1 00:00:00,000 --> 00:00:01,845 MAGDALENA TODA: According to my watch, 2 00:00:01,845 --> 00:00:04,224 we are right on time to start. 3 00:00:04,224 --> 00:00:09,680 I may be one minute early, or something. 4 00:00:09,680 --> 00:00:16,624 Do you have questions out of the material we covered last time? 5 00:00:16,624 --> 00:00:19,592 What I'm planning on doing-- let me tell you 6 00:00:19,592 --> 00:00:20,592 what I'm planning to do. 7 00:00:20,592 --> 00:00:24,064 I will cover triple integrals today. 8 00:00:24,064 --> 00:00:26,730 And this way, you would have accumulated 9 00:00:26,730 --> 00:00:31,950 enough to deal with most of the problems in homework four. 10 00:00:31,950 --> 00:00:36,310 You have mastered the double integration by now, 11 00:00:36,310 --> 00:00:41,020 in all sorts of coordinates, which is a good thing. 12 00:00:41,020 --> 00:00:46,054 Triple integrals are your friend. 13 00:00:46,054 --> 00:00:51,014 If you have understood the double integration, 14 00:00:51,014 --> 00:00:52,998 you will have no problem understanding 15 00:00:52,998 --> 00:00:54,990 the triple integrals. 16 00:00:54,990 --> 00:00:56,790 The idea is the same. 17 00:00:56,790 --> 00:00:58,950 You look at different domains, and then you 18 00:00:58,950 --> 00:01:04,541 realize that there are Fubini-Tunelli type of results. 19 00:01:04,541 --> 00:01:06,400 I'm going to present one right now. 20 00:01:06,400 --> 00:01:12,550 And there are also regions of a certain type, that 21 00:01:12,550 --> 00:01:15,597 can be treated differentially, and then you 22 00:01:15,597 --> 00:01:20,330 have cases in which reversing the order of integration 23 00:01:20,330 --> 00:01:25,939 for those triple integrals is going to help you a lot. 24 00:01:25,939 --> 00:01:30,869 OK, 12.5 is the name of the section, triple integrals. 25 00:01:30,869 --> 00:01:42,710 26 00:01:42,710 --> 00:01:46,420 So what should you imagine? 27 00:01:46,420 --> 00:01:50,600 You should imagine that somebody gives you 28 00:01:50,600 --> 00:01:55,230 a function of three variables. 29 00:01:55,230 --> 00:02:00,308 Let's call that-- it doesn't often have a name as a letter, 30 00:02:00,308 --> 00:02:03,640 but let's call it w. 31 00:02:03,640 --> 00:02:08,199 Being a function of three coordinates, x, y, and z, 32 00:02:08,199 --> 00:02:10,911 where x, y, z is in R3. 33 00:02:10,911 --> 00:02:14,890 34 00:02:14,890 --> 00:02:22,480 And we have some assumptions about the domain D 35 00:02:22,480 --> 00:02:24,725 that you are working over, and you 36 00:02:24,725 --> 00:02:36,640 have D as a closed-bounded domain in R3. 37 00:02:36,640 --> 00:02:39,390 38 00:02:39,390 --> 00:02:44,010 Examples that you're going to do use frequently. 39 00:02:44,010 --> 00:02:46,740 Frequently used. 40 00:02:46,740 --> 00:02:51,880 A sphere, a ball, actually, because in here, 41 00:02:51,880 --> 00:02:56,070 if a sphere is together with a shell, it is the ball. 42 00:02:56,070 --> 00:03:07,170 Then you have some types of polyhedra in r3 of all types. 43 00:03:07,170 --> 00:03:12,110 44 00:03:12,110 --> 00:03:15,180 And by that, I mean the classical polyhedra 45 00:03:15,180 --> 00:03:23,500 whose sides are just polygons. 46 00:03:23,500 --> 00:03:27,110 But you will also have some curvilinear polyhedra, as well. 47 00:03:27,110 --> 00:03:32,954 48 00:03:32,954 --> 00:03:34,960 What do I mean? 49 00:03:34,960 --> 00:03:36,620 I mean, we've seen that already. 50 00:03:36,620 --> 00:03:38,408 For example, somebody give you a graph 51 00:03:38,408 --> 00:03:49,560 of a function, g of x and y, a continuous function, and says, 52 00:03:49,560 --> 00:03:53,591 OK, can you estimate the volume under the graph? 53 00:03:53,591 --> 00:03:55,850 Right? 54 00:03:55,850 --> 00:03:59,050 And until now, we treated this volume under the graph 55 00:03:59,050 --> 00:04:05,610 as double integral of g of x, y, continuous function over d, a, 56 00:04:05,610 --> 00:04:08,860 where d, a was dx dy. 57 00:04:08,860 --> 00:04:15,730 And we said double integral over the projected domain 58 00:04:15,730 --> 00:04:16,360 in the plane. 59 00:04:16,360 --> 00:04:17,620 That's what I have. 60 00:04:17,620 --> 00:04:20,640 But can I treat it, this volume, can I treat it 61 00:04:20,640 --> 00:04:22,062 as a triple integral? 62 00:04:22,062 --> 00:04:25,380 This is the question, and answer is-- 63 00:04:25,380 --> 00:04:27,240 so can I make three snakes? 64 00:04:27,240 --> 00:04:29,690 The answer is yes. 65 00:04:29,690 --> 00:04:32,160 And the way I'm going to define that 66 00:04:32,160 --> 00:04:39,290 would be a triple integral over a 3D domain. 67 00:04:39,290 --> 00:04:48,580 Let's call it curvilinear d in r3, which is the volume-- which 68 00:04:48,580 --> 00:04:52,890 is the body under the graph of this positive function, 69 00:04:52,890 --> 00:04:57,200 and above the projected area in plane. 70 00:04:57,200 --> 00:05:00,240 So it's going to be a cylinder in this case. 71 00:05:00,240 --> 00:05:02,830 And I'll put here 1 dv. 72 00:05:02,830 --> 00:05:07,840 And dv is a mysterious element. 73 00:05:07,840 --> 00:05:11,190 That's the volume element. 74 00:05:11,190 --> 00:05:15,220 And I will talk a little bit about it right now. 75 00:05:15,220 --> 00:05:17,600 So what can you imagine? 76 00:05:17,600 --> 00:05:23,100 They give you a way to look at it in the book. 77 00:05:23,100 --> 00:05:28,250 I mean, we give you a way to look at it in the book. 78 00:05:28,250 --> 00:05:31,330 It's not very thorough in explanations, 79 00:05:31,330 --> 00:05:36,145 but it certainly gives you the general idea of what you want, 80 00:05:36,145 --> 00:05:38,240 what you need. 81 00:05:38,240 --> 00:05:42,630 So somebody gives you a potato. 82 00:05:42,630 --> 00:05:44,860 It doesn't have to be this cylinder. 83 00:05:44,860 --> 00:05:53,390 It's something beautiful, a body inside a compact surface. 84 00:05:53,390 --> 00:05:55,510 Let's say there are no self-intersections. 85 00:05:55,510 --> 00:06:06,340 You have some compact surface, like a sphere or a polyhedron, 86 00:06:06,340 --> 00:06:09,930 assume it's simply connected, and it doesn't 87 00:06:09,930 --> 00:06:12,750 have any self-intersections. 88 00:06:12,750 --> 00:06:16,670 So a beautiful potato that's smooth. 89 00:06:16,670 --> 00:06:19,740 If you imagine a potato that has singularities, 90 00:06:19,740 --> 00:06:23,500 like most potatoes have singularities, boo-boos, 91 00:06:23,500 --> 00:06:26,290 and cuts, so that's bad. 92 00:06:26,290 --> 00:06:31,762 So think about some regular surface 93 00:06:31,762 --> 00:06:35,280 that's closed, no self-intersections, 94 00:06:35,280 --> 00:06:39,230 and that is a potato that is [INAUDIBLE]. 95 00:06:39,230 --> 00:06:42,550 Oh let's call it-- p for potato, no, 96 00:06:42,550 --> 00:06:44,970 because I have got to use p for the partition. 97 00:06:44,970 --> 00:06:48,880 So let me call it D from 3D domain, 98 00:06:48,880 --> 00:06:54,072 because it's a three-dimensional domain, enclosed 99 00:06:54,072 --> 00:07:00,650 by a curve, enclosed by a compact surface. 100 00:07:00,650 --> 00:07:04,160 101 00:07:04,160 --> 00:07:05,170 So think potato. 102 00:07:05,170 --> 00:07:08,710 What do we do in terms of partitions? 103 00:07:08,710 --> 00:07:12,600 Those pixels were pixels for the 2D world in flat line. 104 00:07:12,600 --> 00:07:15,700 But now, we don't have pixels anymore. 105 00:07:15,700 --> 00:07:16,960 Yes we do. 106 00:07:16,960 --> 00:07:20,460 I was watching lots of sci-fi, and the holograms 107 00:07:20,460 --> 00:07:23,810 have the three-dimensional pixels. 108 00:07:23,810 --> 00:07:26,980 I'm going to try and make a partition. 109 00:07:26,980 --> 00:07:31,110 It's going to be a hard way to partition this potato. 110 00:07:31,110 --> 00:07:39,140 But you have to imagine you have a rectangular partition, 111 00:07:39,140 --> 00:07:49,510 so every little pixel will be a-- is not cube. 112 00:07:49,510 --> 00:07:52,740 It has to be a little tiny parallelepiped. 113 00:07:52,740 --> 00:07:59,850 So a 3D pixel, let me put pixel in quotes, 114 00:07:59,850 --> 00:08:01,780 because this is kind of the idea. 115 00:08:01,780 --> 00:08:04,270 Well have what kind of dimensions? 116 00:08:04,270 --> 00:08:08,210 We'll have three dimensions, right? 117 00:08:08,210 --> 00:08:16,730 Three dimensions, a delta xk, a delta yk, and a delta zk 118 00:08:16,730 --> 00:08:18,632 for the pixel number k. 119 00:08:18,632 --> 00:08:21,360 That's pixel number k. 120 00:08:21,360 --> 00:08:23,610 We have to number them, see how many they are. 121 00:08:23,610 --> 00:08:29,580 Where k is from 1 to n, and is the total number of 3D 122 00:08:29,580 --> 00:08:31,930 pixels that I'm covering the whole thing. 123 00:08:31,930 --> 00:08:34,169 So don't think graphing paper, anymore, 124 00:08:34,169 --> 00:08:36,450 because that's outdated. 125 00:08:36,450 --> 00:08:39,190 Don't even think of 2D image. 126 00:08:39,190 --> 00:08:46,082 Think of some hologram, where you cover everything 127 00:08:46,082 --> 00:08:51,500 with tiny, tiny, tiny 3D pixels so 128 00:08:51,500 --> 00:08:58,255 that in the limit, when you pass to the limit, with respect to n 129 00:08:58,255 --> 00:09:04,470 and going to infinity, the discrete image, 130 00:09:04,470 --> 00:09:07,510 you're going to have something like a diamond shaped thingy, 131 00:09:07,510 --> 00:09:10,870 will convert to the smooth potato. 132 00:09:10,870 --> 00:09:16,710 So as n goes to infinity, that surface 133 00:09:16,710 --> 00:09:21,040 made of tiny, tiny squares will convert to the data. 134 00:09:21,040 --> 00:09:27,680 So how do you actually find a triple snake integral f 135 00:09:27,680 --> 00:09:33,460 of x, y, z over the domain D, dx dy dz. 136 00:09:33,460 --> 00:09:38,230 137 00:09:38,230 --> 00:09:42,080 OK, this theoretically should be what? 138 00:09:42,080 --> 00:09:43,430 Think pixels. 139 00:09:43,430 --> 00:09:53,200 Limit as n goes to infinity of the sum of the-- 140 00:09:53,200 --> 00:09:54,625 what do I need to do? 141 00:09:54,625 --> 00:10:01,420 Think the whole partition into pixels. 142 00:10:01,420 --> 00:10:09,430 How many pixels? n pixels total is called a p. 143 00:10:09,430 --> 00:10:12,846 Script p. 144 00:10:12,846 --> 00:10:21,070 And the normal p will be the highest diameter of-- highest 145 00:10:21,070 --> 00:10:27,200 diameter among all pixels. 146 00:10:27,200 --> 00:10:32,030 147 00:10:32,030 --> 00:10:34,620 And you're going to say, Oh, what the heck? 148 00:10:34,620 --> 00:10:36,030 I don't understand it. 149 00:10:36,030 --> 00:10:39,350 I have these three-dimensional cubes, or three-dimensional 150 00:10:39,350 --> 00:10:42,880 barely by p that get tinier, and tinier, and tinier. 151 00:10:42,880 --> 00:10:47,350 What in the world is going to be a diameter of such a pixel? 152 00:10:47,350 --> 00:10:49,040 Well, you have to take this pixel 153 00:10:49,040 --> 00:10:55,240 and magnify it so we can look at it a little bit better. 154 00:10:55,240 --> 00:10:58,215 What do we mean by diameter of this pixel? 155 00:10:58,215 --> 00:11:00,900 156 00:11:00,900 --> 00:11:02,910 Let's call this pixel k. 157 00:11:02,910 --> 00:11:08,180 158 00:11:08,180 --> 00:11:13,090 The maximum of all the distances you can compute between two 159 00:11:13,090 --> 00:11:16,080 arbitrary points inside. 160 00:11:16,080 --> 00:11:17,710 Between two arbitrary points inside, 161 00:11:17,710 --> 00:11:19,130 you have many [INAUDIBLE]. 162 00:11:19,130 --> 00:11:25,342 So the maximum of the distance between points, 163 00:11:25,342 --> 00:11:34,720 let's call them r and q inside the pixel. 164 00:11:34,720 --> 00:11:37,930 So if it were for me to ask you to find 165 00:11:37,930 --> 00:11:40,993 that diameter in this case, what would that be? 166 00:11:40,993 --> 00:11:42,118 STUDENT: It's the diagonal. 167 00:11:42,118 --> 00:11:46,240 MAGDALENA TODA: It's the diagonal between this corner 168 00:11:46,240 --> 00:11:49,290 and the opposite corner. 169 00:11:49,290 --> 00:11:55,710 So this would be the highest distance inside this pixel. 170 00:11:55,710 --> 00:11:58,750 If it's a cube-- you see that I wanted a cube. 171 00:11:58,750 --> 00:12:02,210 If it's a parallelepiped, it's the same idea. 172 00:12:02,210 --> 00:12:08,630 So I have that opposite corner distance kind of thing. 173 00:12:08,630 --> 00:12:10,202 OK. 174 00:12:10,202 --> 00:12:12,018 So I know what I want. 175 00:12:12,018 --> 00:12:15,010 I want n to go to infinity. 176 00:12:15,010 --> 00:12:19,490 That means I'm going to have the p going to 0. 177 00:12:19,490 --> 00:12:24,140 The length of the highest diameter 178 00:12:24,140 --> 00:12:26,273 will go shrinking to 0. 179 00:12:26,273 --> 00:12:29,670 And then I'm going to say here, what do I have inside? 180 00:12:29,670 --> 00:12:31,445 F of some intermediate point. 181 00:12:31,445 --> 00:12:34,300 In every pixel, I take a point. 182 00:12:34,300 --> 00:12:36,420 Another pixel, another point, and so on. 183 00:12:36,420 --> 00:12:39,010 So how many such points do I have? 184 00:12:39,010 --> 00:12:42,110 n, because it's the number of pixels. 185 00:12:42,110 --> 00:12:45,860 So inside, let's call this as pixel p k. 186 00:12:45,860 --> 00:12:49,600 What is the little point that I took out of the [INAUDIBLE] 187 00:12:49,600 --> 00:12:51,630 inside the cube, or inside the pixel? 188 00:12:51,630 --> 00:12:59,650 Let's call that mister x k star, y k star, x k star. 189 00:12:59,650 --> 00:13:01,160 Why do we put a star? 190 00:13:01,160 --> 00:13:02,590 Because he is a star. 191 00:13:02,590 --> 00:13:05,330 He wants to be number one in these little domains, 192 00:13:05,330 --> 00:13:06,880 and says I'm a star. 193 00:13:06,880 --> 00:13:11,520 So we take that intermediate point, x k star, 194 00:13:11,520 --> 00:13:16,365 y k star, x k star. 195 00:13:16,365 --> 00:13:21,700 Then we have this function multiplied by the delta v k. 196 00:13:21,700 --> 00:13:25,428 somebody tell me what this delta v k will mean, 197 00:13:25,428 --> 00:13:27,360 because ir really looks weird. 198 00:13:27,360 --> 00:13:29,290 And then k will be from 1 to n. 199 00:13:29,290 --> 00:13:30,110 So what do you do? 200 00:13:30,110 --> 00:13:35,710 You sum up all these weighted volumes. 201 00:13:35,710 --> 00:13:37,960 This is a weight. 202 00:13:37,960 --> 00:13:40,500 So this is a volume. 203 00:13:40,500 --> 00:13:41,880 All these weighted volumes. 204 00:13:41,880 --> 00:13:46,790 We sum them up for all the pixels k from 1 to n. 205 00:13:46,790 --> 00:13:51,145 We are going to get something like this cover 206 00:13:51,145 --> 00:13:56,485 with tiny parallelepipeds in the limit, 207 00:13:56,485 --> 00:13:59,310 as the partitions' [? norm ?] go to 0, or the number of pixels 208 00:13:59,310 --> 00:14:01,130 goes to infinity. 209 00:14:01,130 --> 00:14:06,732 This discrete surface will converge 210 00:14:06,732 --> 00:14:11,452 to the beautiful smooth potato, and give you 211 00:14:11,452 --> 00:14:12,740 a perfect linear image. 212 00:14:12,740 --> 00:14:16,730 Actually, if we saw a hologram, this is what it is. 213 00:14:16,730 --> 00:14:20,940 Our eyes actually see a bunch of I 214 00:14:20,940 --> 00:14:25,000 tiny-- many, many, many, many, millions of pixels 215 00:14:25,000 --> 00:14:27,900 that are cubes in 3D. 216 00:14:27,900 --> 00:14:29,500 But it's an optical illusion. 217 00:14:29,500 --> 00:14:33,010 We see, OK, it's a curvilinear, it's a smooth body of a person. 218 00:14:33,010 --> 00:14:34,980 It's not smooth at all. 219 00:14:34,980 --> 00:14:37,335 If you get closer and closer to that diagram 220 00:14:37,335 --> 00:14:38,840 and put your eye glasses on, you are 221 00:14:38,840 --> 00:14:41,430 going to see, oh, this is not a real person. 222 00:14:41,430 --> 00:14:44,000 It's made of pixels that are all cubes. 223 00:14:44,000 --> 00:14:47,580 just the same, you see your digital image of your picture 224 00:14:47,580 --> 00:14:49,330 on Facebook, whatever it is. 225 00:14:49,330 --> 00:14:52,350 If you would be able to be enlarge it, 226 00:14:52,350 --> 00:14:57,210 you would see the pixels, being little tiny squares there. 227 00:14:57,210 --> 00:15:00,380 The graphical imaging has improved. 228 00:15:00,380 --> 00:15:05,098 The quality of our digital imaging has improved a lot. 229 00:15:05,098 --> 00:15:08,780 But of course, 20 years ago, when you weren't even born, 230 00:15:08,780 --> 00:15:12,390 we could still see the pixels in the photographic images 231 00:15:12,390 --> 00:15:14,070 in a digital camera. 232 00:15:14,070 --> 00:15:18,984 Those tiny first cameras, what was that, '98? 233 00:15:18,984 --> 00:15:19,860 STUDENT: Kodak. 234 00:15:19,860 --> 00:15:23,720 MAGDALENA TODA: Like AOL cameras that were so cheap. 235 00:15:23,720 --> 00:15:30,246 The cheaper the camera, the worse the resolution. 236 00:15:30,246 --> 00:15:33,029 I remember some resolutions like 400 by 600. 237 00:15:33,029 --> 00:15:34,070 STUDENT: Black and white. 238 00:15:34,070 --> 00:15:35,000 MAGDALENA TODA: Not black and white. 239 00:15:35,000 --> 00:15:36,915 Black and white would have been neat. 240 00:15:36,915 --> 00:15:40,610 But really nasty in the sense that you had the feeling 241 00:15:40,610 --> 00:15:43,290 that the colors were not even-- they 242 00:15:43,290 --> 00:15:45,940 were blending into each other, because the resolution 243 00:15:45,940 --> 00:15:47,940 was so small. 244 00:15:47,940 --> 00:15:50,780 So it was not at all pleasing to the eye. 245 00:15:50,780 --> 00:15:53,220 What was good is that any kind of defects you 246 00:15:53,220 --> 00:15:56,020 would have, something like a pimple 247 00:15:56,020 --> 00:15:58,820 could not be seen in that, because the resolution was 248 00:15:58,820 --> 00:16:02,340 so slow that you couldn't see the boo-boos, 249 00:16:02,340 --> 00:16:06,100 the pimples, the defects of a face or something. 250 00:16:06,100 --> 00:16:08,450 Now, you can see. 251 00:16:08,450 --> 00:16:11,620 With the digital cameras we have now, we can do, 252 00:16:11,620 --> 00:16:14,380 of course, Adobe Photoshop, and all of us 253 00:16:14,380 --> 00:16:18,380 will look great if we photoshop our pictures. 254 00:16:18,380 --> 00:16:18,910 OK. 255 00:16:18,910 --> 00:16:22,240 So this is what it is in the limit. 256 00:16:22,240 --> 00:16:26,600 But in reality, in the everyday reality, 257 00:16:26,600 --> 00:16:30,840 you cannot take Riemann sums like that-- 258 00:16:30,840 --> 00:16:33,284 this is a Riemann approximating sum-- 259 00:16:33,284 --> 00:16:38,660 and then cast to the limit, and get ideal curvilinear domains. 260 00:16:38,660 --> 00:16:39,230 No. 261 00:16:39,230 --> 00:16:40,780 You don't do that. 262 00:16:40,780 --> 00:16:44,190 You have to deal with the equivalent 263 00:16:44,190 --> 00:16:45,875 of the fundamental theorem of calculus 264 00:16:45,875 --> 00:16:51,375 from Calc I, which is called the Fubini-Tonelli type of theorem 265 00:16:51,375 --> 00:16:53,270 in Calc III. 266 00:16:53,270 --> 00:16:54,330 So say it again. 267 00:16:54,330 --> 00:16:58,078 So the Fubini-Tunelli theorem that you 268 00:16:58,078 --> 00:17:04,130 learned for double integrals over a rectangle 269 00:17:04,130 --> 00:17:10,089 can be generalized to the Fubini-Tonelli theorem 270 00:17:10,089 --> 00:17:12,848 over a parallelepiped. 271 00:17:12,848 --> 00:17:14,960 And it's the same thing, practically, 272 00:17:14,960 --> 00:17:20,450 as applying the fundamental theorem of calculus in Calc I. 273 00:17:20,450 --> 00:17:25,210 So somebody says, well, let me start with a simple example. 274 00:17:25,210 --> 00:17:29,355 I give you a-- you will say, Magdalena, 275 00:17:29,355 --> 00:17:30,230 you are offending us. 276 00:17:30,230 --> 00:17:32,280 This is way too easy. 277 00:17:32,280 --> 00:17:35,350 What do you think, that we cannot understand the concept? 278 00:17:35,350 --> 00:17:39,240 I'll just try to start with the simplest possible example 279 00:17:39,240 --> 00:17:40,640 that I think of. 280 00:17:40,640 --> 00:17:44,474 So x is between a and b. 281 00:17:44,474 --> 00:17:46,360 In my case, they will be positive numbers, 282 00:17:46,360 --> 00:17:50,170 because I want everything to be in the first octant. 283 00:17:50,170 --> 00:17:55,710 First octant means x positive, y positive, and z positive 284 00:17:55,710 --> 00:17:58,290 all together. 285 00:17:58,290 --> 00:18:01,870 To make my life easier, I take that example, 286 00:18:01,870 --> 00:18:06,030 and I say, I know the numbers for x, y, z. 287 00:18:06,030 --> 00:18:09,530 I would like you to compute two integrals. 288 00:18:09,530 --> 00:18:13,560 One would be the volume of this object. 289 00:18:13,560 --> 00:18:15,000 Let's call it body. 290 00:18:15,000 --> 00:18:19,030 It's not a dead body, it's just body in 3D. 291 00:18:19,030 --> 00:18:22,690 The volume of the body, we say it 292 00:18:22,690 --> 00:18:27,118 like a mathematician, V of B. What is that by definition? 293 00:18:27,118 --> 00:18:28,500 Who's going to tell me? 294 00:18:28,500 --> 00:18:30,690 Triple snake. 295 00:18:30,690 --> 00:18:32,940 Don't say triple snake to other people, 296 00:18:32,940 --> 00:18:36,000 because other professors are more orthodox than me. 297 00:18:36,000 --> 00:18:39,970 They will laugh-- they will not joke about it. 298 00:18:39,970 --> 00:18:50,890 So triple integral over the body of-- to get the volume, 299 00:18:50,890 --> 00:18:54,600 the weight must be 1. 300 00:18:54,600 --> 00:18:59,070 f integral must be 1, and then you have exactly dV. 301 00:18:59,070 --> 00:19:01,175 How can I convince you what we have here, 302 00:19:01,175 --> 00:19:02,515 in terms of Fubini-Tonelli? 303 00:19:02,515 --> 00:19:04,650 It's really beautiful. 304 00:19:04,650 --> 00:19:11,570 B is going to be a, b segment cross product, c, d segment, 305 00:19:11,570 --> 00:19:12,900 cross product. 306 00:19:12,900 --> 00:19:15,610 What is the altitude? 307 00:19:15,610 --> 00:19:17,370 E, f, e to f. 308 00:19:17,370 --> 00:19:20,380 Interval e to f means the height. 309 00:19:20,380 --> 00:19:23,606 So length, width, height. 310 00:19:23,606 --> 00:19:28,496 This is the box, or carry-on, or USPS parcel, or whatever 311 00:19:28,496 --> 00:19:32,408 box you want to measure. 312 00:19:32,408 --> 00:19:37,730 So how am I going to set up the Fubini-Tonelli integral? 313 00:19:37,730 --> 00:19:44,040 a to b, c to d, e to f, 1. 314 00:19:44,040 --> 00:19:46,100 And now, who counts first? 315 00:19:46,100 --> 00:19:48,260 dz, dy, dx. 316 00:19:48,260 --> 00:19:54,510 So it is like the equivalent of the vertical strip 317 00:19:54,510 --> 00:19:58,570 thingy in double corners, double integrals. 318 00:19:58,570 --> 00:19:59,450 Yes, sir? 319 00:19:59,450 --> 00:20:01,568 STUDENT: Professor, why did you use 1 dV? 320 00:20:01,568 --> 00:20:02,930 Why 1? 321 00:20:02,930 --> 00:20:04,050 MAGDALENA TODA: OK. 322 00:20:04,050 --> 00:20:05,570 You'll see in a second. 323 00:20:05,570 --> 00:20:08,950 This is the same thing we do for areas. 324 00:20:08,950 --> 00:20:14,400 So when you compute an area-- very good question. 325 00:20:14,400 --> 00:20:18,290 If you use 1 here, and you put delta [? ak ?] 326 00:20:18,290 --> 00:20:21,360 that is the graphing paper area. 327 00:20:21,360 --> 00:20:24,660 It's going to be all the tiny areas, 328 00:20:24,660 --> 00:20:27,700 summed up, sum of all the delta [? ak ?] which 329 00:20:27,700 --> 00:20:30,862 means this little pixel, plus this little pixel, 330 00:20:30,862 --> 00:20:33,170 plus this little pixel, plus this little pixel, 331 00:20:33,170 --> 00:20:38,360 plus 1,000 pixels all together will cover up the area. 332 00:20:38,360 --> 00:20:41,710 If you have the volume of a potato, 333 00:20:41,710 --> 00:20:44,470 a body that is alive, but shouldn't move. 334 00:20:44,470 --> 00:20:46,870 OK, it should stay in one place. 335 00:20:46,870 --> 00:20:50,145 Then, to compute the volume of the potato, 336 00:20:50,145 --> 00:20:53,600 you have to say, the potato, the smooth potato, 337 00:20:53,600 --> 00:20:59,280 is the limit of the sum of all the tiny cubes of potato, 338 00:20:59,280 --> 00:21:03,050 if you cut the potato in many cubes, like you cut cheese. 339 00:21:03,050 --> 00:21:06,055 They got a bunch of cheddar cheese into small cubes, 340 00:21:06,055 --> 00:21:10,800 and they feed us with crackers and wine-- OK, no comments. 341 00:21:10,800 --> 00:21:15,350 So you have delta vk, you have 1,000 little cubes, 342 00:21:15,350 --> 00:21:17,800 tiny, tiny, tiny, like that Lego. 343 00:21:17,800 --> 00:21:20,160 OK, forget about the cheese. 344 00:21:20,160 --> 00:21:22,750 The cheese cubes are way too big. 345 00:21:22,750 --> 00:21:26,000 So imagine Legos that are really performing 346 00:21:26,000 --> 00:21:28,905 with millions of little pieces. 347 00:21:28,905 --> 00:21:35,980 Have you seen the exhibit, Lego exhibit with almost invisible 348 00:21:35,980 --> 00:21:40,130 Legos at the Civic Center? 349 00:21:40,130 --> 00:21:42,850 They have that art festival. 350 00:21:42,850 --> 00:21:46,085 How many of you go to the art festival? 351 00:21:46,085 --> 00:21:48,845 Is it every April? 352 00:21:48,845 --> 00:21:50,705 Something like that. 353 00:21:50,705 --> 00:21:53,515 So imagine those little tiny Legos, but being cubes 354 00:21:53,515 --> 00:21:54,520 and put together. 355 00:21:54,520 --> 00:21:56,350 This is what it is. 356 00:21:56,350 --> 00:21:58,170 So f Vy. 357 00:21:58,170 --> 00:22:01,560 Now, can we verify the volume of a box? 358 00:22:01,560 --> 00:22:02,230 It's very easy. 359 00:22:02,230 --> 00:22:03,660 What do we do? 360 00:22:03,660 --> 00:22:08,070 Well, first of all, I would do it in a slow way, 361 00:22:08,070 --> 00:22:10,270 and you are going to shout at me, I know. 362 00:22:10,270 --> 00:22:13,980 But I'll tell you why you need to bear with me. 363 00:22:13,980 --> 00:22:16,670 So integral of 1 dz goes first. 364 00:22:16,670 --> 00:22:18,740 That's z between f and e. 365 00:22:18,740 --> 00:22:21,030 So it's f minus e, am I right? 366 00:22:21,030 --> 00:22:22,820 You say, duh, that's to easy for me. 367 00:22:22,820 --> 00:22:25,610 I'm know it's too easy for me, but I'm 368 00:22:25,610 --> 00:22:26,800 going somewhere with it. 369 00:22:26,800 --> 00:22:29,560 dy dx. 370 00:22:29,560 --> 00:22:34,640 The one inside, f minus e is a constant, pulls out, 371 00:22:34,640 --> 00:22:36,860 completely out of the product. 372 00:22:36,860 --> 00:22:42,890 And then I have integral from a to b of-- what is that left? 373 00:22:42,890 --> 00:22:46,700 1 dy, y between d and c. 374 00:22:46,700 --> 00:22:48,360 So d minus c, right? 375 00:22:48,360 --> 00:22:52,810 d minus c dx. 376 00:22:52,810 --> 00:22:56,510 And so on and so forth, until I get it. 377 00:22:56,510 --> 00:23:02,450 If minus c times d minus c times b minus a, 378 00:23:02,450 --> 00:23:06,230 and goodbye, because this is the volume of the box. 379 00:23:06,230 --> 00:23:08,010 It's the height. 380 00:23:08,010 --> 00:23:10,310 This is the height. 381 00:23:10,310 --> 00:23:12,842 No, excuse me, guys. 382 00:23:12,842 --> 00:23:15,800 The height is-- this one is the height. 383 00:23:15,800 --> 00:23:20,610 This is the width, and this is the length, whatever you want. 384 00:23:20,610 --> 00:23:21,515 All right. 385 00:23:21,515 --> 00:23:25,434 How could I have done it if I were a little bit smarter? 386 00:23:25,434 --> 00:23:27,900 STUDENT: You could have just put it in three integrals. 387 00:23:27,900 --> 00:23:29,900 MAGDALENA TODA: Right Hey, I have 388 00:23:29,900 --> 00:23:35,830 a theorem, just like before, which says 389 00:23:35,830 --> 00:23:37,530 three integrals in a product. 390 00:23:37,530 --> 00:23:40,448 This is what Matt immediately remembered. 391 00:23:40,448 --> 00:23:43,794 We had two integrals in a product last time. 392 00:23:43,794 --> 00:23:46,662 So what have we proved in double integrals 393 00:23:46,662 --> 00:23:50,020 remains valid in triple integrals 394 00:23:50,020 --> 00:23:52,830 if we have something like that. 395 00:23:52,830 --> 00:23:54,790 So I'm going the same theorem. 396 00:23:54,790 --> 00:23:55,770 It's in the book. 397 00:23:55,770 --> 00:23:56,750 We have a proof. 398 00:23:56,750 --> 00:24:02,360 So you have integral from a to b, c to d, e to f. 399 00:24:02,360 --> 00:24:08,500 And then, some guys that you like, f of x, times g of y, 400 00:24:08,500 --> 00:24:10,950 times h of z. 401 00:24:10,950 --> 00:24:15,980 Functions of x, y, z, separated variables. 402 00:24:15,980 --> 00:24:18,650 So f, a function of x only, g a function 403 00:24:18,650 --> 00:24:21,460 of y only, h a function of z only. 404 00:24:21,460 --> 00:24:23,020 This is the complicated case. 405 00:24:23,020 --> 00:24:26,250 And then I have 406 00:24:26,250 --> 00:24:27,710 STUDENT: dz, dy, dx. 407 00:24:27,710 --> 00:24:29,040 MAGDALENA TODA: dz, dy, dx. 408 00:24:29,040 --> 00:24:29,690 Excellent. 409 00:24:29,690 --> 00:24:32,618 Thanks for whispering, because I was a little bit 410 00:24:32,618 --> 00:24:35,900 confused for a second. 411 00:24:35,900 --> 00:24:38,820 So, just as Matt said, go ahead and observe 412 00:24:38,820 --> 00:24:42,440 that you can treat them one at a time like you did here, 413 00:24:42,440 --> 00:24:47,000 and integrate one at a time, and integrate again, and pull out 414 00:24:47,000 --> 00:24:49,710 a constant, integrate again, pull out a constant. 415 00:24:49,710 --> 00:24:53,280 But practically this is exactly the same as integral 416 00:24:53,280 --> 00:25:02,190 from a to b of f of x alone, dx, times integral 417 00:25:02,190 --> 00:25:11,610 from c to d, g of y alone, dy, and times integral from e 418 00:25:11,610 --> 00:25:18,180 to f of h of z, dz, and close. 419 00:25:18,180 --> 00:25:22,220 So you've seen the version of the double integral, 420 00:25:22,220 --> 00:25:26,520 and this is the same result for triple integrals. 421 00:25:26,520 --> 00:25:29,810 And it's practically-- what is the proof? 422 00:25:29,810 --> 00:25:33,604 You just pull out one at a time, so the proof 423 00:25:33,604 --> 00:25:37,520 is that you start working and say, mister z counts here, 424 00:25:37,520 --> 00:25:39,800 and he's the only one that counts. 425 00:25:39,800 --> 00:25:43,540 These guys get out for a walk one at a time outside 426 00:25:43,540 --> 00:25:46,210 of the first integral inside. 427 00:25:46,210 --> 00:25:48,850 And then, integral of h of z, dz, 428 00:25:48,850 --> 00:25:53,445 over the corresponding domain, will be just a constant, c1, 429 00:25:53,445 --> 00:25:55,060 that pulls out. 430 00:25:55,060 --> 00:25:59,710 And that is that-- c1 that pulls out. 431 00:25:59,710 --> 00:26:02,840 Ans since you pull them out in this product one at a time, 432 00:26:02,840 --> 00:26:04,632 that's what you get. 433 00:26:04,632 --> 00:26:08,720 I'm not going to give you this as an exercise in the midterm 434 00:26:08,720 --> 00:26:12,210 with a proof, but this is one of the first exercises 435 00:26:12,210 --> 00:26:16,310 I had as a freshman in my multi-- 436 00:26:16,310 --> 00:26:20,261 I took it as a freshman, as multivariable calculus. 437 00:26:20,261 --> 00:26:23,219 And it was a pop quiz. 438 00:26:23,219 --> 00:26:26,660 My professor just came one day, and said, guys, 439 00:26:26,660 --> 00:26:30,117 you have to try to do this [? before ?] by yourself. 440 00:26:30,117 --> 00:26:33,099 And some of us did, some of us didn't. 441 00:26:33,099 --> 00:26:35,087 To me, it really looked very easy. 442 00:26:35,087 --> 00:26:40,570 I was very happy to prove it, in an elementary way, of course. 443 00:26:40,570 --> 00:26:41,120 OK. 444 00:26:41,120 --> 00:26:45,892 So how hard is it to generalize, to go 445 00:26:45,892 --> 00:26:47,868 to non-rectangular domains? 446 00:26:47,868 --> 00:26:49,350 Of course it's a pain. 447 00:26:49,350 --> 00:26:54,356 It's really a pain, like it was before. 448 00:26:54,356 --> 00:26:59,830 But you will be able to figure out what's going on. 449 00:26:59,830 --> 00:27:02,964 In most cases, you're going to have 450 00:27:02,964 --> 00:27:06,510 a domain that's really not bad, a domain that 451 00:27:06,510 --> 00:27:09,210 has x between fixed values. 452 00:27:09,210 --> 00:27:14,410 For example y between your favorite guys, 453 00:27:14,410 --> 00:27:20,020 something like f of x and g of x, top and bottom. 454 00:27:20,020 --> 00:27:22,280 That's what you had for double integral. 455 00:27:22,280 --> 00:27:26,410 Well, in addition, in this case, you 456 00:27:26,410 --> 00:27:31,260 will have z between-- let's make this guy 457 00:27:31,260 --> 00:27:36,005 big F and big G, other functions. 458 00:27:36,005 --> 00:27:37,920 This is going to be a function of x, y. 459 00:27:37,920 --> 00:27:40,700 This is going to be a function of x, y, 460 00:27:40,700 --> 00:27:44,030 and that's the upper and the lower. 461 00:27:44,030 --> 00:27:49,870 And find the triple integral of, let's say 1 over d dV 462 00:27:49,870 --> 00:27:52,860 will be a volume of the potato. 463 00:27:52,860 --> 00:27:55,890 Now, I'm sick of potatoes, because they're not 464 00:27:55,890 --> 00:27:58,990 my favorite food. 465 00:27:58,990 --> 00:28:02,615 Let me imagine I'm making a tetrahedron, a lot of cheese. 466 00:28:02,615 --> 00:28:08,950 467 00:28:08,950 --> 00:28:13,780 I'm going to draw this same tetrahedron from last time. 468 00:28:13,780 --> 00:28:15,457 So what did we do last time? 469 00:28:15,457 --> 00:28:17,942 We took a plane that was beautiful, 470 00:28:17,942 --> 00:28:21,980 and we said let's cut with that plane. 471 00:28:21,980 --> 00:28:24,910 This is the plane we are cutting the cheese with. 472 00:28:24,910 --> 00:28:25,850 It's a knife. 473 00:28:25,850 --> 00:28:28,039 x plus y plus z equals 1. 474 00:28:28,039 --> 00:28:29,830 Imagine that there's an infinite knife that 475 00:28:29,830 --> 00:28:31,510 comes into the frame. 476 00:28:31,510 --> 00:28:33,150 Everything is cheese. 477 00:28:33,150 --> 00:28:37,020 The space, the universe is covered in solid cheese. 478 00:28:37,020 --> 00:28:42,289 So the whole thing, the Euclidean space 479 00:28:42,289 --> 00:28:43,802 is covered in cheddar cheese. 480 00:28:43,802 --> 00:28:44,510 That's all there. 481 00:28:44,510 --> 00:28:48,310 From everywhere, you come with this knife, 482 00:28:48,310 --> 00:28:52,710 and you cut along this plane-- hi 483 00:28:52,710 --> 00:28:54,960 let's call this [? high plane. ?] 484 00:28:54,960 --> 00:28:59,500 And then you cut the x plane along the x, y plane, 485 00:28:59,500 --> 00:29:03,480 y, z plane and x, z plane. 486 00:29:03,480 --> 00:29:04,430 What are these called? 487 00:29:04,430 --> 00:29:06,522 Planes of coordinates. 488 00:29:06,522 --> 00:29:07,480 And what do you obtain? 489 00:29:07,480 --> 00:29:09,230 Then, you throw everything away, and you 490 00:29:09,230 --> 00:29:15,252 maintain only the tetrahedron made of cheese. 491 00:29:15,252 --> 00:29:17,974 Now, you remember what the corners were. 492 00:29:17,974 --> 00:29:18,640 This is 0, 0, 0. 493 00:29:18,640 --> 00:29:19,700 It's a piece of cake. 494 00:29:19,700 --> 00:29:22,880 But I want to know the vertices. 495 00:29:22,880 --> 00:29:27,580 And you know them, and I don't want to spend time 496 00:29:27,580 --> 00:29:30,180 discussing why you know them. 497 00:29:30,180 --> 00:29:30,680 So 498 00:29:30,680 --> 00:29:30,990 STUDENT: 0-- 499 00:29:30,990 --> 00:29:31,989 MAGDALENA TODA: 1, 0, 0. 500 00:29:31,989 --> 00:29:33,401 Thank you. 501 00:29:33,401 --> 00:29:33,900 Huh? 502 00:29:33,900 --> 00:29:34,847 STUDENT: 0, 1, 0. 503 00:29:34,847 --> 00:29:35,680 MAGDALENA TODA: Yes. 504 00:29:35,680 --> 00:29:38,280 And 0, 0, 1. 505 00:29:38,280 --> 00:29:39,640 All right. 506 00:29:39,640 --> 00:29:40,340 Great. 507 00:29:40,340 --> 00:29:44,020 The only thing is, if we see the cheese being a solid, 508 00:29:44,020 --> 00:29:49,870 we don't see this part, the three axes of corners behind. 509 00:29:49,870 --> 00:29:55,718 so I'm going to make them dotted, and you see the slice, 510 00:29:55,718 --> 00:29:58,596 here, it has to be really planar. 511 00:29:58,596 --> 00:30:01,395 And you ask yourself, how do you set up 512 00:30:01,395 --> 00:30:03,660 the triple integral that represents 513 00:30:03,660 --> 00:30:05,810 the volume of this object? 514 00:30:05,810 --> 00:30:07,360 Is it hard? 515 00:30:07,360 --> 00:30:08,800 It shouldn't be hard. 516 00:30:08,800 --> 00:30:13,080 You just have to think what the domain will be like, 517 00:30:13,080 --> 00:30:16,736 and you say the domain is inside the tetrahedron. 518 00:30:16,736 --> 00:30:18,230 Do you want d or t? 519 00:30:18,230 --> 00:30:20,940 T from tetrahedron. 520 00:30:20,940 --> 00:30:22,190 It doesn't matter. 521 00:30:22,190 --> 00:30:23,260 We have a new name. 522 00:30:23,260 --> 00:30:26,650 We get bored of all sorts of names and notations. 523 00:30:26,650 --> 00:30:28,320 We change them. 524 00:30:28,320 --> 00:30:33,749 Mathematicians have imagination, so we change our notations. 525 00:30:33,749 --> 00:30:35,290 Like we cannot change our identities, 526 00:30:35,290 --> 00:30:37,840 and we suffer because of that. 527 00:30:37,840 --> 00:30:40,745 So you can be a nerd mathematician imagining 528 00:30:40,745 --> 00:30:43,920 you're Spiderman, and you can take, 529 00:30:43,920 --> 00:30:47,720 give any name you want, and you can adopt a new name, 530 00:30:47,720 --> 00:30:51,680 and this is behind our motivation 531 00:30:51,680 --> 00:30:56,320 why we like to change names and change notation so much. 532 00:30:56,320 --> 00:30:56,820 OK? 533 00:30:56,820 --> 00:31:03,250 So we have triple integral of this T. All right, of what? 534 00:31:03,250 --> 00:31:05,940 1 dV. 535 00:31:05,940 --> 00:31:06,630 Good. 536 00:31:06,630 --> 00:31:09,690 Now we understand what we need to do, 537 00:31:09,690 --> 00:31:12,050 just like [? Miteish ?] asked me why. 538 00:31:12,050 --> 00:31:15,600 OK, now we know this is going to be a limit of little cubes. 539 00:31:15,600 --> 00:31:18,150 If were to cover this piece of cheese 540 00:31:18,150 --> 00:31:21,800 in tiny, tiny, infinitesimally small cubes. 541 00:31:21,800 --> 00:31:25,340 But now we know a method to do it. 542 00:31:25,340 --> 00:31:29,510 So according to-- Fubini-Tonelli type of result. 543 00:31:29,510 --> 00:31:39,140 We would have a between-- no, x-- is first, dz. 544 00:31:39,140 --> 00:31:43,220 z is first, y is moving next, x is moving last. 545 00:31:43,220 --> 00:31:46,746 z is constrained to move between a and b. 546 00:31:46,746 --> 00:31:50,650 But in this case, a and b should be prescribed by you guys, 547 00:31:50,650 --> 00:31:55,670 because you should think where everybody lives. 548 00:31:55,670 --> 00:31:59,780 Not you, I mean the coordinates in their imaginary world. 549 00:31:59,780 --> 00:32:03,380 The coordinates represent somebodies. 550 00:32:03,380 --> 00:32:03,880 STUDENT: 0. 551 00:32:03,880 --> 00:32:07,520 MAGDALENA TODA: x, 0 to 1. 552 00:32:07,520 --> 00:32:10,550 How should I give you a feeling for that? 553 00:32:10,550 --> 00:32:11,930 Just draw this line. 554 00:32:11,930 --> 00:32:14,590 This red segment between 0 to 1. 555 00:32:14,590 --> 00:32:18,040 That expresses everything instead of words 556 00:32:18,040 --> 00:32:22,810 into pictures, because every picture is worth 1,000 words. 557 00:32:22,810 --> 00:32:26,740 y is married to x, unfortunately. 558 00:32:26,740 --> 00:32:30,520 y cannot say, oh, I am y, I'm going wherever I want. 559 00:32:30,520 --> 00:32:34,430 He hits his head against this purple line. 560 00:32:34,430 --> 00:32:36,820 He cannot go beyond that purple line. 561 00:32:36,820 --> 00:32:39,780 He's constrained, poor y. 562 00:32:39,780 --> 00:32:41,300 So he says, I'm moving. 563 00:32:41,300 --> 00:32:42,205 I'm mister y. 564 00:32:42,205 --> 00:32:46,270 I'm moving in this direction, but I cannot go past the purple 565 00:32:46,270 --> 00:32:49,174 line in plane here. 566 00:32:49,174 --> 00:32:52,130 567 00:32:52,130 --> 00:32:56,640 I need you, because if you go, I'm lost. 568 00:32:56,640 --> 00:32:58,740 y is between 0 and-- 569 00:32:58,740 --> 00:32:59,652 STUDENT: 1 minus x. 570 00:32:59,652 --> 00:33:00,735 MAGDALENA TODA: 1 minus x. 571 00:33:00,735 --> 00:33:01,820 Excellent Roberto. 572 00:33:01,820 --> 00:33:04,860 How did we think about this? 573 00:33:04,860 --> 00:33:08,330 The purple line has equation-- how do you 574 00:33:08,330 --> 00:33:10,992 get to the equation of the purple line, first of all? 575 00:33:10,992 --> 00:33:15,250 In your imagination, your plug in z equals 0. 576 00:33:15,250 --> 00:33:19,400 So the purple line would be x plus y equals 1. 577 00:33:19,400 --> 00:33:24,756 And so mister y will be 1 minus x here. 578 00:33:24,756 --> 00:33:27,680 That's how you got it. 579 00:33:27,680 --> 00:33:31,960 And finally, z is that-- mister z foes from the floor 580 00:33:31,960 --> 00:33:34,190 all the way-- imagine somebody who 581 00:33:34,190 --> 00:33:39,765 is like-- z is a helium balloon, and he 582 00:33:39,765 --> 00:33:42,850 is left-- you let him go from the floor, 583 00:33:42,850 --> 00:33:44,780 and he goes all the way to the ceiling. 584 00:33:44,780 --> 00:33:48,690 And the ceiling is not flat like our ceiling. 585 00:33:48,690 --> 00:33:55,080 The ceiling is this oblique plane. 586 00:33:55,080 --> 00:33:59,380 So z is going to hit his head against the roof at some point, 587 00:33:59,380 --> 00:34:01,440 and he doesn't know where he is going 588 00:34:01,440 --> 00:34:05,870 to hit his head, unless you tell him where that happens. 589 00:34:05,870 --> 00:34:11,248 So he knows he leaves at 0, and he's going to end up where? 590 00:34:11,248 --> 00:34:12,656 STUDENT: 1 minus y minus x. 591 00:34:12,656 --> 00:34:13,739 MAGDALENA TODA: Excellent. 592 00:34:13,739 --> 00:34:15,620 1 minus x minus y. 593 00:34:15,620 --> 00:34:17,620 How do we do that? 594 00:34:17,620 --> 00:34:22,920 We pull z out of that, and say, 1 minus x minus y. 595 00:34:22,920 --> 00:34:27,309 So that is the equation of the shaded purple plane, 596 00:34:27,309 --> 00:34:29,520 and this is as far as you can go. 597 00:34:29,520 --> 00:34:33,550 You cannot go past the roof of your house, 598 00:34:33,550 --> 00:34:38,130 which is the purple plane, the purple shaded plane. 599 00:34:38,130 --> 00:34:39,840 So here you are. 600 00:34:39,840 --> 00:34:40,480 Is this hard? 601 00:34:40,480 --> 00:34:40,980 No. 602 00:34:40,980 --> 00:34:44,060 In many problems on the final and on the midterm, 603 00:34:44,060 --> 00:34:48,199 we tell you, don't even think about solving that, 604 00:34:48,199 --> 00:34:53,199 because we believe you. 605 00:34:53,199 --> 00:34:56,112 Just set up the integral. 606 00:34:56,112 --> 00:34:58,880 I might give you something like that again, just 607 00:34:58,880 --> 00:35:04,250 set up the integral and you have to do that. 608 00:35:04,250 --> 00:35:07,840 But now, I would like to actually work it out, 609 00:35:07,840 --> 00:35:11,690 see how hard it is. 610 00:35:11,690 --> 00:35:14,650 So is this hard to work this out? 611 00:35:14,650 --> 00:35:18,242 612 00:35:18,242 --> 00:35:20,950 I have to do it one at a time, because you see, 613 00:35:20,950 --> 00:35:23,620 I don't have fixed endpoints. 614 00:35:23,620 --> 00:35:28,260 I cannot say, I'm applying the problem with the integral if f 615 00:35:28,260 --> 00:35:32,420 times the integral of g, times-- so I have to integrate one 616 00:35:32,420 --> 00:35:37,800 at a time, because I don't have fixed endpoints. 617 00:35:37,800 --> 00:35:42,410 And the integral of 1dz is z between that and that. 618 00:35:42,410 --> 00:35:47,450 So z, 1 minus x minus y will be what's left over, 619 00:35:47,450 --> 00:35:49,710 and then I have dy, and then I have dx. 620 00:35:49,710 --> 00:35:53,746 And at this point it looks horrible enough, 621 00:35:53,746 --> 00:35:56,216 but we have to pray that in the end 622 00:35:56,216 --> 00:36:01,601 it's not going to be so hard, and I'm going to keep going. 623 00:36:01,601 --> 00:36:05,417 So we have integral from 0 to 1. 624 00:36:05,417 --> 00:36:12,030 We have integral from 0 to 1 minus x.. 625 00:36:12,030 --> 00:36:16,690 I'll just copy and paste it. 626 00:36:16,690 --> 00:36:19,640 Which is integral from 0 to 1. 627 00:36:19,640 --> 00:36:24,716 Now I have to think, and that's dangerous. 628 00:36:24,716 --> 00:36:27,440 I have 1 minux x with respect to y. 629 00:36:27,440 --> 00:36:28,580 This is going to be ugly. 630 00:36:28,580 --> 00:36:34,480 That's a constant with respect to y, and times y, 631 00:36:34,480 --> 00:36:39,172 minus-- integrate with respect to y, y is [? what? ?] 632 00:36:39,172 --> 00:36:39,880 y squared over 2. 633 00:36:39,880 --> 00:36:45,620 634 00:36:45,620 --> 00:36:49,140 Between y equals 0 down. 635 00:36:49,140 --> 00:36:51,930 That's going to save my life, because for y equals 0, 636 00:36:51,930 --> 00:36:55,440 0 is going to be a great simplification. 637 00:36:55,440 --> 00:37:00,064 And for y equals 1 minus x on top, 638 00:37:00,064 --> 00:37:02,230 hopefully it's not going to be the end of the world. 639 00:37:02,230 --> 00:37:07,460 It looks ugly now, but I'm an optimistic person, 640 00:37:07,460 --> 00:37:11,164 so I hope that this is going to get better. 641 00:37:11,164 --> 00:37:14,314 And I can see it's going to get better. 642 00:37:14,314 --> 00:37:15,730 So I have integral from here to 1. 643 00:37:15,730 --> 00:37:18,270 And now I say, OK, let me think. 644 00:37:18,270 --> 00:37:20,330 Life is not so bad. 645 00:37:20,330 --> 00:37:21,450 Why? 646 00:37:21,450 --> 00:37:25,380 1 minus x, 1 minus x is 1 minus x squared. 647 00:37:25,380 --> 00:37:28,300 I could think faster, you could think faster than me, 648 00:37:28,300 --> 00:37:30,010 but I don't want to rush. 649 00:37:30,010 --> 00:37:36,120 1 minus x squared over 2. 650 00:37:36,120 --> 00:37:37,360 So it's not bad at all. 651 00:37:37,360 --> 00:37:42,994 Look, I'm getting this guy who is beautiful in the end, when 652 00:37:42,994 --> 00:37:45,429 I'm going to integrate, and you have 653 00:37:45,429 --> 00:37:48,351 to keep your fingers crossed for me, 654 00:37:48,351 --> 00:37:53,710 because I don't know what I'm going to get. 655 00:37:53,710 --> 00:38:03,410 So I get integral from 0 to 1, 1/2 out, 1 minus x squared dx. 656 00:38:03,410 --> 00:38:05,806 Is this bad? 657 00:38:05,806 --> 00:38:08,340 Can you do this by yourself without my help? 658 00:38:08,340 --> 00:38:11,430 What are you going to do? 659 00:38:11,430 --> 00:38:16,430 x squared minus 2x plus 1. 660 00:38:16,430 --> 00:38:17,470 That's the square. 661 00:38:17,470 --> 00:38:19,373 STUDENT: Why not just change it? 662 00:38:19,373 --> 00:38:20,206 MAGDALENA TODA: Huh? 663 00:38:20,206 --> 00:38:22,359 STUDENT: Why not just change it? 664 00:38:22,359 --> 00:38:24,150 MAGDALENA TODA: You can do it in many ways. 665 00:38:24,150 --> 00:38:26,310 You can do whatever you want. 666 00:38:26,310 --> 00:38:27,690 I don't care. 667 00:38:27,690 --> 00:38:31,557 I want you to the right answer one way or another. 668 00:38:31,557 --> 00:38:35,469 So I'm going to clean a little bit around here. 669 00:38:35,469 --> 00:38:39,870 670 00:38:39,870 --> 00:38:41,337 It's dirty. 671 00:38:41,337 --> 00:38:42,315 You do it. 672 00:38:42,315 --> 00:38:46,716 You have one minute and a half to finish. 673 00:38:46,716 --> 00:38:49,650 And tell me what you get. 674 00:38:49,650 --> 00:38:52,390 STUDENT: 1 minus x cubed over six negative. 675 00:38:52,390 --> 00:38:53,400 MAGDALENA TODA: No, no. 676 00:38:53,400 --> 00:38:54,590 In the end is the number. 677 00:38:54,590 --> 00:38:56,890 What number? 678 00:38:56,890 --> 00:38:58,740 But you have to go slow. 679 00:38:58,740 --> 00:39:01,500 I need three people to give me the same answer. 680 00:39:01,500 --> 00:39:04,160 Because then it's like in that proverb, if two people tell 681 00:39:04,160 --> 00:39:05,770 you drunk, you go to bed. 682 00:39:05,770 --> 00:39:09,925 I need three people to tell me what the answer is in order 683 00:39:09,925 --> 00:39:12,900 to believe them. 684 00:39:12,900 --> 00:39:14,390 Three witnesses. 685 00:39:14,390 --> 00:39:16,190 STUDENT: 1 [? by ?] 6. 686 00:39:16,190 --> 00:39:18,860 MAGDALENA TODA: Who got 1 over 6, raise hand? 687 00:39:18,860 --> 00:39:20,410 Wow, guys, you're fast. 688 00:39:20,410 --> 00:39:22,780 Can you raise hands again? 689 00:39:22,780 --> 00:39:26,080 OK, being fast doesn't mean you're the best, 690 00:39:26,080 --> 00:39:30,030 but I agree you do a very good job, all of you in general. 691 00:39:30,030 --> 00:39:35,470 So I believe there were eight people or nine people. 692 00:39:35,470 --> 00:39:36,500 1 over 6. 693 00:39:36,500 --> 00:39:41,260 Now, how could I have cheated on this problem on the final? 694 00:39:41,260 --> 00:39:43,020 STUDENT: It's a [? junction ?] from this-- 695 00:39:43,020 --> 00:39:43,936 MAGDALENA TODA: Right. 696 00:39:43,936 --> 00:39:48,350 In this case, being a volume, I would have been lucky enough, 697 00:39:48,350 --> 00:39:50,880 and say, it is the volume of a tetrahedron. 698 00:39:50,880 --> 00:39:55,940 I go, the tetrahedron has area of the base 1/2, 699 00:39:55,940 --> 00:39:57,410 the height is 1. 700 00:39:57,410 --> 00:40:01,110 1/2 times 1 divided by 3 is 1/6. 701 00:40:01,110 --> 00:40:06,152 And just pretend on the final that I actually 702 00:40:06,152 --> 00:40:07,110 computed everything. 703 00:40:07,110 --> 00:40:11,530 I could have done that, from here jump to here, or from here 704 00:40:11,530 --> 00:40:13,170 jump straight to here. 705 00:40:13,170 --> 00:40:15,950 And ask you, how did you get from here to here? 706 00:40:15,950 --> 00:40:18,780 And you say, I'm a genius. 707 00:40:18,780 --> 00:40:20,390 Could I not believe you? 708 00:40:20,390 --> 00:40:22,544 I have to give you full credit. 709 00:40:22,544 --> 00:40:28,310 However, what would you have done if I said compute, 710 00:40:28,310 --> 00:40:30,990 I don't know, something worse, something 711 00:40:30,990 --> 00:40:37,330 like triple integral of x, y, z over the tetrahedron 2. 712 00:40:37,330 --> 00:40:39,600 In that case, you cannot cheat. 713 00:40:39,600 --> 00:40:42,060 You're not lucky enough to cheat. 714 00:40:42,060 --> 00:40:43,680 You're lucky enough to cheat when 715 00:40:43,680 --> 00:40:46,630 you have a volume of a prism, you 716 00:40:46,630 --> 00:40:49,660 have a volume of-- and volume means this should be the number 717 00:40:49,660 --> 00:40:51,870 1 here, number 1. 718 00:40:51,870 --> 00:40:56,190 So if you have number 1, here, or I ask you for the volume, 719 00:40:56,190 --> 00:40:58,610 and it's a prism, or tetrahedron, or sphere, 720 00:40:58,610 --> 00:41:01,970 or something, go ahead and cheat, 721 00:41:01,970 --> 00:41:04,300 and pretend that you're actually solving the integral. 722 00:41:04,300 --> 00:41:05,064 Yes, sir. 723 00:41:05,064 --> 00:41:07,147 STUDENT: What would that represent, geometrically, 724 00:41:07,147 --> 00:41:09,150 the triple integral of x, y, z? 725 00:41:09,150 --> 00:41:12,190 MAGDALENA TODA: It's a weighted triple integral. 726 00:41:12,190 --> 00:41:16,390 I'm going to give you examples later. 727 00:41:16,390 --> 00:41:20,150 When you have mass and momentum, when you compute the center 728 00:41:20,150 --> 00:41:25,140 map, or you compute the mass, and somebody give you 729 00:41:25,140 --> 00:41:26,450 densities. 730 00:41:26,450 --> 00:41:32,840 Let me get -- If you have a triple integral over row at x, 731 00:41:32,840 --> 00:41:36,640 y, z, this could be it, but I [? recall ?] it row 732 00:41:36,640 --> 00:41:39,540 for a reason, not just for fun. 733 00:41:39,540 --> 00:41:42,600 And here, dx, dy, dz. 734 00:41:42,600 --> 00:41:45,020 Very good question, and it's very insightful. 735 00:41:45,020 --> 00:41:48,140 For a physicist or engineer, the guy 736 00:41:48,140 --> 00:41:52,030 needs to know why we take this weighted [? integral. ?] 737 00:41:52,030 --> 00:41:55,390 If row is the density of an object, 738 00:41:55,390 --> 00:41:59,390 if it's everywhere the same, if row is a homogeneous density, 739 00:41:59,390 --> 00:42:02,090 for that piece of cheddar cheese-- Oh my God 740 00:42:02,090 --> 00:42:05,110 I'm so hungry-- row would be constant. 741 00:42:05,110 --> 00:42:08,290 If it's a quality cheddar made in Vermont in the best 742 00:42:08,290 --> 00:42:12,620 factory, whatever, row would be considered to be a constant, 743 00:42:12,620 --> 00:42:13,724 right? 744 00:42:13,724 --> 00:42:15,015 And in that case, what happens? 745 00:42:15,015 --> 00:42:17,830 If it's a constant, it's a gets out, 746 00:42:17,830 --> 00:42:21,000 and then you have row times triple integral 1 747 00:42:21,000 --> 00:42:22,972 dV, which is what? 748 00:42:22,972 --> 00:42:25,258 The volume. 749 00:42:25,258 --> 00:42:28,080 And then the volume times the density of the piece of cheese 750 00:42:28,080 --> 00:42:28,940 will be? 751 00:42:28,940 --> 00:42:29,934 STUDENT: [INAUDIBLE] 752 00:42:29,934 --> 00:42:32,916 MAGDALENA TODA: The mass of the piece of cheese, 753 00:42:32,916 --> 00:42:36,680 in kilograms, because I think in kilograms because I 754 00:42:36,680 --> 00:42:38,420 can eat more. 755 00:42:38,420 --> 00:42:39,040 OK? 756 00:42:39,040 --> 00:42:42,040 Actually, no, I'm just kidding. 757 00:42:42,040 --> 00:42:46,660 You guys have really-- I mean, 2 pounds and 1 kilogram 758 00:42:46,660 --> 00:42:47,640 is not the same thin. 759 00:42:47,640 --> 00:42:50,079 Can somebody tell me why? 760 00:42:50,079 --> 00:42:52,120 I mean, you know it's not the same thing because, 761 00:42:52,120 --> 00:42:53,695 the approximation. 762 00:42:53,695 --> 00:42:57,870 But I'm claiming you cannot compare pounds with kilograms 763 00:42:57,870 --> 00:42:58,460 at all. 764 00:42:58,460 --> 00:43:00,084 STUDENT: Pounds is a measure of weight, 765 00:43:00,084 --> 00:43:01,719 whereas kilograms is a measure of mass. 766 00:43:01,719 --> 00:43:02,802 MAGDALENA TODA: Excellent. 767 00:43:02,802 --> 00:43:05,220 Kilogram is a measure of mass, pound 768 00:43:05,220 --> 00:43:07,980 is a measure of the gravitational force. 769 00:43:07,980 --> 00:43:11,300 It's a force measure. 770 00:43:11,300 --> 00:43:15,990 So OK. 771 00:43:15,990 --> 00:43:20,175 772 00:43:20,175 --> 00:43:22,794 Which reminds me, there was-- I don't 773 00:43:22,794 --> 00:43:24,960 know if you saw this short movie for 15 minutes that 774 00:43:24,960 --> 00:43:29,210 got an award the previous Oscar last year, 775 00:43:29,210 --> 00:43:35,295 and there was an old lady telling another old lady 776 00:43:35,295 --> 00:43:41,970 in Great Britain, get 2 pounds of sausage. 777 00:43:41,970 --> 00:43:45,130 And the other one says, I thought we got metric, 778 00:43:45,130 --> 00:43:47,316 because we are in the European Union. 779 00:43:47,316 --> 00:43:51,030 And she said, then get me just the one meter of sausage, 780 00:43:51,030 --> 00:43:52,610 or something. 781 00:43:52,610 --> 00:43:55,090 So it was funny. 782 00:43:55,090 --> 00:43:56,450 So it can be mass. 783 00:43:56,450 --> 00:44:00,200 But what if this density is not the same? 784 00:44:00,200 --> 00:44:04,430 This is exactly why we need to do the integral. 785 00:44:04,430 --> 00:44:07,910 Imagine that the density is-- we have 786 00:44:07,910 --> 00:44:10,750 a piece of cake with layers. 787 00:44:10,750 --> 00:44:13,120 And again, you see how hungry I am. 788 00:44:13,120 --> 00:44:18,510 So you have a layer, and then cream, or whipped cream, 789 00:44:18,510 --> 00:44:22,070 or mousse, and another layer, and another mousse. 790 00:44:22,070 --> 00:44:25,180 The density will vary. 791 00:44:25,180 --> 00:44:29,351 But then there are bodies in physics where the density is 792 00:44:29,351 --> 00:44:30,825 even a smooth function. 793 00:44:30,825 --> 00:44:38,470 It doesn't matter that you have such a discontinuous function. 794 00:44:38,470 --> 00:44:39,220 What would you do? 795 00:44:39,220 --> 00:44:40,090 You just split. 796 00:44:40,090 --> 00:44:47,380 You have triple row 1 for the first layer, then triple row 2 797 00:44:47,380 --> 00:44:50,420 for the second later, the layer of mousse, 798 00:44:50,420 --> 00:44:52,700 and then let's say it's tiramisu, 799 00:44:52,700 --> 00:44:57,916 you have another layer, row three, dV3 for the top layer 800 00:44:57,916 --> 00:44:58,667 of the tiramisu. 801 00:44:58,667 --> 00:45:00,250 STUDENT: Can any row be kept constant? 802 00:45:00,250 --> 00:45:02,826 MAGDALENA TODA: So these are discontinuous. 803 00:45:02,826 --> 00:45:04,650 They are all constant, though. 804 00:45:04,650 --> 00:45:06,600 That would be the great advantage, 805 00:45:06,600 --> 00:45:11,291 because presumably mousse would have the constant density, 806 00:45:11,291 --> 00:45:14,850 the dough has a constant, homogeneous density, and so on. 807 00:45:14,850 --> 00:45:18,620 But what if the density varies in that body from point 808 00:45:18,620 --> 00:45:19,800 to point? 809 00:45:19,800 --> 00:45:23,060 Then nobody can do it by approximation. 810 00:45:23,060 --> 00:45:26,982 You'd say volume, mass 1 plus mass 2 plus mass 3 plus mass 1. 811 00:45:26,982 --> 00:45:31,250 You have to have a triple integral where this row varies, 812 00:45:31,250 --> 00:45:33,280 constantly varies. 813 00:45:33,280 --> 00:45:35,410 And for an engineer, that would be a puzzle. 814 00:45:35,410 --> 00:45:38,565 Poor engineers says, oh my God, the density 815 00:45:38,565 --> 00:45:40,190 is different from one point to another. 816 00:45:40,190 --> 00:45:42,950 I have to find an approximated function 817 00:45:42,950 --> 00:45:48,040 for that density moving from one point to another on that body. 818 00:45:48,040 --> 00:45:55,060 And then the only way to do it would be to solve an integral. 819 00:45:55,060 --> 00:45:57,445 Imagine that somebody-- now it just occurred, 820 00:45:57,445 --> 00:46:01,710 I never thought about it-- we would 821 00:46:01,710 --> 00:46:05,340 be measured in terms of this type of integral. 822 00:46:05,340 --> 00:46:10,840 Of course, people would be able to measure mass right away. 823 00:46:10,840 --> 00:46:13,390 But then, if you were to know the density-- 824 00:46:13,390 --> 00:46:17,670 you cannot even know the density at every point of the body. 825 00:46:17,670 --> 00:46:21,470 It varies a lot, so every point of our bodies 826 00:46:21,470 --> 00:46:25,930 has a different material and a density. 827 00:46:25,930 --> 00:46:27,114 OK. 828 00:46:27,114 --> 00:46:28,364 STUDENT: Tiramisu. [INAUDIBLE] 829 00:46:28,364 --> 00:46:31,339 830 00:46:31,339 --> 00:46:32,172 MAGDALENA TODA: Huh? 831 00:46:32,172 --> 00:46:33,124 STUDENT: So you use the tiramasu, 832 00:46:33,124 --> 00:46:34,080 you're making me hungry. 833 00:46:34,080 --> 00:46:35,496 MAGDALENA TODA: Yeah, because now, 834 00:46:35,496 --> 00:46:38,020 OK take your mind off the tiramisu. 835 00:46:38,020 --> 00:46:40,250 Think about an exam. 836 00:46:40,250 --> 00:46:42,210 Then you don't-- 837 00:46:42,210 --> 00:46:45,140 STUDENT: Now I'm sick. 838 00:46:45,140 --> 00:46:46,140 MAGDALENA TODA: Exactly. 839 00:46:46,140 --> 00:46:49,775 Now you need something against nausea. 840 00:46:49,775 --> 00:46:53,530 Let's see what else is interesting to do. 841 00:46:53,530 --> 00:46:57,760 842 00:46:57,760 --> 00:47:00,430 I'll give you ten minutes. 843 00:47:00,430 --> 00:47:01,720 How much did I steal from you? 844 00:47:01,720 --> 00:47:07,220 I stole constantly about five minutes of your breaks 845 00:47:07,220 --> 00:47:09,960 for the last few Tuesdays. 846 00:47:09,960 --> 00:47:12,340 STUDENT: So the integral-- 847 00:47:12,340 --> 00:47:14,020 MAGDALENA TODA: The integral of that. 848 00:47:14,020 --> 00:47:18,825 I think I would be fair to give you 10 minutes 849 00:47:18,825 --> 00:47:22,296 as a gift today to compensate. 850 00:47:22,296 --> 00:47:28,272 OK, so remind me to let you go 10 minutes early. 851 00:47:28,272 --> 00:47:32,256 Especially since spring break is coming. 852 00:47:32,256 --> 00:47:37,236 We have a 3D application. 853 00:47:37,236 --> 00:47:40,224 We have several 3D applications. 854 00:47:40,224 --> 00:47:44,115 Let me see which one I want to mimic first. 855 00:47:44,115 --> 00:47:49,290 856 00:47:49,290 --> 00:47:51,370 Yeah. 857 00:47:51,370 --> 00:47:56,483 I'm going to pick my favorite, because I just want to. 858 00:47:56,483 --> 00:48:02,399 859 00:48:02,399 --> 00:48:16,249 So imagine you have a disc that is 860 00:48:16,249 --> 00:48:19,195 x squared plus y squared equals 1 would be the circle. 861 00:48:19,195 --> 00:48:21,650 That's the unit disc on the floor. 862 00:48:21,650 --> 00:48:27,390 863 00:48:27,390 --> 00:48:38,080 And then I have the plane x plus y plus z equals 8. 864 00:48:38,080 --> 00:48:40,060 Then I'm going to draw that plane. 865 00:48:40,060 --> 00:48:41,130 I'll try my best. 866 00:48:41,130 --> 00:48:48,960 867 00:48:48,960 --> 00:48:50,950 It's similar to two examples from the book, 868 00:48:50,950 --> 00:48:53,794 but I did not want to repeat the ones in the book 869 00:48:53,794 --> 00:48:57,066 because I want you to actually read them. 870 00:48:57,066 --> 00:48:59,000 That's kind of the idea. 871 00:48:59,000 --> 00:49:05,840 So you have this picture, and you 872 00:49:05,840 --> 00:49:10,670 realize that we had that in the first octant before. 873 00:49:10,670 --> 00:49:15,650 So I say, I don't want the volume 874 00:49:15,650 --> 00:49:21,000 of the body over the whole disc, only over the part of the disc 875 00:49:21,000 --> 00:49:23,830 which is in the first octant. 876 00:49:23,830 --> 00:49:33,060 So I say, I want this domain D, which is going to be what? 877 00:49:33,060 --> 00:49:35,590 x squared plus y squared less than or equal to 1 878 00:49:35,590 --> 00:49:39,970 in plane, with x positive, y positive. 879 00:49:39,970 --> 00:49:42,740 Do you know what we call that in trigonometry? 880 00:49:42,740 --> 00:49:46,840 881 00:49:46,840 --> 00:49:50,350 Does anybody know what we call this in trigonometry? 882 00:49:50,350 --> 00:49:57,706 883 00:49:57,706 --> 00:50:00,390 Let me put the points while you think. 884 00:50:00,390 --> 00:50:03,144 Hopefully, you are thinking about this. 885 00:50:03,144 --> 00:50:07,000 This is 1 in x-axis. 886 00:50:07,000 --> 00:50:13,910 1, 0, 0, and this is 0, 1, 0, and this is y. 887 00:50:13,910 --> 00:50:18,080 If I were to go up until I meet the plane, 888 00:50:18,080 --> 00:50:20,074 what point would this-- 889 00:50:20,074 --> 00:50:22,260 STUDENT: [INAUDIBLE] 890 00:50:22,260 --> 00:50:27,390 MAGDALENA TODA: What point would this-- on the thing. 891 00:50:27,390 --> 00:50:30,240 892 00:50:30,240 --> 00:50:34,920 STUDENT: 1, 0, 7 and then 0, 1, 7. 893 00:50:34,920 --> 00:50:36,094 MAGDALENA TODA: 1, 0, 7. 894 00:50:36,094 --> 00:50:41,034 895 00:50:41,034 --> 00:50:46,962 This would be you said 0, 1, 6. 896 00:50:46,962 --> 00:50:49,080 And this would be 1, 0, 7. 897 00:50:49,080 --> 00:50:50,800 How did you think about this? 898 00:50:50,800 --> 00:50:52,289 How do you know? 899 00:50:52,289 --> 00:50:53,080 STUDENT: Y plus z-- 900 00:50:53,080 --> 00:50:57,162 MAGDALENA TODA: Because z, because it's on the y-axis, 901 00:50:57,162 --> 00:51:01,025 and since you are on the x-axis here, y has to be 0. 902 00:51:01,025 --> 00:51:01,725 So you're right. 903 00:51:01,725 --> 00:51:02,225 Very good. 904 00:51:02,225 --> 00:51:03,270 Excellent. 905 00:51:03,270 --> 00:51:11,040 Now I'm going to say, I'd like to know 906 00:51:11,040 --> 00:51:34,378 the-- compute the volume of the body that is bounded above 907 00:51:34,378 --> 00:51:52,370 from above by x plus y plus z equals 8, who's projection 908 00:51:52,370 --> 00:52:00,870 on the floor is the domain D. And I'll say 909 00:52:00,870 --> 00:52:02,330 volume of the cylindrical body. 910 00:52:02,330 --> 00:52:10,350 911 00:52:10,350 --> 00:52:15,020 So how could you obtain such a, again-- No, 912 00:52:15,020 --> 00:52:17,460 this is Murphy's Law. 913 00:52:17,460 --> 00:52:25,009 OK, how could you obtain such an object, such a cylinder? 914 00:52:25,009 --> 00:52:27,009 STUDENT: Take a pencil, and cut it into fourths. 915 00:52:27,009 --> 00:52:27,842 MAGDALENA TODA: Huh? 916 00:52:27,842 --> 00:52:31,100 STUDENT: Take like a cylindrical pencil and cut it into fourths. 917 00:52:31,100 --> 00:52:33,910 MAGDALENA TODA: Take a salami, a piece of salami. 918 00:52:33,910 --> 00:52:39,700 Cut that piece of salami into four, into four quarters. 919 00:52:39,700 --> 00:52:43,868 920 00:52:43,868 --> 00:52:50,440 And then we take, we slice, and we slice like that. 921 00:52:50,440 --> 00:52:51,639 So we have something like-- 922 00:52:51,639 --> 00:52:53,680 STUDENT: I tried to think of a non- food example. 923 00:52:53,680 --> 00:52:55,716 MAGDALENA TODA: --a quarter. 924 00:52:55,716 --> 00:52:59,480 How can I draw this? 925 00:52:59,480 --> 00:53:01,360 OK, this is what it means. 926 00:53:01,360 --> 00:53:03,460 You don't see this one. 927 00:53:03,460 --> 00:53:04,860 You don't see this part. 928 00:53:04,860 --> 00:53:06,130 You don't see this part. 929 00:53:06,130 --> 00:53:07,090 This is curved. 930 00:53:07,090 --> 00:53:10,610 And here, instead of cutting with another perpendicular 931 00:53:10,610 --> 00:53:13,140 plane, along the salami-- so this 932 00:53:13,140 --> 00:53:17,503 is the axis of the salami-- instead of taking the knife 933 00:53:17,503 --> 00:53:20,825 and cutting like that, I'm cutting an oblique plane, 934 00:53:20,825 --> 00:53:26,476 and this is what this oblique plane will do. 935 00:53:26,476 --> 00:53:28,145 STUDENT: If you cut that way, then you 936 00:53:28,145 --> 00:53:30,769 would have only squares. 937 00:53:30,769 --> 00:53:31,730 MAGDALENA TODA: Hmm? 938 00:53:31,730 --> 00:53:42,538 So I'm going to have some oblique-- I cannot draw better. 939 00:53:42,538 --> 00:53:45,442 I don't know how to draw better. 940 00:53:45,442 --> 00:53:47,483 So it's going to be an oblique cut in the salami. 941 00:53:47,483 --> 00:53:50,246 942 00:53:50,246 --> 00:53:53,150 Let's think how we do this problem. 943 00:53:53,150 --> 00:53:55,145 Elementary, it will be a piece of cake-- 944 00:53:55,145 --> 00:53:57,716 it would be a piece of-- 945 00:53:57,716 --> 00:53:58,840 STUDENT: A piece of salami. 946 00:53:58,840 --> 00:53:59,631 MAGDALENA TODA: No. 947 00:53:59,631 --> 00:54:01,200 It wouldn't be apiece of salami. 948 00:54:01,200 --> 00:54:02,858 STUDENT: It could be done. 949 00:54:02,858 --> 00:54:05,680 MAGDALENA TODA: How could we do that quickly with the Calculus 950 00:54:05,680 --> 00:54:07,288 III we know? 951 00:54:07,288 --> 00:54:09,204 STUDENT: Find the triple integral. 952 00:54:09,204 --> 00:54:11,599 Oh, you want us to do the double integral? 953 00:54:11,599 --> 00:54:14,450 MAGDALENA TODA: Double, triple, I don't know what to do. 954 00:54:14,450 --> 00:54:16,000 What do you think is best? 955 00:54:16,000 --> 00:54:17,705 Let's do that triple integral first, 956 00:54:17,705 --> 00:54:20,715 and you'll see that it's the same thing as double integral. 957 00:54:20,715 --> 00:54:31,292 Triple integral over B, the body of the salami, 1 dV. 958 00:54:31,292 --> 00:54:34,280 How can we set it up? 959 00:54:34,280 --> 00:54:37,090 Well, this is a little bit tricky. 960 00:54:37,090 --> 00:54:38,900 It's going to be like that. 961 00:54:38,900 --> 00:54:42,152 962 00:54:42,152 --> 00:54:45,420 We can say, I have a double integral over my domain, 963 00:54:45,420 --> 00:54:51,570 D. When it comes to the z, mister z has to be first. 964 00:54:51,570 --> 00:54:54,500 So mister z says, I'm first. 965 00:54:54,500 --> 00:54:56,500 I know where I'm going. 966 00:54:56,500 --> 00:54:59,630 You guys, x and y are bound together, 967 00:54:59,630 --> 00:55:02,730 mired in the element of area of the circles. 968 00:55:02,730 --> 00:55:06,350 This is like dx dy. 969 00:55:06,350 --> 00:55:07,650 But I am independent from you. 970 00:55:07,650 --> 00:55:08,850 I am z. 971 00:55:08,850 --> 00:55:12,960 So I'm going all the way from the floor to what? 972 00:55:12,960 --> 00:55:14,660 You taught me that. 973 00:55:14,660 --> 00:55:20,250 8 minus x minus y, and 1. 974 00:55:20,250 --> 00:55:22,400 This is the way to do it as a triple integral, 975 00:55:22,400 --> 00:55:24,556 but then Alex will say, I could have 976 00:55:24,556 --> 00:55:26,350 done this as a double integral. 977 00:55:26,350 --> 00:55:28,510 Let me show you how. 978 00:55:28,510 --> 00:55:32,710 I could have done it over the domain D in plane. 979 00:55:32,710 --> 00:55:35,840 Put the function, 8 minus x minus y 980 00:55:35,840 --> 00:55:38,250 is [? B and ?] z from the very beginning, 981 00:55:38,250 --> 00:55:42,640 because that's my altitude function, f of x and y. 982 00:55:42,640 --> 00:55:47,080 So then I say dx dy, dx dy, it doesn't matter. 983 00:55:47,080 --> 00:55:48,496 That's the only theory element. 984 00:55:48,496 --> 00:55:48,995 Fine. 985 00:55:48,995 --> 00:55:51,240 It's the same thing. 986 00:55:51,240 --> 00:55:53,579 This is what I wanted you to observe. 987 00:55:53,579 --> 00:55:55,870 Whether you view it like the triple integral like that, 988 00:55:55,870 --> 00:55:58,410 or you view it as the double integral like that, 989 00:55:58,410 --> 00:56:01,810 it's the same thing. 990 00:56:01,810 --> 00:56:03,260 This is not a headache. 991 00:56:03,260 --> 00:56:06,440 The headache is coming next. 992 00:56:06,440 --> 00:56:07,840 This is not a headache. 993 00:56:07,840 --> 00:56:11,080 So you can do it in two ways. 994 00:56:11,080 --> 00:56:13,790 And I'd like to look at the-- check 995 00:56:13,790 --> 00:56:19,670 the two methods of doing this. 996 00:56:19,670 --> 00:56:23,590 997 00:56:23,590 --> 00:56:27,410 And set up the integrals without solving them. 998 00:56:27,410 --> 00:56:37,520 999 00:56:37,520 --> 00:56:38,645 Can you read my mind? 1000 00:56:38,645 --> 00:56:41,959 Do you realize what I'm asking? 1001 00:56:41,959 --> 00:56:43,500 Imagine that would be on the midterm. 1002 00:56:43,500 --> 00:56:46,520 What do you think I'm asking, the two methods? 1003 00:56:46,520 --> 00:56:49,250 This can be interpreted in many ways. 1004 00:56:49,250 --> 00:56:50,250 There are two methods. 1005 00:56:50,250 --> 00:56:53,670 I mean, one method by doing it with Cartesian 1006 00:56:53,670 --> 00:56:56,300 coordinates x and y. 1007 00:56:56,300 --> 00:56:58,420 The other method is switching to polar coordinates 1008 00:56:58,420 --> 00:57:01,420 and set up the integral without solving. 1009 00:57:01,420 --> 00:57:03,790 And you say, why not solving? 1010 00:57:03,790 --> 00:57:05,450 Because I'm going to cheat. 1011 00:57:05,450 --> 00:57:07,630 I'm going to use a TI-92 to solve it, 1012 00:57:07,630 --> 00:57:11,244 or I'm going to use a Matlab or Maple. 1013 00:57:11,244 --> 00:57:12,825 If it looks a little bit complicated, 1014 00:57:12,825 --> 00:57:15,260 then I don't want to spend my time. 1015 00:57:15,260 --> 00:57:18,742 Actually, engineers, after taking Calc III, 1016 00:57:18,742 --> 00:57:19,690 they know a lot. 1017 00:57:19,690 --> 00:57:22,570 They understand a lot about volumes, areas. 1018 00:57:22,570 --> 00:57:27,170 But do you think if you work on a real-life problem like that, 1019 00:57:27,170 --> 00:57:29,170 that your boss will let you waste your time 1020 00:57:29,170 --> 00:57:30,720 and do the integral by hand? 1021 00:57:30,720 --> 00:57:31,220 STUDENT: No. 1022 00:57:31,220 --> 00:57:33,540 MAGDALENA TODA: Most integrals are really complicated 1023 00:57:33,540 --> 00:57:34,700 in everyday life. 1024 00:57:34,700 --> 00:57:36,390 So what you're going to do is going 1025 00:57:36,390 --> 00:57:40,180 to be a scientific software, like Matlab, which is primarily 1026 00:57:40,180 --> 00:57:44,040 for engineers, Mathematica, which is similar to Matlab, 1027 00:57:44,040 --> 00:57:46,180 but is mainly for mathematicians. 1028 00:57:46,180 --> 00:57:48,165 It was invented at the University 1029 00:57:48,165 --> 00:57:50,665 of Illinois Urbana-Champaign, and they're still 1030 00:57:50,665 --> 00:57:51,790 very proud of it. 1031 00:57:51,790 --> 00:57:54,475 I prefer Matlab because I feel Matlab 1032 00:57:54,475 --> 00:57:58,000 is stronger, has higher capabilities than Mathematica. 1033 00:57:58,000 --> 00:57:59,190 You can use Maple. 1034 00:57:59,190 --> 00:58:05,180 Maple lets you set up the endpoints even as functions. 1035 00:58:05,180 --> 00:58:08,500 And then it's user friendly, you type in this, 1036 00:58:08,500 --> 00:58:10,405 you type in the endpoints. 1037 00:58:10,405 --> 00:58:12,320 It has little windows, here. 1038 00:58:12,320 --> 00:58:14,300 You don't need to know any programming. 1039 00:58:14,300 --> 00:58:17,995 It's made for people who have no programming skills. 1040 00:58:17,995 --> 00:58:20,385 So it's going to show a little window on top, 1041 00:58:20,385 --> 00:58:22,146 here, here, here and here. 1042 00:58:22,146 --> 00:58:24,260 You [? have ?] those, and you press Enter, 1043 00:58:24,260 --> 00:58:26,750 and it's going to spit the answer back at you. 1044 00:58:26,750 --> 00:58:29,000 So this is how engineers actually 1045 00:58:29,000 --> 00:58:30,830 solve the everyday integrals. 1046 00:58:30,830 --> 00:58:32,680 Not by hand. 1047 00:58:32,680 --> 00:58:36,210 I want to be able to set it up in both ways 1048 00:58:36,210 --> 00:58:39,200 before I go home or eat something, right? 1049 00:58:39,200 --> 00:58:43,915 So we don't have to spend a lot of time on it. 1050 00:58:43,915 --> 00:58:47,660 But if you want to tell me how I am going to set it up, 1051 00:58:47,660 --> 00:58:49,730 I would be very grateful. 1052 00:58:49,730 --> 00:58:53,720 So this is Cartesian, and this is polar. 1053 00:58:53,720 --> 00:59:05,710 1054 00:59:05,710 --> 00:59:07,000 All right. 1055 00:59:07,000 --> 00:59:07,840 Who helps me? 1056 00:59:07,840 --> 00:59:10,337 In Cartesian-- which one do you prefer? 1057 00:59:10,337 --> 00:59:11,420 I mean, it doesn't matter. 1058 00:59:11,420 --> 00:59:13,840 You guys are good and smart, and you'll 1059 00:59:13,840 --> 00:59:15,900 figure out what I need to do. 1060 00:59:15,900 --> 00:59:19,380 If I want to do it in terms of vertical strip-- so 1061 00:59:19,380 --> 00:59:21,270 for vertical strip method-- first 1062 00:59:21,270 --> 00:59:25,014 I integrate with respect to y, and then with respect to x. 1063 00:59:25,014 --> 00:59:27,350 And maybe, to test your understanding, 1064 00:59:27,350 --> 00:59:29,640 let me change the order of integrals 1065 00:59:29,640 --> 00:59:33,280 and see how much you understood from that last time. 1066 00:59:33,280 --> 00:59:35,280 STUDENT: [INAUDIBLE] 1067 00:59:35,280 --> 00:59:39,074 MAGDALENA TODA: So x is between what and what? 1068 00:59:39,074 --> 00:59:39,783 STUDENT: 0 and 1. 1069 00:59:39,783 --> 00:59:41,324 MAGDALENA TODA: Look at this picture. 1070 00:59:41,324 --> 00:59:43,493 I have to reproduce this picture like that. 1071 00:59:43,493 --> 00:59:46,930 0 to 1, says Alex, and he's right. 1072 00:59:46,930 --> 00:59:49,242 And why will he decide against-- 1073 00:59:49,242 --> 00:59:50,367 STUDENT: 1 minus x squared. 1074 00:59:50,367 --> 00:59:52,822 MAGDALENA TODA: --square root 1 minus x squared. 1075 00:59:52,822 --> 00:59:55,277 So we know very well what we are going to do, 1076 00:59:55,277 --> 00:59:57,260 what Maple is going to do for us. 1077 00:59:57,260 --> 00:59:59,990 1 square root 1 minus x squared. 1078 00:59:59,990 --> 01:00:01,610 And then what do I put here? 1079 01:00:01,610 --> 01:00:03,450 8 minus x minus y. 1080 01:00:03,450 --> 01:00:06,040 Can I do it by hand? 1081 01:00:06,040 --> 01:00:08,200 Yes, I guarantee to you I can do it by hand. 1082 01:00:08,200 --> 01:00:10,290 Let me tell you why. 1083 01:00:10,290 --> 01:00:13,880 Because when we integrate with respect to y, I get xy. 1084 01:00:13,880 --> 01:00:17,603 So I get xy, and y will be plugged in 1 minus x squared. 1085 01:00:17,603 --> 01:00:23,600 How am I going to solve an integral like this? 1086 01:00:23,600 --> 01:00:28,085 I can the first one with a table, the second one with a u 1087 01:00:28,085 --> 01:00:30,500 substitution. 1088 01:00:30,500 --> 01:00:33,470 On the last one is a little bit painful. 1089 01:00:33,470 --> 01:00:34,970 I'm going to have y squared over 2-- 1090 01:00:34,970 --> 01:00:36,345 STUDENT: That's the easiest part. 1091 01:00:36,345 --> 01:00:38,915 MAGDALENA TODA: According to Alex, yes, you're right. 1092 01:00:38,915 --> 01:00:40,567 Maybe that is the easiest. 1093 01:00:40,567 --> 01:00:42,900 STUDENT: That's the [INAUDIBLE] part you can integrate-- 1094 01:00:42,900 --> 01:00:45,000 MAGDALENA TODA: And I can integrate one at a time, 1095 01:00:45,000 --> 01:00:46,990 and I'm going to waste all my time. 1096 01:00:46,990 --> 01:00:49,250 So if I want to be an efficient engineer, 1097 01:00:49,250 --> 01:00:53,540 and my boss is waiting for the end-of-the-day project, 1098 01:00:53,540 --> 01:00:56,160 of course I'm not going to do this by hand. 1099 01:00:56,160 --> 01:00:59,010 How about the other integral? 1100 01:00:59,010 --> 01:01:01,640 Same integral. 1101 01:01:01,640 --> 01:01:04,790 Same idea, y between 0 and 1. 1102 01:01:04,790 --> 01:01:06,690 And x between 0 and 1103 01:01:06,690 --> 01:01:08,440 STUDENT: Square root of 1 minus y squared. 1104 01:01:08,440 --> 01:01:10,960 MAGDALENA TODA: Square root of 1 minus y squared. 1105 01:01:10,960 --> 01:01:16,340 Because I'll do this guy with horizontal strips, 1106 01:01:16,340 --> 01:01:19,450 and forget about the vertical strips. 1107 01:01:19,450 --> 01:01:23,330 And here's the y-- I rotate my head and it cracks, 1108 01:01:23,330 --> 01:01:27,710 so that means that I need some yoga. 1109 01:01:27,710 --> 01:01:30,530 y is between 0 and 1. 1110 01:01:30,530 --> 01:01:31,406 Or gymnastics. 1111 01:01:31,406 --> 01:01:35,766 So x is between 0 and square root 1112 01:01:35,766 --> 01:01:40,470 1 minus y squared. [INAUDIBLE]. 1113 01:01:40,470 --> 01:01:41,970 And I'll leave it here on the meter. 1114 01:01:41,970 --> 01:01:46,160 And I'm going to make a sample like I promised. 1115 01:01:46,160 --> 01:01:48,530 OK, good. 1116 01:01:48,530 --> 01:01:53,230 How would you do this to set up the polar coordinate integral? 1117 01:01:53,230 --> 01:01:57,780 And that is why Alex said maybe that's a pain because 1118 01:01:57,780 --> 01:02:00,040 of a reason. 1119 01:02:00,040 --> 01:02:03,860 And he's right, it's a little bit painful to solve by hand. 1120 01:02:03,860 --> 01:02:06,730 But again, once you switch to polar, 1121 01:02:06,730 --> 01:02:10,560 you can solve it with a calculator or a computer 1122 01:02:10,560 --> 01:02:14,270 software, scientific software in no time. 1123 01:02:14,270 --> 01:02:19,180 In Maple, you just have to plug in the numbers. 1124 01:02:19,180 --> 01:02:22,059 You cannot plug in theta, I think, as a symbol. 1125 01:02:22,059 --> 01:02:22,600 I'm not sure. 1126 01:02:22,600 --> 01:02:26,630 But you can put theta as t and r will be r, 1127 01:02:26,630 --> 01:02:28,695 or you can use whatever letters you 1128 01:02:28,695 --> 01:02:31,280 want that are roman letters. 1129 01:02:31,280 --> 01:02:35,430 So you have to integrate smartly, here, 1130 01:02:35,430 --> 01:02:38,220 switching to r and theta, and think 1131 01:02:38,220 --> 01:02:41,010 about the meaning of that. 1132 01:02:41,010 --> 01:02:45,490 So first of all, if I put dr d theta, 1133 01:02:45,490 --> 01:02:48,650 I'm not worried that you won't be able to get r and theta, 1134 01:02:48,650 --> 01:02:51,120 because I know you can do it. 1135 01:02:51,120 --> 01:02:56,380 You can prove it to me right now. r between 0 and 1136 01:02:56,380 --> 01:02:56,880 STUDENT: 1. 1137 01:02:56,880 --> 01:02:57,963 MAGDALENA TODA: Excellent. 1138 01:02:57,963 --> 01:03:01,495 And theta, pay attention, between 0 and 1139 01:03:01,495 --> 01:03:02,245 STUDENT: pi over 2 1140 01:03:02,245 --> 01:03:03,328 MAGDALENA TODA: Excellent. 1141 01:03:03,328 --> 01:03:03,970 I'm proud. 1142 01:03:03,970 --> 01:03:04,500 Yes, sir? 1143 01:03:04,500 --> 01:03:06,574 STUDENT: Is it supposed to be r dr 2 theta, 1144 01:03:06,574 --> 01:03:08,035 or are you going to add that later? 1145 01:03:08,035 --> 01:03:10,470 MAGDALENA TODA: I will add it here. 1146 01:03:10,470 --> 01:03:13,403 So the integrand will contain the r. 1147 01:03:13,403 --> 01:03:16,740 Now what do I put in terms of this? 1148 01:03:16,740 --> 01:03:19,080 I left enough room. 1149 01:03:19,080 --> 01:03:23,080 STUDENT: Is it pi over 2, or is it negative pi over 2? 1150 01:03:23,080 --> 01:03:24,600 MAGDALENA TODA: It doesn't matter, 1151 01:03:24,600 --> 01:03:30,960 because I'll have to take that-- we assume always theta to go 1152 01:03:30,960 --> 01:03:35,275 counterclockwise, and go between 0 and pi over 2, 1153 01:03:35,275 --> 01:03:38,760 so that when you start-- let me make this motion. 1154 01:03:38,760 --> 01:03:41,340 You are here at theta equals 0. 1155 01:03:41,340 --> 01:03:42,040 STUDENT: Oh, OK. 1156 01:03:42,040 --> 01:03:42,320 Sorry. 1157 01:03:42,320 --> 01:03:43,799 I got my coordinates mixed around-- 1158 01:03:43,799 --> 01:03:46,417 MAGDALENA TODA: --and counterclockwise to pi over 2. 1159 01:03:46,417 --> 01:03:47,250 [INTERPOSING VOICES] 1160 01:03:47,250 --> 01:03:49,415 1161 01:03:49,415 --> 01:03:50,290 MAGDALENA TODA: Yeah. 1162 01:03:50,290 --> 01:03:54,550 So you go in the trigonometric-- Here, you have 8 minus, 1163 01:03:54,550 --> 01:03:57,750 and who tells me what I'm supposed to type? 1164 01:03:57,750 --> 01:03:58,980 STUDENT: r over x. 1165 01:03:58,980 --> 01:04:07,920 MAGDALENA TODA: r cosine theta minus 1166 01:04:07,920 --> 01:04:08,503 STUDENT: Sine. 1167 01:04:08,503 --> 01:04:11,270 MAGDALENA TODA: r sine theta. 1168 01:04:11,270 --> 01:04:13,830 And let mister whatever his name is, 1169 01:04:13,830 --> 01:04:17,354 the computer, find the answer. 1170 01:04:17,354 --> 01:04:18,880 Can I do it by hand? 1171 01:04:18,880 --> 01:04:20,710 Actually, I can. 1172 01:04:20,710 --> 01:04:27,200 I can, but again, it's not worth it, because it drives me crazy. 1173 01:04:27,200 --> 01:04:29,530 How would I do it by hand? 1174 01:04:29,530 --> 01:04:32,175 I would split the integral into three, 1175 01:04:32,175 --> 01:04:36,090 and I would easily compute 8 times r, 1176 01:04:36,090 --> 01:04:37,340 integrand is going to be easy. 1177 01:04:37,340 --> 01:04:37,839 Right? 1178 01:04:37,839 --> 01:04:39,020 Agree with me? 1179 01:04:39,020 --> 01:04:40,620 Then what am I going to do? 1180 01:04:40,620 --> 01:04:46,936 I'm going to say, an r out times an r, out comes r squared. 1181 01:04:46,936 --> 01:04:50,850 And I have integral of r squared times a function 1182 01:04:50,850 --> 01:04:53,500 of theta only, which is going to be 1183 01:04:53,500 --> 01:04:56,420 sine theta plus cosine theta. 1184 01:04:56,420 --> 01:04:58,810 We are going to say, yes, with a minus, with a minus. 1185 01:04:58,810 --> 01:05:01,660 1186 01:05:01,660 --> 01:05:06,590 Now, when I compute r and theta thingy, 1187 01:05:06,590 --> 01:05:09,800 theta will be between 0 and pi over 2. 1188 01:05:09,800 --> 01:05:12,260 r will be between 0 and 1. 1189 01:05:12,260 --> 01:05:16,350 But I don't care, because Matthew reminded me, 1190 01:05:16,350 --> 01:05:19,160 if you have a product of separate variables, 1191 01:05:19,160 --> 01:05:22,320 life becomes all of the sudden easier for you. 1192 01:05:22,320 --> 01:05:24,930 STUDENT: You've also got to add your integral of [? 8r ?] 1193 01:05:24,930 --> 01:05:25,430 [? dr. ?] 1194 01:05:25,430 --> 01:05:25,900 MAGDALENA TODA: Yeah. 1195 01:05:25,900 --> 01:05:28,180 At the end, I'm going to add the integral of 8r. 1196 01:05:28,180 --> 01:05:30,120 So I take them separately. 1197 01:05:30,120 --> 01:05:32,970 I just look at one chunk. 1198 01:05:32,970 --> 01:05:35,080 And this chunk will be what? 1199 01:05:35,080 --> 01:05:39,280 Can you even see how easy it's going to be with the naked eye? 1200 01:05:39,280 --> 01:05:41,470 Firs of all, integral from 0 to 1, 1201 01:05:41,470 --> 01:05:43,910 r squared dr is a piece of cake. 1202 01:05:43,910 --> 01:05:45,979 How much is that-- piece of salami. 1203 01:05:45,979 --> 01:05:46,520 STUDENT: 1/3. 1204 01:05:46,520 --> 01:05:49,105 MAGDALENA TODA: 1/3. 1205 01:05:49,105 --> 01:05:49,605 Right? 1206 01:05:49,605 --> 01:05:52,080 Because it's r cubed over 3. 1207 01:05:52,080 --> 01:05:53,070 Then you have 1/3. 1208 01:05:53,070 --> 01:05:54,700 That's easy. 1209 01:05:54,700 --> 01:05:58,470 With a minus in front, but I don't care about it in the end. 1210 01:05:58,470 --> 01:06:03,790 What is the integral of sine theta cosine theta? 1211 01:06:03,790 --> 01:06:06,240 STUDENT: Negative [INAUDIBLE]. 1212 01:06:06,240 --> 01:06:11,450 MAGDALENA TODA: Minus cosine theta plus sine theta 1213 01:06:11,450 --> 01:06:14,280 taken between 0 and pi over 2. 1214 01:06:14,280 --> 01:06:15,940 Will this be hard? 1215 01:06:15,940 --> 01:06:19,590 Who's going to tell me what, or how I'm going to get what-- 1216 01:06:19,590 --> 01:06:22,688 we don't compute it now, but I just give you. 1217 01:06:22,688 --> 01:06:24,160 Cosine of pi over 3 is? 1218 01:06:24,160 --> 01:06:24,660 STUDENT: 0. 1219 01:06:24,660 --> 01:06:25,409 MAGDALENA TODA: 0. 1220 01:06:25,409 --> 01:06:26,866 Sine of pi over 2 is? 1221 01:06:26,866 --> 01:06:27,574 STUDENT: Oh yeah. 1222 01:06:27,574 --> 01:06:27,700 1. 1223 01:06:27,700 --> 01:06:28,450 MAGDALENA TODA: 1. 1224 01:06:28,450 --> 01:06:30,970 So this is going to be 1 minus, what's 1225 01:06:30,970 --> 01:06:34,556 the whole thingy computed at 0? 1226 01:06:34,556 --> 01:06:35,430 STUDENT: [INAUDIBLE]. 1227 01:06:35,430 --> 01:06:38,710 MAGDALENA TODA: It's going to be minus 1, but minus minus 1 1228 01:06:38,710 --> 01:06:40,500 is plus 1. 1229 01:06:40,500 --> 01:06:42,930 So I have 2. 1230 01:06:42,930 --> 01:06:46,310 So only this chunk of the integral would be easy. 1231 01:06:46,310 --> 01:06:47,210 Minus 2/3. 1232 01:06:47,210 --> 01:06:48,320 OK? 1233 01:06:48,320 --> 01:06:51,090 So it can be done by hand, but why waste the time when 1234 01:06:51,090 --> 01:06:52,520 you can do it with Maple? 1235 01:06:52,520 --> 01:06:53,110 Yes, sir? 1236 01:06:53,110 --> 01:06:56,330 STUDENT: Where did you get rid of 8? 1237 01:06:56,330 --> 01:06:58,230 On the second, after the 8-- 1238 01:06:58,230 --> 01:06:59,730 MAGDALENA TODA: No, I didn't. 1239 01:06:59,730 --> 01:07:01,870 That's exactly what we were talking. 1240 01:07:01,870 --> 01:07:05,650 Alex says, but you just talked about integral of 8r, 1241 01:07:05,650 --> 01:07:06,970 but you didn't want to do it. 1242 01:07:06,970 --> 01:07:09,100 I said, I didn't want to do it. 1243 01:07:09,100 --> 01:07:12,760 This is just the second chunk of this integral. 1244 01:07:12,760 --> 01:07:17,130 So I know that I can do integral of integral of 8r in no time. 1245 01:07:17,130 --> 01:07:20,138 Then I would need to take this and add that, 1246 01:07:20,138 --> 01:07:21,112 and get the number. 1247 01:07:21,112 --> 01:07:23,060 I don't care about the number. 1248 01:07:23,060 --> 01:07:24,508 I just care about the method. 1249 01:07:24,508 --> 01:07:25,008 Yes, sir? 1250 01:07:25,008 --> 01:07:27,613 STUDENT: Why are the limits from 0 to 1 instead of like 0 1251 01:07:27,613 --> 01:07:29,884 to r squared? 1252 01:07:29,884 --> 01:07:32,866 Because didn't we say earlier the domain 1253 01:07:32,866 --> 01:07:34,854 is x squared plus y squared? 1254 01:07:34,854 --> 01:07:37,549 Wouldn't that be r squared? 1255 01:07:37,549 --> 01:07:38,340 MAGDALENA TODA: No. 1256 01:07:38,340 --> 01:07:39,050 No, wait. 1257 01:07:39,050 --> 01:07:40,208 This is r squared. 1258 01:07:40,208 --> 01:07:40,833 STUDENT: Right. 1259 01:07:40,833 --> 01:07:43,660 Why didn't we plug r squared into the 1 again. 1260 01:07:43,660 --> 01:07:47,324 MAGDALENA TODA: And that means r is between 0 and 1, right? 1261 01:07:47,324 --> 01:07:47,990 STUDENT: Oh, OK. 1262 01:07:47,990 --> 01:07:49,823 MAGDALENA TODA: r squared being less than 1. 1263 01:07:49,823 --> 01:07:51,770 That means r is between 0 and 1. 1264 01:07:51,770 --> 01:07:52,970 OK? 1265 01:07:52,970 --> 01:07:56,570 And one last problem-- no. 1266 01:07:56,570 --> 01:07:58,970 No last problem. 1267 01:07:58,970 --> 01:08:00,770 We have barely 10 minutes. 1268 01:08:00,770 --> 01:08:04,070 So you read from the book some. 1269 01:08:04,070 --> 01:08:07,670 I will come back to this section, and I'll do review. 1270 01:08:07,670 --> 01:08:10,970 Have a wonderful spring break, and I'm 1271 01:08:10,970 --> 01:08:13,670 going to see you after spring break on Tuesday. 1272 01:08:13,670 --> 01:08:16,420 [INTERPOSING VOICES] 1273 01:08:16,420 --> 01:08:18,158