1 00:00:00 --> 00:00:00 2 00:00:00 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:09 offer high quality educational resources for free. 6 00:00:09 --> 00:00:12 To make a donation, or to view additional materials from 7 00:00:12 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:26 PROFESSOR: Now, today I need to get started by finishing 10 00:00:26 --> 00:00:28 up what I did last time. 11 00:00:28 --> 00:00:31 Namely, talking about numerical methods. 12 00:00:31 --> 00:00:34 And I want to just carry out one example. 13 00:00:34 --> 00:00:40 And then I want to fill in one loose end. 14 00:00:40 --> 00:00:48 And then we'll talk about the unit overall. 15 00:00:48 --> 00:01:01 We were talking, last time, about numerical integration. 16 00:01:01 --> 00:01:05 I'm going to illustrate this just with the simplest 17 00:01:05 --> 00:01:07 example that I can. 18 00:01:07 --> 00:01:14 We're going to look at the integral from 1 to 2 of dx / x. 19 00:01:14 --> 00:01:18 Which we know perfectly well already is the log of x 20 00:01:18 --> 00:01:23 evaluated between 1 and 2, which is ln 2 - ln 1. 21 00:01:23 --> 00:01:26 Which is just ln 2. 22 00:01:26 --> 00:01:30 Now, if you punch that into your calculator, you're going 23 00:01:30 --> 00:01:32 to get something like this. 24 00:01:32 --> 00:01:41 I hope I saved it here. 25 00:01:41 --> 00:01:41 Yeah. 26 00:01:41 --> 00:01:53 It's about 0.693147. 27 00:01:53 --> 00:01:55 That's more digits than we're going to get in 28 00:01:55 --> 00:01:56 our discussion here. 29 00:01:56 --> 00:02:02 Anyway, that's about how big this number is. 30 00:02:02 --> 00:02:07 And the numerical integration methods will give you about 31 00:02:07 --> 00:02:13 as much accuracy as you can get on the function itself. 32 00:02:13 --> 00:02:15 And, of course, some functions we may have more 33 00:02:15 --> 00:02:17 trouble approximating. 34 00:02:17 --> 00:02:19 But the function 1 / x, we know pretty well how to do, because 35 00:02:19 --> 00:02:21 we know how to divide. 36 00:02:21 --> 00:02:25 So since the function that we're integrating here is 1 / 37 00:02:25 --> 00:02:28 x, it's going to be not too difficult to get 38 00:02:28 --> 00:02:29 some arithmetic. 39 00:02:29 --> 00:02:32 Nevertheless, I'm going to do this in the simplest 40 00:02:32 --> 00:02:34 possible case. 41 00:02:34 --> 00:02:44 Namely, just with two intervals. 42 00:02:44 --> 00:02:47 Now, you really can't expect things to work so well 43 00:02:47 --> 00:02:48 with two intervals. 44 00:02:48 --> 00:02:52 That's a pretty ridiculous approximation to your function. 45 00:02:52 --> 00:02:55 When you have two intervals, that means you're looking at 46 00:02:55 --> 00:02:58 the graph of this hyperbola. 47 00:02:58 --> 00:03:04 And you have 1 here, and you have 2 here and you have 3/2. 48 00:03:04 --> 00:03:07 And you're really only keeping track of the values 49 00:03:07 --> 00:03:10 at these three spots. 50 00:03:10 --> 00:03:14 So the idea that you can approximate the area just by 51 00:03:14 --> 00:03:18 knowing the values of three places is a little bit of a 52 00:03:18 --> 00:03:21 stretch of the imagination. 53 00:03:21 --> 00:03:24 But we're going to try it anyway. 54 00:03:24 --> 00:03:33 Now, the trapezoidal rule is the following formula. 55 00:03:33 --> 00:03:38 It's delta x ( 1/2 the first value + the second value 56 00:03:38 --> 00:03:42 + 1/2 the third value). 57 00:03:42 --> 00:03:47 In this case, the pattern is 1/2, 1, 1, 1, 1, 1, 1/2. 58 00:03:47 --> 00:03:52 And in this case, delta x = 1/2 because this 59 00:03:52 --> 00:03:55 interval's of length 1. 60 00:03:55 --> 00:03:57 The b - a, right. 61 00:03:57 --> 00:03:59 Let's just point that out here. 62 00:03:59 --> 00:04:05 Here, b = 2. a = 1. b - a = 1. 63 00:04:05 --> 00:04:08 And the number n = 2. 64 00:04:08 --> 00:04:16 And so, delta x, which is b - a / n = 1/2. 65 00:04:16 --> 00:04:17 So here's what we get. 66 00:04:17 --> 00:04:19 And let's just see what this number is. 67 00:04:19 --> 00:04:23 It's 1/2 of the value at here. 68 00:04:23 --> 00:04:25 Well, so let's just check what these values are. 69 00:04:25 --> 00:04:29 This value is 1, this value over here is 2/3, and 70 00:04:29 --> 00:04:33 the last value is 1/2. 71 00:04:33 --> 00:04:36 Because the function, of course, was y = 1 / x. 72 00:04:36 --> 00:04:38 And those were the three values that we have. 73 00:04:38 --> 00:04:47 So y0, this one is y0, this one is y1, and this one is y2. 74 00:04:47 --> 00:05:00 Now, here we have (1/2* 1 + 2/3 + 1/2 * 1/2). 75 00:05:00 --> 00:05:05 Now, on an exam, I don't expect you to add up long messes 76 00:05:05 --> 00:05:07 of numbers like this. 77 00:05:07 --> 00:05:10 When you have two numbers, I expect you to add them up if 78 00:05:10 --> 00:05:12 they're reasonable, or subtract them. 79 00:05:12 --> 00:05:14 Just as we do when we take antiderivatives. 80 00:05:14 --> 00:05:18 Like, for example, I don't want you to leave the answer to 81 00:05:18 --> 00:05:19 an integration like this in this form. 82 00:05:19 --> 00:05:21 I want you to simplify it at least down to here. 83 00:05:21 --> 00:05:23 And I of course don't expect you to know the 84 00:05:23 --> 00:05:24 numerical approximation. 85 00:05:24 --> 00:05:27 But I certainly expect you to be able to do that. 86 00:05:27 --> 00:05:29 On the other hand, when the arithmetic gets a little bit 87 00:05:29 --> 00:05:32 long, you can relax a little bit. 88 00:05:32 --> 00:05:35 But I did carry this out on my calculator. 89 00:05:35 --> 00:05:41 Unless I'm mistaken, it's about 0.96. 90 00:05:41 --> 00:05:44 It's pretty far off. 91 00:05:44 --> 00:05:47 So remember what it was. 92 00:05:47 --> 00:05:49 It's what you get when you get these straight lines. 93 00:05:49 --> 00:05:52 And there are these little extra pieces of junk there. 94 00:05:52 --> 00:05:55 Now, don't trust that too much, but the point 95 00:05:55 --> 00:06:00 is that it's far off. 96 00:06:00 --> 00:06:09 So now, let's take a look at Simpson's Rule. 97 00:06:09 --> 00:06:13 And I claim that Simpson's Rule is surprisingly accurate. 98 00:06:13 --> 00:06:15 In this case, really, even a little more than 99 00:06:15 --> 00:06:17 it deserves to be. 100 00:06:17 --> 00:06:29 The formula is (delta x / 3) ( y0 + 4 y1 + y2). 101 00:06:29 --> 00:06:33 So the pattern is 1, 4, 1, or 1, 4 and then it alternates 102 00:06:33 --> 00:06:39 2's and 4's until 4, 1 at the very end. 103 00:06:39 --> 00:06:44 And if I just plug in the numbers now, what I get is 1/6, 104 00:06:44 --> 00:06:47 because delta x = 1/2 again. 105 00:06:47 --> 00:06:50 And the value for y0 = 1. 106 00:06:50 --> 00:06:55 And the value for y1 = 2/3. 107 00:06:55 --> 00:06:59 And the value for y2 = 1/2. 108 00:06:59 --> 00:07:03 So here's the estimate in this case. 109 00:07:03 --> 00:07:08 And this one I did carry out carefully. 110 00:07:08 --> 00:07:16 And it came out to 0.69444. 111 00:07:16 --> 00:07:19 Which is actually pretty impressive, if you 112 00:07:19 --> 00:07:20 think about it. 113 00:07:20 --> 00:07:26 Given what the logarithm is. 114 00:07:26 --> 00:07:30 Now, what's going on with Simpson's Rule 115 00:07:30 --> 00:07:33 in general is this. 116 00:07:33 --> 00:07:43 If you -- Simpson's minus the exact answer. 117 00:07:43 --> 00:07:46 In absolute value, is approximately of the 118 00:07:46 --> 00:07:49 size of (delta x)^ 4. 119 00:07:50 --> 00:07:52 That's really the way it behaves. 120 00:07:52 --> 00:08:03 Which means that if delta x is about 1/10, so if we had 121 00:08:03 --> 00:08:07 divided this up into 10, intervals which we didn't, but 122 00:08:07 --> 00:08:09 if we'd divided it up into 10 intervals, then you could 123 00:08:09 --> 00:08:15 expect that delta x, the error would be about 10 ^ - 4. 124 00:08:15 --> 00:08:21 In other words, four digits of accuracy here for this thing. 125 00:08:21 --> 00:08:26 But the exact analysis of this, a more careful analysis of 126 00:08:26 --> 00:08:27 this, is in your textbook. 127 00:08:27 --> 00:08:30 And I'm not going to do. 128 00:08:30 --> 00:08:33 But I just want to point out that it is an effective method. 129 00:08:33 --> 00:08:36 It really does give you nice four-digit with manageable, 130 00:08:36 --> 00:08:38 you could even really do it by hand. 131 00:08:38 --> 00:08:40 It's so convenient. 132 00:08:40 --> 00:08:42 The Simpson's Rule. 133 00:08:42 --> 00:08:46 Whereas the other rules aren't really that impressive as 134 00:08:46 --> 00:08:51 far as giving fairly accurate answers. 135 00:08:51 --> 00:08:55 The last little remark to make is that the reason 136 00:08:55 --> 00:08:59 is that Simpson's Rule is matching a parabola. 137 00:08:59 --> 00:09:03 And somehow the parabola follows this curve better. 138 00:09:03 --> 00:09:05 It's giving the exact answer. 139 00:09:05 --> 00:09:07 So I'll mention this. 140 00:09:07 --> 00:09:27 Simpson's Rule is derived using the exact answer for all 141 00:09:27 --> 00:09:32 degree 2 polynomials. 142 00:09:32 --> 00:09:36 In other words, parabolas. 143 00:09:36 --> 00:09:38 All parabolas. 144 00:09:38 --> 00:09:40 But even all the ones of lower degree. 145 00:09:40 --> 00:09:43 So straight lines would work, and constants 146 00:09:43 --> 00:09:44 would work as well. 147 00:09:44 --> 00:09:47 Whereas the other ones only work for, say, straight lines. 148 00:09:47 --> 00:09:51 The trapezoidal rule only works for straight lines. 149 00:09:51 --> 00:09:53 But ther isn't a weird accident. 150 00:09:53 --> 00:09:55 It turns out that it also works for cubics. 151 00:09:55 --> 00:09:59 Once you get the formulas, it works for cubics. 152 00:09:59 --> 00:10:06 So it's also exact for cubics. 153 00:10:06 --> 00:10:12 And that's what explains the fourth order validity. 154 00:10:12 --> 00:10:15 The last thing that I want to point out is that this 155 00:10:15 --> 00:10:17 is extremely vague, that I said there. 156 00:10:17 --> 00:10:20 And you should be a little bit cautious about it. 157 00:10:20 --> 00:10:30 You need to watch out for 1 / x for x near 0. 158 00:10:30 --> 00:10:34 All bets are off if the function is singular. 159 00:10:34 --> 00:10:36 And there's a lot of area under there. 160 00:10:36 --> 00:10:40 And it's also true that if the derivative messes up, 161 00:10:40 --> 00:10:41 you're in trouble too. 162 00:10:41 --> 00:10:44 You really need for the function to be nice and smooth 163 00:10:44 --> 00:10:46 in order for Simpson's Rule to work. 164 00:10:46 --> 00:10:48 This is woth out. 165 00:10:48 --> 00:10:54 That's a real woth out, but try to. 166 00:10:54 --> 00:10:55 Watch out. 167 00:10:55 --> 00:10:58 Watch out for whenever x near 0. 168 00:10:58 --> 00:11:00 Then this thing doesn't work. 169 00:11:00 --> 00:11:03 This thing really depends on bounds on derivatives. 170 00:11:03 --> 00:11:05 But I'm going to be relatively vague about that. 171 00:11:05 --> 00:11:12 I'm not attempting to give you an error analysis here. 172 00:11:12 --> 00:11:16 OK, so if you were doing this on an exam, how do 173 00:11:16 --> 00:11:21 you remember this strange pattern of numbers? 174 00:11:21 --> 00:11:27 The one thing that I want to recommend to you is, as a way 175 00:11:27 --> 00:11:37 of remembering it, so the one mnemonic device, we'll call it 176 00:11:37 --> 00:11:41 a mnemonic device here, for remembering what it is that 177 00:11:41 --> 00:11:48 you're doing, is to remind yourself of what happens for 178 00:11:48 --> 00:11:50 the simplest possible case. 179 00:11:50 --> 00:11:53 Which is f ( x) = 1. 180 00:11:53 --> 00:11:57 It seems very modest, but if it doesn't give you the exact 181 00:11:57 --> 00:12:02 answer for f (x) = 1, you've got the wrong weightings. 182 00:12:02 --> 00:12:06 And here, if you check out what happens in the first formula 183 00:12:06 --> 00:12:13 here, y0 / 2 + y1 +..., well, we'll go all the way 184 00:12:13 --> 00:12:17 to yn - 1 + yn / 2. 185 00:12:17 --> 00:12:19 If you check that formula out here, this is the 186 00:12:19 --> 00:12:21 trapezoidal rule. 187 00:12:21 --> 00:12:26 If you check it out for this case, then what you get is 188 00:12:26 --> 00:12:29 that this is equal to delta x times what? 189 00:12:29 --> 00:12:30 Well, all of these are 1's. 190 00:12:30 --> 00:12:32 And how many are there in the middle? 191 00:12:32 --> 00:12:34 There are n - 1 of them in the middle. 192 00:12:34 --> 00:12:38 So it's 1/2 + n - 1 + 1/2. 193 00:12:38 --> 00:12:40 At the tail end. 194 00:12:40 --> 00:12:43 So all told it's (delta x)( n). 195 00:12:43 --> 00:12:50 And I remind you that delta x = b - a / n. 196 00:12:50 --> 00:12:57 So, delta x, this thing, is equal to b - a. 197 00:12:57 --> 00:12:58 And that's just as it should be. 198 00:12:58 --> 00:13:02 What we just calculated is an approximation to 199 00:13:02 --> 00:13:04 this integral here. 200 00:13:04 --> 00:13:07 Which is just the area of the rectangle of base 201 00:13:07 --> 00:13:09 b - a and height 1. 202 00:13:09 --> 00:13:12 Which of course is b - a. 203 00:13:12 --> 00:13:15 So this is the check that you got your weighted 204 00:13:15 --> 00:13:18 average correct here. 205 00:13:18 --> 00:13:20 You've put the correct weightings on everything. 206 00:13:20 --> 00:13:23 And you can do this same thing with Simpson's Rule. 207 00:13:23 --> 00:13:31 And match up those quantities. 208 00:13:31 --> 00:13:36 There was a question in the room at some point. 209 00:13:36 --> 00:13:38 No, OK. 210 00:13:38 --> 00:13:45 So now, the next thing I want to do for you is the loose 211 00:13:45 --> 00:13:49 end which I left hanging. 212 00:13:49 --> 00:13:54 Namely, I want to compute that mysterious constant 213 00:13:54 --> 00:13:56 square root of pi / 2. 214 00:13:56 --> 00:14:00 This is really one of the most famous computations 215 00:14:00 --> 00:14:02 in calculus. 216 00:14:02 --> 00:14:04 And it's a very, very clever trick. 217 00:14:04 --> 00:14:09 I certainly don't expect you to come up with this trick. 218 00:14:09 --> 00:14:11 I certainly wouldn't have myself. 219 00:14:11 --> 00:14:15 But it's an important thing to calculate. 220 00:14:15 --> 00:14:17 And it's just very useful. 221 00:14:17 --> 00:14:19 So I'm going to tell you about it. 222 00:14:19 --> 00:14:22 And it's just on the subject that we're dealing with in 223 00:14:22 --> 00:14:28 this unit; namely, slicing. 224 00:14:28 --> 00:14:30 Or adding up. 225 00:14:30 --> 00:14:36 So the first step, which is just something that we already 226 00:14:36 --> 00:14:50 did, was that we found the volume under this curve. 227 00:14:50 --> 00:14:54 This bell-shaped curve, e ^ - r ^2. 228 00:14:54 --> 00:15:00 But rotated around an axis. 229 00:15:00 --> 00:15:09 Rotated around this axis. 230 00:15:09 --> 00:15:10 Around this way. 231 00:15:10 --> 00:15:12 So we figured that out. 232 00:15:12 --> 00:15:15 And that was a relatively short computation. 233 00:15:15 --> 00:15:18 I'm just going to remind you, it goes by shells. 234 00:15:18 --> 00:15:23 We integrate the whole range from 0 to infinity. 235 00:15:23 --> 00:15:29 And we have 2 pi r e^ - r ^2 dr. So this again is the 236 00:15:29 --> 00:15:31 circumference of the shell. 237 00:15:31 --> 00:15:32 This is the height of the shell, and this is the 238 00:15:32 --> 00:15:34 thickness of the shell. 239 00:15:34 --> 00:15:45 Circumference, height, thickness. 240 00:15:45 --> 00:15:47 So we're just taking a little piece here and 241 00:15:47 --> 00:15:48 sweeping it around. 242 00:15:48 --> 00:15:52 And then adding up. 243 00:15:52 --> 00:15:58 And then this antiderivative is pi, - pi e^ - r ^2, 244 00:15:58 --> 00:16:00 evaluated at 0 and infinity. 245 00:16:00 --> 00:16:03 And we worked this out last time. 246 00:16:03 --> 00:16:06 This is pi. 247 00:16:06 --> 00:16:12 It's pi (1 - 0). 248 00:16:12 --> 00:16:13 Which is pi. 249 00:16:13 --> 00:16:15 So the conclusion is that V = pi. 250 00:16:15 --> 00:16:20 We already know that. 251 00:16:20 --> 00:16:23 Now, the problem that we want to deal with now is the problem 252 00:16:23 --> 00:16:31 not of a volume, but an area. 253 00:16:31 --> 00:16:32 And this looks quite different. 254 00:16:32 --> 00:16:37 And of course the answer is going to be different. 255 00:16:37 --> 00:16:38 But let's do it. 256 00:16:38 --> 00:16:40 So this is this question mark here. 257 00:16:40 --> 00:16:42 And I'm going to do the one from minus 258 00:16:42 --> 00:16:47 infinity to infinity. 259 00:16:47 --> 00:16:52 And I'll relate it to what we talked about earlier in this 260 00:16:52 --> 00:16:57 unit, in just a couple of minutes when I show you the 261 00:16:57 --> 00:17:00 procedure that we're going to follow. 262 00:17:00 --> 00:17:04 So here's the quantity and now, what this is interpreted as is 263 00:17:04 --> 00:17:07 the area under this bell curve. 264 00:17:07 --> 00:17:22 This time, Q is really an area. 265 00:17:22 --> 00:17:26 Now, what's going to turn out to happen, is this. 266 00:17:26 --> 00:17:27 This is the trick. 267 00:17:27 --> 00:17:31 We're going to compute V in a different way. 268 00:17:31 --> 00:17:35 And you'll see it laid out in just a second. 269 00:17:35 --> 00:17:42 We will compute V by slices. 270 00:17:42 --> 00:17:45 We're going to slice it like a piece of bread here. 271 00:17:45 --> 00:17:48 We're going to solve for that same thing here. 272 00:17:48 --> 00:17:51 And then, amazingly, what's going to happen is that we 273 00:17:51 --> 00:17:57 will discover that V = Q ^2. 274 00:17:57 --> 00:18:03 That's going to be what's going to come out. 275 00:18:03 --> 00:18:07 And that's the end of the computation that we want. 276 00:18:07 --> 00:18:08 Because actually we already know what V is. 277 00:18:08 --> 00:18:11 We don't want to read this equation forward. 278 00:18:11 --> 00:18:12 We want to read it the other way. 279 00:18:12 --> 00:18:18 We want to say Q^2 = V, which we already know is pi. 280 00:18:18 --> 00:18:29 And so Q = the square root of pi. 281 00:18:29 --> 00:18:33 I haven't shown this yet, this is the weird part. 282 00:18:33 --> 00:18:36 And I'm going to put it in a little box so that we know that 283 00:18:36 --> 00:18:38 this is what we need to check. 284 00:18:38 --> 00:18:43 We need to check this fact here. 285 00:18:43 --> 00:18:45 We haven't done that yet. 286 00:18:45 --> 00:18:51 Now, let me connect this with what we did a few days ago. 287 00:18:51 --> 00:18:56 With what I called one of the important functions of 288 00:18:56 --> 00:19:02 mathematics besides the ones you already know. 289 00:19:02 --> 00:19:05 And so the function that we were faced with, and that we 290 00:19:05 --> 00:19:15 discussed, was this one. 291 00:19:15 --> 00:19:19 And then, we were interested in the value at infinity. 292 00:19:19 --> 00:19:23 We were interested in this. 293 00:19:23 --> 00:19:28 Which, if you draw a picture of it, and you draw the same bell 294 00:19:28 --> 00:19:32 curve, that's the area under half. of it. 295 00:19:32 --> 00:19:35 That's the area starting from 0 and going to infinity. 296 00:19:35 --> 00:19:37 That's the area under half. 297 00:19:37 --> 00:19:46 So this chunk is F of infinity. 298 00:19:46 --> 00:19:50 And now I hope that this part of the connection is 299 00:19:50 --> 00:19:52 not meant to be fancy. 300 00:19:52 --> 00:19:58 The idea here is that Q = 2 F(infinity). 301 00:19:58 --> 00:20:00 This number here. 302 00:20:00 --> 00:20:05 And so F of infinity = to the square root of pi / 2, if we 303 00:20:05 --> 00:20:10 believe what we said on the last panel. 304 00:20:10 --> 00:20:14 And that was the thing that I drew a picture of on the board. 305 00:20:14 --> 00:20:18 Namely, the graph of F looked like this. 306 00:20:18 --> 00:20:23 And there was this asymptote, which was the limit F (x) tends 307 00:20:23 --> 00:20:26 to square root of pi / 2. 308 00:20:26 --> 00:20:27 As x goes to infinity. 309 00:20:27 --> 00:20:30 That was that limiting value. 310 00:20:30 --> 00:20:33 Which is F of infinity. 311 00:20:33 --> 00:20:40 So this is the asymptote. 312 00:20:40 --> 00:20:43 And now I've explained the connection between what we 313 00:20:43 --> 00:20:48 claimed to be 4, which was quite mysterious, and what 314 00:20:48 --> 00:20:50 we're actually going to be able to check now. 315 00:20:50 --> 00:21:07 Concretely, by making this computation. 316 00:21:07 --> 00:21:09 So how in the world can you get something like this. 317 00:21:09 --> 00:21:14 What's in that orange box there, that V = Q ^2. 318 00:21:14 --> 00:21:19 Again, the technique is to use slices here. 319 00:21:19 --> 00:21:21 And I'm going to have to draw you a 3-D picture 320 00:21:21 --> 00:21:24 to visualize the slice. 321 00:21:24 --> 00:21:27 Let's do that. 322 00:21:27 --> 00:21:31 I'm going to draw three axes now, because we're now going to 323 00:21:31 --> 00:21:36 be in three-dimensional space, and I want you to imagine the x 324 00:21:36 --> 00:21:39 axis as coming out of the blackboard, the y axis is 325 00:21:39 --> 00:21:42 horizontal, and there's a new axis, which I'll call the 326 00:21:42 --> 00:21:45 z axis which is going up. 327 00:21:45 --> 00:21:49 So what's happening here is that I'm thinking of this, 328 00:21:49 --> 00:21:51 this is, if you like, some kind of side view. 329 00:21:51 --> 00:21:52 And this is a view where I've tilted things a 330 00:21:52 --> 00:21:57 little bit up to the top. 331 00:21:57 --> 00:22:01 Now, the distribution, or you could think of this target in 332 00:22:01 --> 00:22:03 the plane, where the most likely places to hit were in 333 00:22:03 --> 00:22:05 the middle and it died off. 334 00:22:05 --> 00:22:07 As we went down. 335 00:22:07 --> 00:22:11 Now, I want to draw a picture of this graph. 336 00:22:11 --> 00:22:15 I'm going to draw a picture of e^ - r squared. 337 00:22:15 --> 00:22:18 And it's basically a hump. 338 00:22:18 --> 00:22:22 So I'm going to take the first, the slice along y = 0. 339 00:22:22 --> 00:22:23 The y = 0 slice. 340 00:22:23 --> 00:22:26 And I claim that that goes up like this. 341 00:22:26 --> 00:22:28 And then comes back down. 342 00:22:28 --> 00:22:32 Let me shade this in, so that you can see what 343 00:22:32 --> 00:22:34 kind of a slice this is. 344 00:22:34 --> 00:22:39 This is supposed to be along this vertical plane here. 345 00:22:39 --> 00:22:40 Which is coming out of the blackboard and 346 00:22:40 --> 00:22:42 coming towards you. 347 00:22:42 --> 00:22:43 And that's a slice. 348 00:22:43 --> 00:22:46 Now, I'm going to draw one more slice so that you 349 00:22:46 --> 00:22:49 can see what's happening. 350 00:22:49 --> 00:22:52 I'm going to draw a slice at another place. 351 00:22:52 --> 00:22:52 Along here. 352 00:22:52 --> 00:22:55 This will be y = b. 353 00:22:55 --> 00:22:56 Some other level. 354 00:22:56 --> 00:22:58 And now I'm going to show you what happens. 355 00:22:58 --> 00:23:02 What happens is that the hump dies down a little bit. 356 00:23:02 --> 00:23:06 So the bump is just a little bit lower. 357 00:23:06 --> 00:23:10 And it's going to look a little bit the same way. 358 00:23:10 --> 00:23:11 But it's just going to be a bit smaller. 359 00:23:11 --> 00:23:15 So there's another slice here. 360 00:23:15 --> 00:23:16 Like that. 361 00:23:16 --> 00:23:20 And I want to give a name to these slices. 362 00:23:20 --> 00:23:25 I'm going to call this A ( b). 363 00:23:25 --> 00:23:36 That is, the area of the b slice. 364 00:23:36 --> 00:23:39 Under the surface. 365 00:23:39 --> 00:23:41 OK Yes, question. 366 00:23:41 --> 00:23:47 STUDENT: [INAUDIBLE] 367 00:23:47 --> 00:23:48 PROFESSOR: Yeah, the solid. 368 00:23:48 --> 00:23:48 Yeah. 369 00:23:48 --> 00:23:53 We're trying to figure out this volume here, which is the one 370 00:23:53 --> 00:23:56 we started out with, by slices. 371 00:23:56 --> 00:23:59 So first I have to think of, I'm going to visualize. 372 00:23:59 --> 00:24:01 So here I didn't even visualize. 373 00:24:01 --> 00:24:04 I took a cross section and I thought about how to spin it 374 00:24:04 --> 00:24:06 around without actually doing that in 375 00:24:06 --> 00:24:08 three-dimensional space. 376 00:24:08 --> 00:24:10 But now I'm going to take a different kind of slice. 377 00:24:10 --> 00:24:12 I'm going to take that same bump, which is a 378 00:24:12 --> 00:24:13 three-dimensional object. 379 00:24:13 --> 00:24:16 I'm going to lay it down on a plane. 380 00:24:16 --> 00:24:17 Which looks like this. 381 00:24:17 --> 00:24:20 And then it's a bump here. 382 00:24:20 --> 00:24:22 It's a hump. 383 00:24:22 --> 00:24:27 And now I'm going to try to slice it by various planes. 384 00:24:27 --> 00:24:29 STUDENT: [INAUDIBLE] 385 00:24:29 --> 00:24:36 PROFESSOR: So one way of defining the bump, as you just 386 00:24:36 --> 00:24:38 suggested, is you take this curve and you rotate it 387 00:24:38 --> 00:24:39 around this z-axis. 388 00:24:39 --> 00:24:41 So in other words, you make this the axis of rotation, 389 00:24:41 --> 00:24:43 you spin it around. 390 00:24:43 --> 00:24:44 That's correct. 391 00:24:44 --> 00:24:48 So that shows you that the peaks as you go down here are 392 00:24:48 --> 00:24:50 going to descend the same way. 393 00:24:50 --> 00:24:54 But I don't want to draw those lines. 394 00:24:54 --> 00:24:57 I want to imagine what the parallel slices are. 395 00:24:57 --> 00:24:59 Because I don't want to get cross slices. 396 00:24:59 --> 00:25:01 I want all slices parallel to the same thing. 397 00:25:01 --> 00:25:05 STUDENT: [INAUDIBLE] 398 00:25:05 --> 00:25:05 PROFESSOR: OK. 399 00:25:05 --> 00:25:13 This is not particularly easy to visualize. 400 00:25:13 --> 00:25:17 Now, here's the formula for volume by slices. 401 00:25:17 --> 00:25:23 The formula for volume by slices is that you add up 402 00:25:23 --> 00:25:28 the areas of the slices. 403 00:25:28 --> 00:25:31 That's how you do it. 404 00:25:31 --> 00:25:32 You take each slice. 405 00:25:32 --> 00:25:34 You add the cross-sectional area, and then you take a 406 00:25:34 --> 00:25:38 little thickness, dy, and then you add all of them up. 407 00:25:38 --> 00:25:41 Because this is extending over the whole plane, we're going to 408 00:25:41 --> 00:25:43 have to go all the way from minus infinity to 409 00:25:43 --> 00:25:45 plus infinity. 410 00:25:45 --> 00:25:56 And this is the formula for volumes by slicing. 411 00:25:56 --> 00:26:00 And now our goal, in order to do this calculation, we're 412 00:26:00 --> 00:26:04 going to just fix y = some b. 413 00:26:04 --> 00:26:06 We're just going to fix one of these slices. 414 00:26:06 --> 00:26:13 And we're going to calculate A ( b). 415 00:26:13 --> 00:26:16 That's what we need to do in order to make this 416 00:26:16 --> 00:26:23 procedure succeed. 417 00:26:23 --> 00:26:27 This is the only place where this method works. 418 00:26:27 --> 00:26:32 But it's an important one. 419 00:26:32 --> 00:26:35 In order to make it work, I'm going to have to again 420 00:26:35 --> 00:26:37 draw the plot from a different point of view. 421 00:26:37 --> 00:26:43 I'm going to do the top view. 422 00:26:43 --> 00:26:48 So I want to look down on this x-y plane here. 423 00:26:48 --> 00:26:52 This is the x direction, and here's the y direction. 424 00:26:52 --> 00:26:56 And then again I want to draw my slice. 425 00:26:56 --> 00:27:00 My slice is here. 426 00:27:00 --> 00:27:04 At y = b. 427 00:27:04 --> 00:27:07 So we're just right on top of it. 428 00:27:07 --> 00:27:09 And it's coming up at some kind of bump. 429 00:27:09 --> 00:27:11 Here, with a little higher in the middle and going 430 00:27:11 --> 00:27:15 down on the sides. 431 00:27:15 --> 00:27:20 Now, the formula for the height is this. 432 00:27:20 --> 00:27:25 If I take a distance r here, the formula for the height 433 00:27:25 --> 00:27:31 of the bump is e ^ - r ^2. 434 00:27:31 --> 00:27:35 I'll store that over here. e ^ - r ^2 is the 435 00:27:35 --> 00:27:37 height at this place. 436 00:27:37 --> 00:27:41 If this distance to the origin is r. 437 00:27:41 --> 00:27:43 That's true all the way around. 438 00:27:43 --> 00:27:48 And in terms of b and x, we can figure out that 439 00:27:48 --> 00:27:49 by this right triangle. 440 00:27:49 --> 00:27:52 This height is b, and this distance is x. 441 00:27:52 --> 00:28:02 So r ^2 = b ^2 + x ^2. 442 00:28:02 --> 00:28:02 Question. 443 00:28:02 --> 00:28:15 STUDENT: [INAUDIBLE] 444 00:28:15 --> 00:28:19 PROFESSOR: The question is, is that the x-y plane. 445 00:28:19 --> 00:28:23 So the answer is that over here I cleverly used the letter r. 446 00:28:23 --> 00:28:27 I avoided using y's and z's or anything. 447 00:28:27 --> 00:28:28 And over here, this is the distance r. 448 00:28:28 --> 00:28:31 And you like this, is z, going up. 449 00:28:31 --> 00:28:33 That's the way to think of it. 450 00:28:33 --> 00:28:36 So that all of the letters are consistent. 451 00:28:36 --> 00:28:39 So I just avoided giving it a name. 452 00:28:39 --> 00:28:41 That's good, that's exactly the point. 453 00:28:41 --> 00:28:43 Alright. 454 00:28:43 --> 00:28:46 So now, I claim I have a formula for r ^2. 455 00:28:46 --> 00:28:48 And so I can write this down. 456 00:28:48 --> 00:28:58 This is e ^ - (b ^2 + x ^2). 457 00:28:58 --> 00:29:02 But now I'm going to use the rule of exponents. 458 00:29:02 --> 00:29:07 Which is that this is the same as (e ^ - b ^2) ( e^ - x^2). 459 00:29:07 --> 00:29:11 And that's going to be the main way in which we use the 460 00:29:11 --> 00:29:15 particular function that we're dealing with here. 461 00:29:15 --> 00:29:20 That's really the main step, amazingly. 462 00:29:20 --> 00:29:33 So now I get to compute what A ( b) is. 463 00:29:33 --> 00:29:37 A( b) is the area under a curve. 464 00:29:37 --> 00:29:41 So it's going to be, let me write it over here, A(b) is the 465 00:29:41 --> 00:29:47 area under this curve here. 466 00:29:47 --> 00:29:51 Which is some constant times, so if you imagine, call 467 00:29:51 --> 00:29:53 this thing the name c. 468 00:29:53 --> 00:29:57 Under some curve, c e ^ - x ^2. 469 00:29:57 --> 00:30:06 Where the c = e^ - b ^2. 470 00:30:06 --> 00:30:07 That's what our slice is. 471 00:30:07 --> 00:30:09 In fact, it looks like one of those. 472 00:30:09 --> 00:30:12 It looks like one of those bumps. 473 00:30:12 --> 00:30:14 Here's its formula again. 474 00:30:14 --> 00:30:19 It's the integral from minus infinity to infinity of (e^ 475 00:30:19 --> 00:30:28 - b ^2)( e ^ - x ^2) dx. 476 00:30:28 --> 00:30:31 We just recopied what I had up there. 477 00:30:31 --> 00:30:37 And this is the height at each value of x, with b fixed. 478 00:30:37 --> 00:30:42 And now, so we have a lot of steps here. 479 00:30:42 --> 00:30:44 But each of them is very elementary. 480 00:30:44 --> 00:30:47 The first one was just that law of exponents. 481 00:30:47 --> 00:30:49 That we could split the two into products. 482 00:30:49 --> 00:30:51 Now I'm going to make that splitting even further. 483 00:30:51 --> 00:30:53 This is a constant. 484 00:30:53 --> 00:30:55 It's not varying with x. 485 00:30:55 --> 00:30:59 So I'm going to factor it out of the integral. 486 00:30:59 --> 00:31:02 This is e ^ - b ^2 times the integral from minus infinity 487 00:31:02 --> 00:31:06 to infinity of e^ -x^2 dx. 488 00:31:06 --> 00:31:10 So this might look frightening, but actually it's just the 489 00:31:10 --> 00:31:11 property of an integral. 490 00:31:11 --> 00:31:13 All integrals have this kind of property. 491 00:31:13 --> 00:31:17 You can always factor out a constant. 492 00:31:17 --> 00:31:20 And now here comes the remarkable thing. 493 00:31:20 --> 00:31:24 This is e ^ - b^2 times a number which is 494 00:31:24 --> 00:31:26 now familiar to us. 495 00:31:26 --> 00:31:27 What is this number? 496 00:31:27 --> 00:31:29 This is what we're looking for. 497 00:31:29 --> 00:31:38 This is our unknown, Q. 498 00:31:38 --> 00:31:41 So I've computed A(b), and now I'm ready to finish 499 00:31:41 --> 00:31:43 the problem off. 500 00:31:43 --> 00:31:47 A (b) = ( e^ - b^2) Q. 501 00:31:47 --> 00:31:49 Q is that strange number which we don't know yet. 502 00:31:49 --> 00:31:51 What it is. 503 00:31:51 --> 00:31:55 So now I'm going to compute the whole volume. 504 00:31:55 --> 00:31:58 The whole volume, remember, it's over there, it's minus 505 00:31:58 --> 00:32:02 infinity to infinity, A ( y) dy. 506 00:32:02 --> 00:32:04 And now I'm just going to plug in the formula 507 00:32:04 --> 00:32:06 that we've found for a. 508 00:32:06 --> 00:32:08 Now I'm doing this for each b, so I'm doing it 509 00:32:08 --> 00:32:09 varying over all b's. 510 00:32:09 --> 00:32:12 So I have the integral from minus infinity to infinity. 511 00:32:12 --> 00:32:15 And here I have e^ - y ^2. 512 00:32:15 --> 00:32:17 I've replaced b by y. 513 00:32:17 --> 00:32:19 And now I have Q. 514 00:32:19 --> 00:32:21 And I have dy. 515 00:32:21 --> 00:32:24 I just recopied what I had over there into the 516 00:32:24 --> 00:32:27 formula for slicing. 517 00:32:27 --> 00:32:30 And now, I'm going to do this trick of factoring out the 518 00:32:30 --> 00:32:32 constant a second time. 519 00:32:32 --> 00:32:33 This is a constant. 520 00:32:33 --> 00:32:35 It doesn't depend on y. 521 00:32:35 --> 00:32:38 It's the same for all y, it just will factor out. 522 00:32:38 --> 00:32:40 So this is the same as Q times the integral from minus 523 00:32:40 --> 00:32:47 infinity to infinity, e ^ - y ^2 dy. 524 00:32:47 --> 00:32:52 And now, lo and behold, this expression here. 525 00:32:52 --> 00:32:54 Of course, notice how I defined Q. 526 00:32:54 --> 00:32:56 Let's go back carefully to where Q is defined. 527 00:32:56 --> 00:33:01 Here's Q. 528 00:33:01 --> 00:33:02 This t is a dummy variable. 529 00:33:02 --> 00:33:04 It doesn't matter what I call it. 530 00:33:04 --> 00:33:06 I can call it x, I can call it u, I can call it v. 531 00:33:06 --> 00:33:10 In this case, I've given it two different names. 532 00:33:10 --> 00:33:13 At this stage, I called it x. 533 00:33:13 --> 00:33:14 And at this stage I'm calling it y. 534 00:33:14 --> 00:33:17 But it's the same variable. 535 00:33:17 --> 00:33:20 And so this little chunk is Q and altogether I have two of 536 00:33:20 --> 00:33:23 them, for Q ^2 being the total. 537 00:33:23 --> 00:33:32 And that's the end of the argument. 538 00:33:32 --> 00:33:33 It's a real miracle. 539 00:33:33 --> 00:33:45 STUDENT: [INAUDIBLE] 540 00:33:45 --> 00:33:47 PROFESSOR: Great question. 541 00:33:47 --> 00:33:49 The question is, wait a minute. 542 00:33:49 --> 00:33:53 As y changes, doesn't x change. 543 00:33:53 --> 00:33:58 And so then this wouldn't be a constant. 544 00:33:58 --> 00:34:01 So that's the way in which we've used the letters x and 545 00:34:01 --> 00:34:04 y in this whole course. 546 00:34:04 --> 00:34:07 When you get to 18.02, you'll almost never do that. 547 00:34:07 --> 00:34:11 Always y and x will be different variables. 548 00:34:11 --> 00:34:12 And they won't have to depend on each other. 549 00:34:12 --> 00:34:15 Now, let me show you where on this picture the 550 00:34:15 --> 00:34:17 x and the y are. 551 00:34:17 --> 00:34:20 We've got a whole x-y plain, and here I'm fixing y 552 00:34:20 --> 00:34:23 = b, y isn't varying. 553 00:34:23 --> 00:34:25 Whereas x is changing. 554 00:34:25 --> 00:34:27 So, in other words, I don't have a relationship between 555 00:34:27 --> 00:34:29 x and y, unless I fix it. 556 00:34:29 --> 00:34:32 In this case I've decided that y is going to be constant. 557 00:34:32 --> 00:34:35 For all x. 558 00:34:35 --> 00:34:39 Over here, I made a computation. 559 00:34:39 --> 00:34:41 And I have a Q, which is just a single number. 560 00:34:41 --> 00:34:44 No matter which b I took, it didn't matter which. 561 00:34:44 --> 00:34:46 No matter which y = b. 562 00:34:46 --> 00:34:48 Of course, I changed the name to b so it wouldn't 563 00:34:48 --> 00:34:50 be so jarring to you. 564 00:34:50 --> 00:34:53 But in fact this b was y all along. 565 00:34:53 --> 00:34:56 It's just that the x varied completely 566 00:34:56 --> 00:34:57 independently of the y. 567 00:34:57 --> 00:35:00 I could fix the y and vary the x, I could fix 568 00:35:00 --> 00:35:03 the x and vary the y. 569 00:35:03 --> 00:35:06 So it's a different use of the letters. 570 00:35:06 --> 00:35:07 From what you're used to. 571 00:35:07 --> 00:35:10 It happens that y is not a function of x. 572 00:35:10 --> 00:35:11 In this case. 573 00:35:11 --> 00:35:11 Yes. 574 00:35:11 --> 00:35:16 STUDENT: [INAUDIBLE] 575 00:35:16 --> 00:35:17 PROFESSOR: Yes. 576 00:35:17 --> 00:35:23 STUDENT: [INAUDIBLE] 577 00:35:23 --> 00:35:26 PROFESSOR: The question is, because I'm rotating around 578 00:35:26 --> 00:35:30 the z axis, doesn't x change exactly as much as y does. 579 00:35:30 --> 00:35:33 What happens is that x and y are symmetric variables. 580 00:35:33 --> 00:35:36 They can be treated equally. 581 00:35:36 --> 00:35:41 But if I decide to take slices with respect to y being fixed 582 00:35:41 --> 00:35:45 and x varying, then of course they're now separated, and I 583 00:35:45 --> 00:35:48 have a separate role for the x and a separate role for the y. 584 00:35:48 --> 00:35:50 Or if I'd sliced it the other way, I would have 585 00:35:50 --> 00:35:51 gotten the same answer. 586 00:35:51 --> 00:35:54 I just would have reversed the roles of x and y. 587 00:35:54 --> 00:35:57 So what's happening is that x and y are on equal 588 00:35:57 --> 00:35:59 footing with each other. 589 00:35:59 --> 00:36:02 In this picture, and I could've sliced the other way. 590 00:36:02 --> 00:36:04 It would have gotten the same answer. 591 00:36:04 --> 00:36:07 That's more or less the answer to your question. 592 00:36:07 --> 00:36:13 OK. 593 00:36:13 --> 00:36:17 Now I have given you a review sheet, and I want to run 594 00:36:17 --> 00:36:23 through, briefly, what's going to be on the exam. 595 00:36:23 --> 00:36:28 And this list of exam questions is what's 596 00:36:28 --> 00:36:30 going to be on the exam. 597 00:36:30 --> 00:36:33 There are, sorry this is not displayed correctly. 598 00:36:33 --> 00:36:35 So, exam questions, but now I'm just going to show 599 00:36:35 --> 00:36:36 you what they are. 600 00:36:36 --> 00:36:38 There are five questions on the exam. 601 00:36:38 --> 00:36:46 They are completely parallel to what you got last year. 602 00:36:46 --> 00:36:48 So you should look at that test. 603 00:36:48 --> 00:36:50 It's worth looking at. 604 00:36:50 --> 00:36:54 And you'll see in the descriptions on this sheet 605 00:36:54 --> 00:37:00 that what I'm describing is what's on that test. 606 00:37:00 --> 00:37:04 So what's going to happen is, and this is also posted on the 607 00:37:04 --> 00:37:10 Web, is that you'll be expected to calculate some definite 608 00:37:10 --> 00:37:13 integrals using the fundamental theorem of calculus. 609 00:37:13 --> 00:37:16 Do a numerical approximation. 610 00:37:16 --> 00:37:17 There'll be a Riemann, a trapezoidal rule 611 00:37:17 --> 00:37:19 and a Simpson's Rule. 612 00:37:19 --> 00:37:22 Calculate areas and volumes. 613 00:37:22 --> 00:37:25 And then some other cumulative sum. 614 00:37:25 --> 00:37:29 Either an average value or probability or perhaps work. 615 00:37:29 --> 00:37:33 And sketch a function which is given in this 616 00:37:33 --> 00:37:37 form as an integral. 617 00:37:37 --> 00:37:42 So those are the questions, and you'll see by the 618 00:37:42 --> 00:37:46 example of last year's exam exactly the style. 619 00:37:46 --> 00:37:48 They're really going to be very similar. 620 00:37:48 --> 00:37:49 Yes, question. 621 00:37:49 --> 00:38:01 STUDENT: [INAUDIBLE] 622 00:38:01 --> 00:38:02 PROFESSOR: OK, good question. 623 00:38:02 --> 00:38:06 So the question is, for Riemann sums, what's the difference 624 00:38:06 --> 00:38:19 between upper and lower, and right and left? 625 00:38:19 --> 00:38:25 So here we have a Riemann sum. 626 00:38:25 --> 00:38:30 And I'm going to give you a picture which is, maybe this 627 00:38:30 --> 00:38:33 function y = 1 / x, which was the one that we were 628 00:38:33 --> 00:38:37 discussing earlier. 629 00:38:37 --> 00:38:41 If you take the function y = 1 / x and you break it up into 630 00:38:41 --> 00:38:45 pieces here, however it doesn't matter how many pieces, let's 631 00:38:45 --> 00:38:48 just say there are four of them. 632 00:38:48 --> 00:38:53 Then the lower Riemann sum is the staircase which 633 00:38:53 --> 00:38:55 fits underneath. 634 00:38:55 --> 00:39:01 So this one is a picture of the lower sum. 635 00:39:01 --> 00:39:03 It's always less. 636 00:39:03 --> 00:39:09 And in the case of a decreasing function, it's going to be, so 637 00:39:09 --> 00:39:20 since if you like, since 1 / x decreases, the lower sum 638 00:39:20 --> 00:39:27 equals the right sum. 639 00:39:27 --> 00:39:29 You can see that visually on this picture. 640 00:39:29 --> 00:39:33 The values you're going to select are going to be the 641 00:39:33 --> 00:39:38 right ends of the rectangles. 642 00:39:38 --> 00:39:40 The upper sum is the left one. 643 00:39:40 --> 00:39:44 Now, if the function wiggles up and down, then you have to pick 644 00:39:44 --> 00:39:45 whichever side is appropriate. 645 00:39:45 --> 00:39:47 Or maybe it'll be a point in the middle, if the maximum 646 00:39:47 --> 00:39:54 is achieved in the middle. 647 00:39:54 --> 00:39:55 Yeah, another question. 648 00:39:55 --> 00:40:02 STUDENT: [INAUDIBLE] 649 00:40:02 --> 00:40:03 PROFESSOR: Correct. 650 00:40:03 --> 00:40:09 If the function is increasing, then the lower sum 651 00:40:09 --> 00:40:11 is the left sum. 652 00:40:11 --> 00:40:13 So it just exactly reverses what's here. 653 00:40:13 --> 00:40:17 So this is decreasing, lower sum is right-hand sum. 654 00:40:17 --> 00:40:24 Increasing, lower sum is left-hand sum. 655 00:40:24 --> 00:40:26 STUDENT: [INAUDIBLE] 656 00:40:26 --> 00:40:26 PROFESSOR: Yes. 657 00:40:26 --> 00:40:44 STUDENT: [INAUDIBLE] 658 00:40:44 --> 00:40:45 PROFESSOR: Good question. 659 00:40:45 --> 00:40:47 Suppose you're faced with a function like this 660 00:40:47 --> 00:40:48 in this last problem. 661 00:40:48 --> 00:40:52 Which, generally, these are the trickiest problems. 662 00:40:52 --> 00:40:54 And the question is, how are you ever going to be able to 663 00:40:54 --> 00:40:58 decide on an asymptote, even whether there is an asymptote. 664 00:40:58 --> 00:41:01 And the answer is, you're not. 665 00:41:01 --> 00:41:05 It's going to be pretty tricky to get keep track 666 00:41:05 --> 00:41:07 of what's happening as it goes to infinity. 667 00:41:07 --> 00:41:09 We had an example on the homework where is was 668 00:41:09 --> 00:41:12 oscillating and it's very unclear what's going on. 669 00:41:12 --> 00:41:15 You have to do a very long analysis for that. 670 00:41:15 --> 00:41:21 So in fact, just don't worry about that now. 671 00:41:21 --> 00:41:23 At the very end of the class, we'll talk a little bit 672 00:41:23 --> 00:41:24 about these asymptotes. 673 00:41:24 --> 00:41:27 And really, the first issue is whether they exist or not. 674 00:41:27 --> 00:41:30 And that's even something. 675 00:41:30 --> 00:41:32 That's a serious question which we'll address at the 676 00:41:32 --> 00:41:33 very end of this course. 677 00:41:33 --> 00:41:36 STUDENT: [INAUDIBLE] 678 00:41:36 --> 00:41:36 PROFESSOR: That's right. 679 00:41:36 --> 00:41:40 It's not going to be anything that complicated. 680 00:41:40 --> 00:41:42 Other questions? 681 00:41:42 --> 00:41:44 We we still have a five whole minutes, and I have an example 682 00:41:44 --> 00:41:50 to give, if nobody has a question. 683 00:41:50 --> 00:41:50 Yeah. 684 00:41:50 --> 00:41:51 STUDENT: [INAUDIBLE] 685 00:41:51 --> 00:41:56 PROFESSOR: The question, uh, will I tell you which 686 00:41:56 --> 00:42:00 one of what to use? 687 00:42:00 --> 00:42:00 STUDENT: [INAUDIBLE] 688 00:42:00 --> 00:42:02 PROFESSOR: When I tell you the numeric approximation is, 689 00:42:02 --> 00:42:04 you'll see on the exam. 690 00:42:04 --> 00:42:06 The practice exam that you have. 691 00:42:06 --> 00:42:09 I will ask you for all three. 692 00:42:09 --> 00:42:11 I will ask you for the Riemann sum, the trapezoidal rule, 693 00:42:11 --> 00:42:12 and the Simpson's rule. 694 00:42:12 --> 00:42:16 I'm guaranteeing you they'll all three be on the exam. 695 00:42:16 --> 00:42:18 I'm guaranteeing that every single thing which is on that 696 00:42:18 --> 00:42:20 piece of paper is on the exam. 697 00:42:20 --> 00:42:23 And you'll see it on the exam that you've got. 698 00:42:23 --> 00:42:27 It's exactly parallel to what's there. 699 00:42:27 --> 00:42:31 STUDENT: [INAUDIBLE] 700 00:42:31 --> 00:42:34 PROFESSOR: So with areas and volume, the question is will I 701 00:42:34 --> 00:42:36 tell you which method to use. 702 00:42:36 --> 00:42:48 So let's discuss that. 703 00:42:48 --> 00:42:58 So with areas and volumes, there's basically, so this 704 00:42:58 --> 00:43:00 is always true with areas. 705 00:43:00 --> 00:43:07 And it's true with volumes of revolution. 706 00:43:07 --> 00:43:08 By the way you should read this sheet. 707 00:43:08 --> 00:43:12 Not everything that's on here is, have I said. 708 00:43:12 --> 00:43:15 But you should read it. 709 00:43:15 --> 00:43:16 Because it's all relevant. 710 00:43:16 --> 00:43:19 So with volumes of revolution, you always work your way 711 00:43:19 --> 00:43:25 back to some 2-D diagram. 712 00:43:25 --> 00:43:31 So there's some 2-D diagram which is always two-dimensional 713 00:43:31 --> 00:43:34 diagram, which is always connected with these problems. 714 00:43:34 --> 00:43:37 I mean, something this hard is really just too hard 715 00:43:37 --> 00:43:39 to do on an exam, right? 716 00:43:39 --> 00:43:40 I mean, I'm not going to ask you something this 717 00:43:40 --> 00:43:42 complicated on the exam. 718 00:43:42 --> 00:43:44 Because this involves a three-dimensional 719 00:43:44 --> 00:43:47 visualization. 720 00:43:47 --> 00:43:50 But once you're down to 2-D, you're supposed to 721 00:43:50 --> 00:43:52 be able to handle it. 722 00:43:52 --> 00:43:54 Now, what's the main issue after you've 723 00:43:54 --> 00:43:56 got your 2-D diagram? 724 00:43:56 --> 00:43:59 The main issue is, do you want to integrate with 725 00:43:59 --> 00:44:06 respect to dx or dy? 726 00:44:06 --> 00:44:13 And the answer is that it will depend. 727 00:44:13 --> 00:44:18 And if there's one that's going to cause you incredible 728 00:44:18 --> 00:44:23 difficulty, and I feel that you're not able to dodge it, 729 00:44:23 --> 00:44:26 then I might give you a hint and say you'd better use 730 00:44:26 --> 00:44:30 shells, or you'd better use disks or washers or 731 00:44:30 --> 00:44:31 something like that. 732 00:44:31 --> 00:44:35 But if I feel that you're grown up enough to figure out which 733 00:44:35 --> 00:44:37 one it is, because one of them is so ridiculous you say 734 00:44:37 --> 00:44:39 forget it, immediately. 735 00:44:39 --> 00:44:40 After thinking about it. 736 00:44:40 --> 00:44:43 Then I won't tell you which one. 737 00:44:43 --> 00:44:45 Because I figure, in other words, I don't want you 738 00:44:45 --> 00:44:47 to waste your time. 739 00:44:47 --> 00:44:50 But I'm willing to waste a minute or two of your time 740 00:44:50 --> 00:44:57 on a wild goose chase. 741 00:44:57 --> 00:45:01 So let me give you an example of this. 742 00:45:01 --> 00:45:14 Suppose you're looking at the curve y < 0 < - x^3. 743 00:45:14 --> 00:45:17 So this is some kind of lump. 744 00:45:17 --> 00:45:18 Like this. 745 00:45:18 --> 00:45:22 It goes from 0 to 1, because the right-hand side 746 00:45:22 --> 00:45:26 is 0 at 0 and 1 here. 747 00:45:26 --> 00:45:26 It's some kind of thing. 748 00:45:26 --> 00:45:29 And there are these two possibilities. 749 00:45:29 --> 00:45:33 One of them is to do shells. 750 00:45:33 --> 00:45:36 And then, so this is supposed to be rotated 751 00:45:36 --> 00:45:41 around the y axis. 752 00:45:41 --> 00:45:45 In this case. 753 00:45:45 --> 00:45:49 And the same would apply, actually, to the area problem. 754 00:45:49 --> 00:45:50 So I'm doing a slightly more complicated problem. 755 00:45:50 --> 00:45:52 But you could ask for the area underneath this. 756 00:45:52 --> 00:45:53 So forth. 757 00:45:53 --> 00:45:54 OK. 758 00:45:54 --> 00:45:57 So we can integrate this dx, or we can integrate this dy. 759 00:45:57 --> 00:46:00 This indicates that I'm deciding that this is going 760 00:46:00 --> 00:46:02 to be of thickness dx, and I'm integrating dx. 761 00:46:02 --> 00:46:04 So that's a choice that I'm making. 762 00:46:04 --> 00:46:08 Now, the minute I made that choice I know 763 00:46:08 --> 00:46:10 that these are shells. 764 00:46:10 --> 00:46:12 Because they sweep around this way and that makes them shells. 765 00:46:12 --> 00:46:15 Cylindrical shells. 766 00:46:15 --> 00:46:19 And if I do that, the setup is this. 767 00:46:19 --> 00:46:25 It's 2 pi x ( x - x ^3) dx. 768 00:46:25 --> 00:46:28 Now, I claim that when you get to this point, you 769 00:46:28 --> 00:46:30 already know you've won. 770 00:46:30 --> 00:46:32 Because this is an easy integral to calculate. 771 00:46:32 --> 00:46:34 So you're done here. 772 00:46:34 --> 00:46:37 You're happy. 773 00:46:37 --> 00:46:42 Now, if you happened to say, oh gee, I hate to do this. 774 00:46:42 --> 00:46:48 I want to do something clever, you could try to do it 775 00:46:48 --> 00:46:54 with cutting this way. 776 00:46:54 --> 00:46:56 Let's do this. 777 00:46:56 --> 00:47:01 And this would be the dy thickness. 778 00:47:01 --> 00:47:05 And then when you sweep this around, you get 779 00:47:05 --> 00:47:10 what we call a washer. 780 00:47:10 --> 00:47:14 Which is really just the difference of two disks. 781 00:47:14 --> 00:47:24 So the shape here is this thing swung around this axis. 782 00:47:24 --> 00:47:26 And it looks like this. 783 00:47:26 --> 00:47:29 So it's going to be the difference of radii. 784 00:47:29 --> 00:47:32 So what's the formula for this? 785 00:47:32 --> 00:47:37 It's some integral of pi times the right end, which I'll 786 00:47:37 --> 00:47:40 call x2, and here the left end, which I'll call x1. 787 00:47:40 --> 00:47:47 So this is pi (x2 ^2 - x1 ^2) dy. 788 00:47:47 --> 00:47:50 Now, already at this stage, you think to yourself 789 00:47:50 --> 00:47:54 this is more complicated than the other method. 790 00:47:54 --> 00:47:56 So you've already abandoned it. 791 00:47:56 --> 00:47:59 But I'm just going to go one step further into this one 792 00:47:59 --> 00:48:01 to see what it is that's happening. 793 00:48:01 --> 00:48:08 If you try to figure out what these values x1 and x2 are, 794 00:48:08 --> 00:48:14 that corresponds to solving for x1 and x2 in terms of y. 795 00:48:14 --> 00:48:21 So that's the following equation. x1 and x2 solve 796 00:48:21 --> 00:48:32 the equation that the curve, x - x^3 = y. 797 00:48:32 --> 00:48:33 Now, look at this equation. 798 00:48:33 --> 00:48:42 That's the equation x ^3 - sorry, x ^3 - x + y, I guess. 799 00:48:42 --> 00:48:43 Let's see. 800 00:48:43 --> 00:48:46 Yeah, that's right, is equal to 0. 801 00:48:46 --> 00:48:53 This is a cubic equation. 802 00:48:53 --> 00:48:55 Although there is a formula for this. 803 00:48:55 --> 00:48:57 You've never been taught the formula for this equation. 804 00:48:57 --> 00:49:00 So therefore, you will never, ever be able to get a formula 805 00:49:00 --> 00:49:02 for x2 and x1 as a function of y. 806 00:49:02 --> 00:49:05 And you'll never be able to compute this one. 807 00:49:05 --> 00:49:08 This is more than just a dead end, it's like crash, burn, 808 00:49:08 --> 00:49:11 and, you know self-destruct. 809 00:49:11 --> 00:49:14 So there may be such a thing, so do the other way. 810 00:49:14 --> 00:49:17 Good luck, folks. 811 00:49:17 --> 00:49:18