --> 01:40:04.772 PROFESSOR: Right? 01:40:04.772 --> 01:40:10.160 So from x equals 0 through x equals y. 01:40:10.160 --> 01:40:15.430 And you have the same individual e to the y squared that before 01:40:15.430 --> 01:40:17.220 went on your nerves. 01:40:17.220 --> 01:40:20.130 Now he's not so bad, actually. 01:40:20.130 --> 01:40:21.786 Why is he not so bad? 01:40:21.786 --> 01:40:24.720 Look what happens in the first parentheses. 01:40:24.720 --> 01:40:27.280 This is so beautiful that it's something 01:40:27.280 --> 01:40:29.100 you didn't even hope for. 01:40:29.100 --> 01:40:34.450 So we copy and paste it from 0 to 1 dy. 01:40:34.450 --> 01:40:38.330 These guys stay outside and they wait. 01:40:38.330 --> 01:40:40.370 Inside, it's our business what we do. 01:40:40.370 --> 01:40:44.790 So Mr. X is independent from e to the y squared. 01:40:44.790 --> 01:40:46.830 So e to the y squared pulls out. 01:40:46.830 --> 01:40:48.342 He's a constant. 01:40:48.342 --> 01:40:53.390 And you have integral of 1 dx between 0 and y. 01:40:53.390 --> 01:40:56.670 How much is that? 01:40:56.670 --> 01:40:57.920 1. 01:40:57.920 --> 01:41:03.550 x between x equals 0 and x equals y. 01:41:03.550 --> 01:41:05.001 So it's y. 01:41:05.001 --> 01:41:05.875 So I'm being serious. 01:41:05.875 --> 01:41:07.600 So I should have said y. 01:41:07.600 --> 01:41:11.830 01:41:11.830 --> 01:41:21.050 Now, if your professor would have given you, in Calc 2, 01:41:21.050 --> 01:41:25.050 this, how would you have done it? 01:41:25.050 --> 01:41:26.927 STUDENT: U-substitution. 01:41:26.927 --> 01:41:28.010 PROFESSOR: U-substitution. 01:41:28.010 --> 01:41:28.776 Excellent. 01:41:28.776 --> 01:41:32.130 What kind of u-substitution [INAUDIBLE]? 01:41:32.130 --> 01:41:35.100 STUDENT: y squared equals u. 01:41:35.100 --> 01:41:40.410 PROFESSOR: y squared equals u, du equals 2y dy. 01:41:40.410 --> 01:41:44.400 So y dy together. 01:41:44.400 --> 01:41:45.973 They stick together. 01:41:45.973 --> 01:41:47.332 They stick together. 01:41:47.332 --> 01:41:49.560 They attract each other as magnets. 01:41:49.560 --> 01:41:56.932 So y dy is going to be 1/2 du-- 1/2 pulls out-- 01:41:56.932 --> 01:42:00.190 integral e to the u du. 01:42:00.190 --> 01:42:00.690 Attention. 01:42:00.690 --> 01:42:03.430 When y is moving between 0 and 1, 01:42:03.430 --> 01:42:06.211 u is moving also between 0 and 1. 01:42:06.211 --> 01:42:11.990 So it really should be a piece of cake. 01:42:11.990 --> 01:42:13.495 Are you guys with me? 01:42:13.495 --> 01:42:15.880 Do you understand what I did? 01:42:15.880 --> 01:42:18.970 Do you understand the words coming out of my mouth? 01:42:18.970 --> 01:42:24.415 01:42:24.415 --> 01:42:25.405 It's easy. 01:42:25.405 --> 01:42:29.490 01:42:29.490 --> 01:42:29.990 Good. 01:42:29.990 --> 01:42:35.460 So what is integral of e to the u du? 01:42:35.460 --> 01:42:39.220 e to the u between 1 up and 0 down. 01:42:39.220 --> 01:42:43.983 So e to the u de to the 1 minus e to the 0 over 2. 01:42:43.983 --> 01:42:47.930 01:42:47.930 --> 01:42:51.045 That is e minus 1 over 2. 01:42:51.045 --> 01:42:54.320 01:42:54.320 --> 01:42:59.215 I could not have solved this if I tried it by integration 01:42:59.215 --> 01:43:02.670 with y first and then x. 01:43:02.670 --> 01:43:04.600 The only way I could have done this 01:43:04.600 --> 01:43:07.700 is by changing the order of integration. 01:43:07.700 --> 01:43:11.556 So how many times have I seen this in the past 12 years 01:43:11.556 --> 01:43:12.510 on the final? 01:43:12.510 --> 01:43:15.228 At least six times. 01:43:15.228 --> 01:43:18.220 It's a problem that could be a little bit 01:43:18.220 --> 01:43:21.150 hard if the student has never seen it before 01:43:21.150 --> 01:43:23.810 and doesn't know what to do [? at that point. ?] 01:43:23.810 --> 01:43:27.075 Let's do a few more in the same category. 01:43:27.075 --> 01:43:36.648 01:43:36.648 --> 01:43:37.612 STUDENT: Professor? 01:43:37.612 --> 01:43:38.266 PROFESSOR: Yes? 01:43:38.266 --> 01:43:40.890 STUDENT: Where did this shape-- where did this graph come from? 01:43:40.890 --> 01:43:43.720 Were we just saying it was with the same-- 01:43:43.720 --> 01:43:44.640 PROFESSOR: OK. 01:43:44.640 --> 01:43:46.750 I read it from here. 01:43:46.750 --> 01:43:50.460 So this and that are the key. 01:43:50.460 --> 01:43:55.160 This is telling me x is between 0 and 1, and at the same, 01:43:55.160 --> 01:43:59.010 time y is between x and 1. 01:43:59.010 --> 01:44:02.510 And when I read this information on the graph, 01:44:02.510 --> 01:44:05.570 I say, well, x is between 0 and 1. 01:44:05.570 --> 01:44:09.070 Mr. Y has the freedom to go between the first bisector, 01:44:09.070 --> 01:44:14.070 which is that, and the cap, his cap, y equals 1. 01:44:14.070 --> 01:44:17.450 So that's how I got to the line strips. 01:44:17.450 --> 01:44:21.265 And from the line strips, I said that I need horizontal strips. 01:44:21.265 --> 01:44:23.970 So I changed the color and I said 01:44:23.970 --> 01:44:27.692 the blue strips go between x. 01:44:27.692 --> 01:44:31.856 x will be x equals 0 and x equals y. 01:44:31.856 --> 01:44:36.910 And then y between 0 and 1, just the same. 01:44:36.910 --> 01:44:38.080 It's a little bit tricky. 01:44:38.080 --> 01:44:42.360 That's why I want to do one or two more problems like that, 01:44:42.360 --> 01:44:46.840 because I know that I remember 20-something years ago, 01:44:46.840 --> 01:44:52.500 I myself needed a little bit of time understanding 01:44:52.500 --> 01:44:56.710 the meaning of reversing the order of integration. 01:44:56.710 --> 01:44:58.970 STUDENT: Does it matter which way you put it? 01:44:58.970 --> 01:45:02.410 PROFESSOR: In this case, it's important that you do reverse. 01:45:02.410 --> 01:45:05.890 But in general, it's doable both ways. 01:45:05.890 --> 01:45:10.018 I mean, in the other problems I'm going to give you today, 01:45:10.018 --> 01:45:11.890 you should be able to do either way. 01:45:11.890 --> 01:45:19.134 So I'm looking for a problem that you could eventually 01:45:19.134 --> 01:45:20.610 do another one. 01:45:20.610 --> 01:45:25.530 01:45:25.530 --> 01:45:27.990 We don't have so many. 01:45:27.990 --> 01:45:31.716 I'm going to go ahead and look into the homework. 01:45:31.716 --> 01:45:32.215 Yeah. 01:45:32.215 --> 01:45:34.835 01:45:34.835 --> 01:45:41.395 So it says, you have this integral, 01:45:41.395 --> 01:45:44.170 the integral from 0 to 4 of the integral 01:45:44.170 --> 01:45:49.880 from x squared to 4y dy dx. 01:45:49.880 --> 01:45:55.990 Draw, compute, and also compute with reversing 01:45:55.990 --> 01:45:58.990 the order of integration to check your work. 01:45:58.990 --> 01:46:01.160 When I say that, it sounds horrible. 01:46:01.160 --> 01:46:04.280 But in reality, the more you work on 01:46:04.280 --> 01:46:08.116 that one, the more familiar you're going to feel. 01:46:08.116 --> 01:46:10.500 So what did I just say? 01:46:10.500 --> 01:46:12.740 Problem number 26. 01:46:12.740 --> 01:46:18.200 You have integral from 0 to 4, integral 01:46:18.200 --> 01:46:24.560 from x squared to 4x dy dx. 01:46:24.560 --> 01:46:27.400 01:46:27.400 --> 01:46:31.310 Interpret geometrically, whatever that means, 01:46:31.310 --> 01:46:35.255 and then compute the integral in two ways, 01:46:35.255 --> 01:46:37.867 with this given order integration, which 01:46:37.867 --> 01:46:40.005 is what kind of strips, guys? 01:46:40.005 --> 01:46:41.910 Vertical strips. 01:46:41.910 --> 01:46:45.010 Or reversing the order of integration. 01:46:45.010 --> 01:46:50.222 And check that the answer is the same just to check your work. 01:46:50.222 --> 01:46:51.960 STUDENT: So first-- 01:46:51.960 --> 01:46:53.100 PROFESSOR: First you draw. 01:46:53.100 --> 01:46:55.890 First you draw because if you don't draw, 01:46:55.890 --> 01:47:00.490 you don't understand what the problem is about. 01:47:00.490 --> 01:47:01.730 And you say, wait a minute. 01:47:01.730 --> 01:47:05.290 But couldn't I go ahead and do it without drawing? 01:47:05.290 --> 01:47:08.400 Yeah, but you're not going to get too far. 01:47:08.400 --> 01:47:11.870 So let's see what kind of problem you have. 01:47:11.870 --> 01:47:13.210 y and x. 01:47:13.210 --> 01:47:16.800 y equals x squared is a what? 01:47:16.800 --> 01:47:18.758 It's a pa-- 01:47:18.758 --> 01:47:19.746 STUDENT: Parabola. 01:47:19.746 --> 01:47:20.734 PROFESSOR: Parabola. 01:47:20.734 --> 01:47:24.192 And this parabola should be nice and sassy. 01:47:24.192 --> 01:47:25.680 Is it fat enough? 01:47:25.680 --> 01:47:27.470 I think it is. 01:47:27.470 --> 01:47:34.140 And the other one will be 4x, y equals 4x. 01:47:34.140 --> 01:47:36.070 What does that look like? 01:47:36.070 --> 01:47:39.340 It looks like a line passing through the origin that 01:47:39.340 --> 01:47:42.700 has slope 4, so the slope is really high. 01:47:42.700 --> 01:47:43.690 STUDENT: Just straight. 01:47:43.690 --> 01:47:48.150 01:47:48.150 --> 01:47:51.870 PROFESSOR: y equals 4x versus y equals x squared. 01:47:51.870 --> 01:47:53.690 Now, do they meet? 01:47:53.690 --> 01:47:57.343 01:47:57.343 --> 01:47:57.884 STUDENT: Yes. 01:47:57.884 --> 01:47:58.508 PROFESSOR: Yes. 01:47:58.508 --> 01:47:59.750 Exactly where do they meet? 01:47:59.750 --> 01:48:00.300 Exactly here. 01:48:00.300 --> 01:48:00.800 STUDENT: 4. 01:48:00.800 --> 01:48:04.190 PROFESSOR: So 4x equals x squared, where do they meet? 01:48:04.190 --> 01:48:06.930 01:48:06.930 --> 01:48:13.480 They meet at-- it has two possible roots. 01:48:13.480 --> 01:48:18.420 One is x equals 0, which is here, 01:48:18.420 --> 01:48:21.270 and one is x equals 4, which is here. 01:48:21.270 --> 01:48:26.720 So really, my graph looks just the way it should look, 01:48:26.720 --> 01:48:29.450 only my parabola is a little bit too fat. 01:48:29.450 --> 01:48:33.840 01:48:33.840 --> 01:48:44.100 This is the point of coordinates 4 and 16. 01:48:44.100 --> 01:48:46.410 Are you guys with me? 01:48:46.410 --> 01:48:52.350 And Mr. X is moving between 0 and 4. 01:48:52.350 --> 01:48:56.910 This is the maximum level x can get. 01:48:56.910 --> 01:49:01.730 And where he stops here at 4, a miracle happens. 01:49:01.730 --> 01:49:06.680 The two curves intersect each other exactly at that point. 01:49:06.680 --> 01:49:11.904 So this looks like a leaf, a slice of orange. 01:49:11.904 --> 01:49:12.404 Oh my god. 01:49:12.404 --> 01:49:12.945 I don't know. 01:49:12.945 --> 01:49:17.825 I'm already hungry so I cannot wait to get out of here. 01:49:17.825 --> 01:49:20.675 I bet you're hungry as well. 01:49:20.675 --> 01:49:24.000 Let's do this problem both ways and then go 01:49:24.000 --> 01:49:26.537 home or to have something to eat. 01:49:26.537 --> 01:49:31.958 How are you going to advise me to solve it first? 01:49:31.958 --> 01:49:34.220 It's already set up to be solved. 01:49:34.220 --> 01:49:35.476 So it's vertical strips. 01:49:35.476 --> 01:49:37.910 And I will say integral from 0 to 4, 01:49:37.910 --> 01:49:40.790 copy and paste the outer part. 01:49:40.790 --> 01:49:46.090 Take the inner part, and do the inner part because it's easy. 01:49:46.090 --> 01:49:50.402 And if it's easy, you tell me how I'm going to do it. 01:49:50.402 --> 01:49:53.556 Integral of 1 dy is y. 01:49:53.556 --> 01:49:58.820 y measured at 4x is 4x, and y measured at x squared 01:49:58.820 --> 01:50:01.120 is x squared. 01:50:01.120 --> 01:50:01.790 Oh thank god. 01:50:01.790 --> 01:50:05.728 This is so beautiful and so easy. 01:50:05.728 --> 01:50:08.650 Let's integrate again. 01:50:08.650 --> 01:50:16.370 4 x squared over 2 times x cubed over 3 between x equals 0 down 01:50:16.370 --> 01:50:17.920 and x equals 4 up. 01:50:17.920 --> 01:50:22.190 01:50:22.190 --> 01:50:23.810 What do I get? 01:50:23.810 --> 01:50:30.050 I get 4 cubed over 2 minus 4 cubed over 3. 01:50:30.050 --> 01:50:31.840 This 4 cubed is an obsession. 01:50:31.840 --> 01:50:33.814 Kick him out. 01:50:33.814 --> 01:50:35.742 1/2 minus 1/3. 01:50:35.742 --> 01:50:39.610 01:50:39.610 --> 01:50:41.450 How much is 1/2 minus 1/3? 01:50:41.450 --> 01:50:42.430 My son knows that. 01:50:42.430 --> 01:50:43.577 STUDENT: 1/6. 01:50:43.577 --> 01:50:44.160 PROFESSOR: OK. 01:50:44.160 --> 01:50:46.060 1/6, yes. 01:50:46.060 --> 01:50:48.710 So we simply take it. 01:50:48.710 --> 01:50:49.965 We can leave it like that. 01:50:49.965 --> 01:50:55.530 If you leave it like that on the exam, I don't mind at all. 01:50:55.530 --> 01:50:58.681 But you could always put 64 over 6 and simplify it. 01:50:58.681 --> 01:51:01.507 01:51:01.507 --> 01:51:03.391 Are you guys with me? 01:51:03.391 --> 01:51:07.070 You can simplify it and get what? 01:51:07.070 --> 01:51:08.202 32 over 3. 01:51:08.202 --> 01:51:10.980 01:51:10.980 --> 01:51:12.530 Don't give me decimals. 01:51:12.530 --> 01:51:14.634 I'm not impressed. 01:51:14.634 --> 01:51:16.383 You're not supposed to use the calculator. 01:51:16.383 --> 01:51:21.295 You are supposed to leave this is exact fraction 01:51:21.295 --> 01:51:24.850 form like that, irreducible. 01:51:24.850 --> 01:51:26.320 Let's do it the other way around, 01:51:26.320 --> 01:51:30.020 and that will be the last thing we do. 01:51:30.020 --> 01:51:34.020 The other way around means I'll take another color. 01:51:34.020 --> 01:51:36.980 I'll do the horizontal stripes. 01:51:36.980 --> 01:51:40.010 01:51:40.010 --> 01:51:44.110 And I will have to rewrite the meaning of these two 01:51:44.110 --> 01:51:49.530 branches of functions with x expressed in terms of y. 01:51:49.530 --> 01:51:51.710 That's the only thing I need to do, right? 01:51:51.710 --> 01:51:55.910 So what is this? 01:51:55.910 --> 01:51:59.210 If y is x squared, what is x? 01:51:59.210 --> 01:52:00.150 STUDENT: Root y. 01:52:00.150 --> 01:52:03.735 PROFESSOR: The inverse function. x will be root of y. 01:52:03.735 --> 01:52:06.000 You said very well. 01:52:06.000 --> 01:52:07.360 So I have to write. 01:52:07.360 --> 01:52:10.470 In [INAUDIBLE], I have what I need 01:52:10.470 --> 01:52:13.082 to have for the line horizontal strip method. 01:52:13.082 --> 01:52:16.060 01:52:16.060 --> 01:52:19.655 And then for the other one, x is going to be y over 4. 01:52:19.655 --> 01:52:22.610 01:52:22.610 --> 01:52:23.600 So what do I do? 01:52:23.600 --> 01:52:32.382 So integral, integral, a 1 that was here hidden, 01:52:32.382 --> 01:52:35.826 but I'll put it because that's the integral. 01:52:35.826 --> 01:52:38.560 And then I go dx dy. 01:52:38.560 --> 01:52:44.990 All I have to care about is the endpoints of the integration. 01:52:44.990 --> 01:52:48.240 Now, pay attention a little bit because Mr. Y is not 01:52:48.240 --> 01:52:49.660 between 0 and 4. 01:52:49.660 --> 01:52:53.210 I had very good students under stress 01:52:53.210 --> 01:52:55.620 in the final putting 0 and 4. 01:52:55.620 --> 01:52:56.760 Don't do that. 01:52:56.760 --> 01:52:59.245 So pay attention to the limits of integration. 01:52:59.245 --> 01:53:01.030 What are the limits? 01:53:01.030 --> 01:53:01.750 0 and-- 01:53:01.750 --> 01:53:02.407 STUDENT: 16. 01:53:02.407 --> 01:53:02.990 PROFESSOR: 16. 01:53:02.990 --> 01:53:04.970 Very good. 01:53:04.970 --> 01:53:09.610 And x will be between root y-- well, which one is on top? 01:53:09.610 --> 01:53:11.610 Which one is on the bottom? 01:53:11.610 --> 01:53:17.110 Because if I move my head, I'll say that's on top 01:53:17.110 --> 01:53:18.610 and that's on the bottom. 01:53:18.610 --> 01:53:22.210 STUDENT: The right side is always on the top. 01:53:22.210 --> 01:53:25.730 PROFESSOR: So the one that looks higher is this one. 01:53:25.730 --> 01:53:29.210 This is more than that in this frame. 01:53:29.210 --> 01:53:36.583 So square of y is on top and y over 4 is on the bottom. 01:53:36.583 --> 01:53:38.978 I should get the same answer. 01:53:38.978 --> 01:53:40.420 If I don't, then I'm in trouble. 01:53:40.420 --> 01:53:43.216 So what do I get? 01:53:43.216 --> 01:53:49.084 Integral from 0 to 16. 01:53:49.084 --> 01:53:51.500 Tonight, when I go home, I'm going 01:53:51.500 --> 01:53:57.024 to cook up the homework for 12.1 and 12.1 at least. 01:53:57.024 --> 01:53:59.175 I'll put some problems similar to that 01:53:59.175 --> 01:54:02.590 because I want to emphasize the same type of problem 01:54:02.590 --> 01:54:05.010 in at least two or three applications for the homework 01:54:05.010 --> 01:54:07.200 for the midterm. 01:54:07.200 --> 01:54:10.575 And maybe one like that will be on the final as well. 01:54:10.575 --> 01:54:13.285 It's very important for you to understand how, 01:54:13.285 --> 01:54:15.273 with this kind of domain, you reverse 01:54:15.273 --> 01:54:16.834 the order of integration. 01:54:16.834 --> 01:54:19.760 Who's helping me here? 01:54:19.760 --> 01:54:22.210 Root y. 01:54:22.210 --> 01:54:26.070 What is root y when-- y to the 1/2. 01:54:26.070 --> 01:54:28.070 I need to integrate. 01:54:28.070 --> 01:54:33.552 So I need minus y over 4 and dy. 01:54:33.552 --> 01:54:39.030 01:54:39.030 --> 01:54:42.188 Can you help me integrate? 01:54:42.188 --> 01:54:44.180 STUDENT: [INAUDIBLE]. 01:54:44.180 --> 01:54:49.830 PROFESSOR: 2/3 y to the 3/2 minus-- 01:54:49.830 --> 01:54:51.120 STUDENT: y squared. 01:54:51.120 --> 01:54:56.480 PROFESSOR: y squared over 8, y equals 0 01:54:56.480 --> 01:54:58.490 on the bottom, piece of cake. 01:54:58.490 --> 01:55:00.220 That will give me 0. 01:55:00.220 --> 01:55:00.966 I'm so happy. 01:55:00.966 --> 01:55:04.560 And y equals 16 on top. 01:55:04.560 --> 01:55:09.830 So for 16, I have 2/3. 01:55:09.830 --> 01:55:12.210 And who's telling me what else? 01:55:12.210 --> 01:55:13.170 STUDENT: 64. 01:55:13.170 --> 01:55:13.910 PROFESSOR: 64. 01:55:13.910 --> 01:55:14.480 4 cubed. 01:55:14.480 --> 01:55:22.700 I can leave it 4 cubed if I want to minus another-- well here, 01:55:22.700 --> 01:55:24.740 I have to pay attention. 01:55:24.740 --> 01:55:27.350 So I have 16 here. 01:55:27.350 --> 01:55:31.390 I got square root of 16, which is 4, cubed. 01:55:31.390 --> 01:55:38.720 Here, I put minus 4 squared, which was there. 01:55:38.720 --> 01:55:40.635 How do you want me to do this simplification? 01:55:40.635 --> 01:55:41.910 STUDENT: [INAUDIBLE]. 01:55:41.910 --> 01:55:44.840 PROFESSOR: I can do 4 to the fourth. 01:55:44.840 --> 01:55:47.190 Are you guys with me? 01:55:47.190 --> 01:55:52.400 I can put, like you prefer, 16 squared over 8. 01:55:52.400 --> 01:55:57.900 01:55:57.900 --> 01:55:59.390 Is it the same answer? 01:55:59.390 --> 01:56:00.150 I don't know. 01:56:00.150 --> 01:56:02.410 Let's see. 01:56:02.410 --> 01:56:09.050 This is really 4 to the 4, so I have 4 times 4 cubed. 01:56:09.050 --> 01:56:19.788 4 cubed gets out and I have 2/3 minus 1/2. 01:56:19.788 --> 01:56:24.060 01:56:24.060 --> 01:56:27.640 And how much is that? 01:56:27.640 --> 01:56:28.765 Again 1/6. 01:56:28.765 --> 01:56:30.670 Are you guys with me? 01:56:30.670 --> 01:56:31.620 1/6. 01:56:31.620 --> 01:56:36.860 So again, I get 4 cubed over 6, so I'm done. 01:56:36.860 --> 01:56:40.420 4 cubed over 6 equals 32 over 3. 01:56:40.420 --> 01:56:42.960 I am happy that I checked my work 01:56:42.960 --> 01:56:44.420 through two different methods. 01:56:44.420 --> 01:56:45.710 I got the same answer. 01:56:45.710 --> 01:56:49.220 01:56:49.220 --> 01:56:51.500 Now, let me tell you something. 01:56:51.500 --> 01:56:55.220 There were also times when on the midterm 01:56:55.220 --> 01:56:59.800 or on the final, due to lack of time and everything, 01:56:59.800 --> 01:57:02.930 we put the following kind of problem. 01:57:02.930 --> 01:57:11.290 Without solving this integral-- without solving-- indicate 01:57:11.290 --> 01:57:16.290 the corresponding integral with the order reversed. 01:57:16.290 --> 01:57:19.680 So all you have to do-- don't do that. 01:57:19.680 --> 01:57:24.715 Just from here, write this and stop. 01:57:24.715 --> 01:57:27.579 Don't waste your time. 01:57:27.579 --> 01:57:29.620 If you do the whole thing, it's going to take you 01:57:29.620 --> 01:57:30.585 10 minutes, 15 minutes. 01:57:30.585 --> 01:57:33.902 If you do just reversing the order of integration, 01:57:33.902 --> 01:57:38.265 I don't know what it takes, a minute and a half, two minutes. 01:57:38.265 --> 01:57:42.250 So in order to save time, at times, 01:57:42.250 --> 01:57:46.166 we gave you just don't solve the problem. reverse 01:57:46.166 --> 01:57:47.618 the order of integration. 01:57:47.618 --> 01:57:54.400 01:57:54.400 --> 01:57:55.750 One last one. 01:57:55.750 --> 01:57:58.230 One last one. 01:57:58.230 --> 01:57:59.730 But I don't want to finish it. 01:57:59.730 --> 01:58:03.226 I want to give you the answer at home, 01:58:03.226 --> 01:58:05.681 or maybe you can finish it. 01:58:05.681 --> 01:58:07.645 It should be shorter. 01:58:07.645 --> 01:58:13.537 You have a circular parabola, but only the first quadrant. 01:58:13.537 --> 01:58:16.490 01:58:16.490 --> 01:58:19.253 So x is positive. 01:58:19.253 --> 01:58:20.260 STUDENT: Question. 01:58:20.260 --> 01:58:21.260 PROFESSOR: I don't know. 01:58:21.260 --> 01:58:22.480 I have to find it. 01:58:22.480 --> 01:58:23.840 Find the volume. 01:58:23.840 --> 01:58:25.470 Example 4, page 934. 01:58:25.470 --> 01:58:28.560 Find the volume of the solid bound 01:58:28.560 --> 01:58:32.690 in the above-- this is a little tricky-- by the plane z 01:58:32.690 --> 01:58:38.010 equals y and below in the xy plane 01:58:38.010 --> 01:58:42.460 by the part of the disk in the first quadrant. 01:58:42.460 --> 01:58:47.940 So z equals y means this is your f of x and y. 01:58:47.940 --> 01:58:50.550 So they gave it to you. 01:58:50.550 --> 01:58:54.430 But then they say, but also, in the xy plane, 01:58:54.430 --> 01:59:00.120 you have to have the part of the disk in the first quadrant. 01:59:00.120 --> 01:59:01.790 This is not so easy. 01:59:01.790 --> 01:59:04.637 They draw it for you to make your life easier. 01:59:04.637 --> 01:59:08.050 The first quadrant is that. 01:59:08.050 --> 01:59:13.600 How do you write the unit circle, x squared equals 1, 01:59:13.600 --> 01:59:16.720 x squared plus y squared less than or equal to 1, 01:59:16.720 --> 01:59:19.350 and x and y are both positive. 01:59:19.350 --> 01:59:21.123 This is the first quadrant. 01:59:21.123 --> 01:59:22.512 How do you compute? 01:59:22.512 --> 01:59:26.680 So they say compute the volume, and I say just 01:59:26.680 --> 01:59:27.850 set up the volume. 01:59:27.850 --> 01:59:30.064 Forget about computing it. 01:59:30.064 --> 01:59:33.473 I could put it in the midterm just like that. 01:59:33.473 --> 01:59:36.380 Set up an integral without solving it 01:59:36.380 --> 01:59:46.110 that indicates the volume under z equals f of xy-- that's 01:59:46.110 --> 01:59:50.980 the geography of z-- and above a certain domain in plane, 01:59:50.980 --> 01:59:55.510 above D in plane. 01:59:55.510 --> 01:59:58.090 So you have, OK, what this should teach you? 01:59:58.090 --> 02:00:08.660 Should teach you that double integral over d f of xy da 02:00:08.660 --> 02:00:10.990 can be solved. 02:00:10.990 --> 02:00:12.550 Do I ask to be solved? 02:00:12.550 --> 02:00:13.720 No. 02:00:13.720 --> 02:00:14.445 Why? 02:00:14.445 --> 02:00:18.100 Because you can finish it later, finish at home. 02:00:18.100 --> 02:00:27.230 Or maybe, I don't even want you to compute on the final. 02:00:27.230 --> 02:00:29.280 So how do we do that? 02:00:29.280 --> 02:00:32.970 f is y. 02:00:32.970 --> 02:00:36.520 Would I be able to choose whichever order integration I 02:00:36.520 --> 02:00:38.380 want? 02:00:38.380 --> 02:00:40.000 It shouldn't matter which order. 02:00:40.000 --> 02:00:43.020 It should be more or less the same. 02:00:43.020 --> 02:00:44.695 What if I do dy dx? 02:00:44.695 --> 02:00:47.630 02:00:47.630 --> 02:00:52.248 Then I have to do the Fubini. 02:00:52.248 --> 02:00:54.230 But it's not a rectangular domain. 02:00:54.230 --> 02:00:54.730 Aha. 02:00:54.730 --> 02:00:56.630 So Magdalena, be a little bit careful 02:00:56.630 --> 02:01:00.410 because this is going to be two finite numbers, 02:01:00.410 --> 02:01:01.865 but these are functions. 02:01:01.865 --> 02:01:04.340 STUDENT: It will be an x function. 02:01:04.340 --> 02:01:08.068 PROFESSOR: So the x is between 0 and 1, 02:01:08.068 --> 02:01:10.020 and that's going to be z. 02:01:10.020 --> 02:01:11.484 You do vertical strips. 02:01:11.484 --> 02:01:13.924 That's a piece of cake. 02:01:13.924 --> 02:01:17.890 But if you do the vertical strips, 02:01:17.890 --> 02:01:21.980 you have to pay attention to the endpoints for x and y, 02:01:21.980 --> 02:01:23.426 and one is easy. 02:01:23.426 --> 02:01:24.487 Which one is trivial? 02:01:24.487 --> 02:01:25.070 STUDENT: Zero. 02:01:25.070 --> 02:01:26.403 PROFESSOR: The bottom one, zero. 02:01:26.403 --> 02:01:29.350 The one that's nontrivial is the upper one. 02:01:29.350 --> 02:01:31.290 STUDENT: There will be 1 minus-- 02:01:31.290 --> 02:01:33.595 STUDENT: Square root of 1 minus y squared. 02:01:33.595 --> 02:01:34.470 PROFESSOR: Very good. 02:01:34.470 --> 02:01:36.210 Square root of 1 minus y squared. 02:01:36.210 --> 02:01:41.110 02:01:41.110 --> 02:01:46.640 So if I were to go one more step further without solving this, 02:01:46.640 --> 02:01:51.400 I'm going to ask you, could this be solved by hand? 02:01:51.400 --> 02:01:57.890 Well, so you have it in the book-- 02:01:57.890 --> 02:02:00.390 STUDENT: Professor, should be a [INAUDIBLE] minus x squared? 02:02:00.390 --> 02:02:03.009 02:02:03.009 --> 02:02:03.800 PROFESSOR: Oh yeah. 02:02:03.800 --> 02:02:04.860 1 minus x squared. 02:02:04.860 --> 02:02:06.590 Excuse me. 02:02:06.590 --> 02:02:08.190 Didn't I write it? 02:02:08.190 --> 02:02:11.996 Yeah, here I should have written y equals square root of 1 02:02:11.996 --> 02:02:14.300 minus x squared. 02:02:14.300 --> 02:02:21.263 So when you do it-- thank you so much-- you go integrate, 02:02:21.263 --> 02:02:26.790 and you have y squared over 2. 02:02:26.790 --> 02:02:29.605 And you evaluate between y equals 0 02:02:29.605 --> 02:02:33.466 and y equals square root 1 minus x squared, 02:02:33.466 --> 02:02:34.918 and then you do the [INAUDIBLE]. 02:02:34.918 --> 02:02:41.220 02:02:41.220 --> 02:02:44.764 In the book, they do it differently. 02:02:44.764 --> 02:02:50.328 They do it with respect to dx and dy and integrate. 02:02:50.328 --> 02:02:52.798 But it doesn't matter how you do it. 02:02:52.798 --> 02:02:54.774 You should get the same answer. 02:02:54.774 --> 02:02:58.330 02:02:58.330 --> 02:03:00.135 All right? 02:03:00.135 --> 02:03:01.065 [INAUDIBLE]? 02:03:01.065 --> 02:03:03.855 STUDENT: [INAUDIBLE] in that way, 02:03:03.855 --> 02:03:06.820 doesn't the square root work out better because there's already 02:03:06.820 --> 02:03:07.690 a y there? 02:03:07.690 --> 02:03:09.036 PROFESSOR: In the other case-- 02:03:09.036 --> 02:03:10.900 STUDENT: Doing dy dx. 02:03:10.900 --> 02:03:12.720 PROFESSOR: Yeah, in the other way, 02:03:12.720 --> 02:03:14.340 it works a little bit differently. 02:03:14.340 --> 02:03:17.420 You can do u-substitution, I think. 02:03:17.420 --> 02:03:20.460 So if you do it the other way, it will be what? 02:03:20.460 --> 02:03:24.430 Integral from 0 to 1, integral form 0 02:03:24.430 --> 02:03:32.432 to square root of 1 minus y squared, y dx dy. 02:03:32.432 --> 02:03:35.166 And what do you do in this case? 02:03:35.166 --> 02:03:37.430 You have integral from 0 to 1. 02:03:37.430 --> 02:03:42.770 Integral of y dx is going to be y is a constant. 02:03:42.770 --> 02:03:48.860 x between the two values will be simply 1 minus y squared dy. 02:03:48.860 --> 02:03:49.920 So you're right. 02:03:49.920 --> 02:03:52.580 Matthew saw that, because he's a prophet, 02:03:52.580 --> 02:03:56.090 and he could see two steps ahead. 02:03:56.090 --> 02:03:57.850 This is very nice what you observed. 02:03:57.850 --> 02:03:59.100 What do you do? 02:03:59.100 --> 02:04:02.590 You take a u-substitution when you go home. 02:04:02.590 --> 02:04:06.010 You get u equals 1 minus y squared. 02:04:06.010 --> 02:04:12.798 du will be minus 2y dy, and you go on. 02:04:12.798 --> 02:04:17.172 So in the book, we got 1/3. 02:04:17.172 --> 02:04:19.602 If you continue with this method, 02:04:19.602 --> 02:04:20.907 I think it's the same answer. 02:04:20.907 --> 02:04:21.490 STUDENT: Yeah. 02:04:21.490 --> 02:04:21.989 I got 1/3. 02:04:21.989 --> 02:04:23.060 PROFESSOR: You got 1/3. 02:04:23.060 --> 02:04:26.070 So sounds good. 02:04:26.070 --> 02:04:28.140 We will stop here. 02:04:28.140 --> 02:04:29.730 You will get homework. 02:04:29.730 --> 02:04:32.560 How long should I leave that homework on? 02:04:32.560 --> 02:04:35.810 Because I'm thinking maybe another month, but please 02:04:35.810 --> 02:04:38.190 don't procrastinate. 02:04:38.190 --> 02:04:41.430 So let's say until the end of March. 02:04:41.430 --> 02:04:44.480 And keep in mind that we have included 02:04:44.480 --> 02:04:48.080 one week of spring break here, which you 02:04:48.080 --> 02:04:51.360 can do whatever you want with. 02:04:51.360 --> 02:04:57.740 Some of you may be in Florida swimming and working on a tan, 02:04:57.740 --> 02:04:59.240 and not working on homework. 02:04:59.240 --> 02:05:01.940 So no matter how, plan ahead. 02:05:01.940 --> 02:05:03.440 Plan ahead and you will do well. 02:05:03.440 --> 02:05:10.390 31st of March for the whole chapter. 02:05:10.390 --> 02:05:11.399