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 --> 00:00:13 To make a donation, or to view additional materials from 7 00:00:13 --> 00:00:17 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:17 --> 00:00:21 at ocw.mit.edu. 9 00:00:21 --> 00:00:24 PROFESSOR: So again, welcome back. 10 00:00:24 --> 00:00:28 And today's topic is a continuation of what 11 00:00:28 --> 00:00:29 we did last time. 12 00:00:29 --> 00:00:32 We still have a little bit of work and thinking to do 13 00:00:32 --> 00:00:38 concerning polar coordinates. 14 00:00:38 --> 00:00:50 So we're going to talk about polar coordinates. 15 00:00:50 --> 00:00:58 And my first job today is to talk a little bit about area. 16 00:00:58 --> 00:01:01 That's something we didn't mention last time. 17 00:01:01 --> 00:01:06 And since we're all back from Thanksgiving, we can certainly 18 00:01:06 --> 00:01:10 talk about it in terms of a pie. 19 00:01:10 --> 00:01:14 Which is the basic idea for area in polar coordinates. 20 00:01:14 --> 00:01:21 Here's our pie, and here's a slice of the pie. 21 00:01:21 --> 00:01:25 The slice has a piece of arc length on it, which I'm 22 00:01:25 --> 00:01:28 going to call delta theta. 23 00:01:28 --> 00:01:31 And the area of that shaded-in slice, I'm 24 00:01:31 --> 00:01:35 going to call delta A. 25 00:01:35 --> 00:01:38 And let's suppose that the radius is a. 26 00:01:38 --> 00:01:39 Little a. 27 00:01:39 --> 00:01:45 So this is a pie of radius a. 28 00:01:45 --> 00:01:48 That's our picture. 29 00:01:48 --> 00:01:51 Now, it's pretty easy to figure out what the area 30 00:01:51 --> 00:01:54 that slice of pie is. 31 00:01:54 --> 00:01:58 The total area is, of course, pi a ^2. 32 00:01:58 --> 00:02:00 We know that. 33 00:02:00 --> 00:02:05 And to get this fraction, delta A, all we have to do is take 34 00:02:05 --> 00:02:10 the percentage of the arc of the total circumference. 35 00:02:10 --> 00:02:13 That's delta theta / 2 pi. 36 00:02:13 --> 00:02:18 This is the fraction of area -- sorry, fraction of the total 37 00:02:18 --> 00:02:20 circumference, the total length around the rim. 38 00:02:20 --> 00:02:24 And then we multiply that by pi a ^2. 39 00:02:24 --> 00:02:27 And that's giving us the total area. 40 00:02:27 --> 00:02:30 And if you work that out, that's delta A is equal to, 41 00:02:30 --> 00:02:36 the pi's cancel and we have 1/2 a ^2 delta theta. 42 00:02:36 --> 00:02:44 So here's the basic formula. 43 00:02:44 --> 00:02:49 And now what we need to do is to talk about 44 00:02:49 --> 00:02:54 a variable pie here. 45 00:02:54 --> 00:02:58 That would be a pie with a kind of a wavy crust. 46 00:02:58 --> 00:03:01 Which is coming around like this. 47 00:03:01 --> 00:03:04 So r = r (theta). 48 00:03:04 --> 00:03:10 The distance from the center is varying with the place where we 49 00:03:10 --> 00:03:13 are, the angle where we're shooting out. 50 00:03:13 --> 00:03:22 And now I want to subdivide that into little chunks here. 51 00:03:22 --> 00:03:26 Now, the idea for adding up the area, the total area of this 52 00:03:26 --> 00:03:31 piece that's swept out, is to break it up into little slices 53 00:03:31 --> 00:03:37 whose areas are almost easy to calculate. 54 00:03:37 --> 00:03:43 Namely, what we're going to do is to take, and I'm 55 00:03:43 --> 00:03:46 going to label it this way. 56 00:03:46 --> 00:03:49 I'm going to take these little circular arcs, which go. 57 00:03:49 --> 00:03:55 So I'm going to extend past where this goes. 58 00:03:55 --> 00:03:58 And then I'm going to take each circular arc here. 59 00:03:58 --> 00:04:00 So here's a circular arc. 60 00:04:00 --> 00:04:04 And then here's another circular arc. 61 00:04:04 --> 00:04:05 And here's another circular arc. 62 00:04:05 --> 00:04:08 It's just right on the nose in that case. 63 00:04:08 --> 00:04:12 Now, in these two cases, so basically the picture that I'm 64 00:04:12 --> 00:04:14 trying to draw for you is this. 65 00:04:14 --> 00:04:17 I have some sector. 66 00:04:17 --> 00:04:19 And then I have some circular arc. 67 00:04:19 --> 00:04:23 And maybe it takes a little extra. 68 00:04:23 --> 00:04:26 There's a little extra area, I'm making an 69 00:04:26 --> 00:04:26 error in the area. 70 00:04:26 --> 00:04:28 This is a little extra area. 71 00:04:28 --> 00:04:32 And maybe to draw it the other way. 72 00:04:32 --> 00:04:34 I'm a little short on this one. 73 00:04:34 --> 00:04:37 And let's say on this one I'm right on the nose. 74 00:04:37 --> 00:04:42 I have the same arc as the curve of the surface. 75 00:04:42 --> 00:04:45 Now this is a little bit like the step functions that 76 00:04:45 --> 00:04:47 we used in Riemann sums. 77 00:04:47 --> 00:04:48 It's practically the same. 78 00:04:48 --> 00:04:52 Eventually, this little band of stuff that we're missing by, if 79 00:04:52 --> 00:04:56 we take very, very narrow little slices here, is 80 00:04:56 --> 00:04:58 going to be negligible. 81 00:04:58 --> 00:05:01 It'll get closer and closer to the curve itself. 82 00:05:01 --> 00:05:04 So that area will tend to 0 in the limit. 83 00:05:04 --> 00:05:05 So we don't have to worry about it. 84 00:05:05 --> 00:05:09 And the approximate relationship is sitting here. 85 00:05:09 --> 00:05:12 Where this distance now is r. 86 00:05:12 --> 00:05:14 So this radius is r. 87 00:05:14 --> 00:05:18 And this is this delta theta. 88 00:05:18 --> 00:05:23 And so in the approximate case, what we have is that delta A is 89 00:05:23 --> 00:05:28 approximately 1/2 r^2 delta theta. 90 00:05:28 --> 00:05:30 Which is practically the same thing we had here. 91 00:05:30 --> 00:05:34 Except that that r is replacing the constant there. 92 00:05:34 --> 00:05:39 And it's approximately true, because r is varying. 93 00:05:39 --> 00:05:42 And then in the limit, we have the exact formula 94 00:05:42 --> 00:05:43 for the differential. 95 00:05:43 --> 00:05:46 Which is this one. 96 00:05:46 --> 00:05:49 So this is the main formula for area. 97 00:05:49 --> 00:05:52 And if you like, the total area then is going to be the 98 00:05:52 --> 00:05:55 integral from some starting place to some end place 99 00:05:55 --> 00:06:00 of 1/2 r ^2 d theta. 100 00:06:00 --> 00:06:04 Now, this is only useful in the situation that we're in. 101 00:06:04 --> 00:06:07 Namely, so this is the other important formula. 102 00:06:07 --> 00:06:11 And this is only useful when r is a function of theta. 103 00:06:11 --> 00:06:17 When this is the way in which the region is presented to us. 104 00:06:17 --> 00:06:20 So that's the setup. 105 00:06:20 --> 00:06:23 And that's our main formula. 106 00:06:23 --> 00:06:28 Let's do what example. 107 00:06:28 --> 00:06:30 The example that I'm going to take is the one that we did 108 00:06:30 --> 00:06:33 at the end of last time, or near the end of last time. 109 00:06:33 --> 00:06:39 Which was this formula here. r = 2a cos theta. 110 00:06:39 --> 00:06:46 Remember, that was the same as (x - a) ^2 + y ^2 = a ^2. 111 00:06:46 --> 00:06:49 So this is what we did last time. 112 00:06:49 --> 00:06:54 We connected this rectangular representation to that 113 00:06:54 --> 00:06:54 polar representation. 114 00:06:54 --> 00:07:02 And the picture is of a circle. 115 00:07:02 --> 00:07:11 Where this is the point (2a, 0). 116 00:07:11 --> 00:07:16 So let's figure out what the area is. 117 00:07:16 --> 00:07:21 Well, first of all, we have to figure out when we sweep out 118 00:07:21 --> 00:07:23 the area, we have to realize that we only go from 119 00:07:23 --> 00:07:27 - pi / 2 to pi / 2. 120 00:07:27 --> 00:07:30 So that's something we can get from the picture. 121 00:07:30 --> 00:07:33 You can also get it directly from this formula if you 122 00:07:33 --> 00:07:38 realize that cosine is positive in this range here. 123 00:07:38 --> 00:07:40 And at the ends, it's 0. 124 00:07:40 --> 00:07:46 So the thing encloses a region at these ends. 125 00:07:46 --> 00:07:54 So at the ends, cosine of plus or minus pi / 2 = 0. 126 00:07:54 --> 00:08:02 That's what synchs this up like a little sack, if you like. 127 00:08:02 --> 00:08:06 So the area is now going to be the integral from - pi / 2 to 128 00:08:06 --> 00:08:11 pi / 2 of 1/2 times the square of r, that's (2a cos 129 00:08:11 --> 00:08:18 theta)^2 d theta. 130 00:08:18 --> 00:08:18 Question. 131 00:08:18 --> 00:08:25 STUDENT: [INAUDIBLE] 132 00:08:25 --> 00:08:27 PROFESSOR: How do I know from looking at the picture that 133 00:08:27 --> 00:08:33 I'm going from - pi / 2 to pi / 2, is the question. 134 00:08:33 --> 00:08:36 I do it with my whole body. 135 00:08:36 --> 00:08:39 I say, here I am pointing down. 136 00:08:39 --> 00:08:41 That's - pi / 2. 137 00:08:41 --> 00:08:43 I sweep up, that's 0. 138 00:08:43 --> 00:08:47 And I get all the way up to here. that's pi / 2. 139 00:08:47 --> 00:08:48 So that's the way I do it. 140 00:08:48 --> 00:08:51 That's really the way I do it, I'm being honest. 141 00:08:51 --> 00:08:54 Now if you're a machine, you can't actually look. 142 00:08:54 --> 00:08:57 And you don't have a body, so you can't point your arms. 143 00:08:57 --> 00:08:59 Then you would have to go by the formulas. 144 00:08:59 --> 00:09:02 And you'd have to actually use something like 145 00:09:02 --> 00:09:03 this formula here. 146 00:09:03 --> 00:09:07 The fact that this is where the loop syncs up. 147 00:09:07 --> 00:09:10 This is where the radius comes into 0. 148 00:09:10 --> 00:09:11 At pi / 2. 149 00:09:11 --> 00:09:17 So you need to know that in order to understand the range. 150 00:09:17 --> 00:09:17 Another question. 151 00:09:17 --> 00:09:23 STUDENT: [INAUDIBLE] 152 00:09:23 --> 00:09:26 PROFESSOR: So when we're doing each, and we just guess that 153 00:09:26 --> 00:09:28 it's going to be a loop. 154 00:09:28 --> 00:09:30 I'm probably going to give you some clues as 155 00:09:30 --> 00:09:31 to what's going on. 156 00:09:31 --> 00:09:33 Because it's very hard to figure these things out. 157 00:09:33 --> 00:09:37 Sometimes it'll be bounded by one curb and another curb, and 158 00:09:37 --> 00:09:39 I'll say it's the thing in between those two curbs. 159 00:09:39 --> 00:09:42 That's the kind of thing that I could do. 160 00:09:42 --> 00:09:47 Here, you really should know this one in advance. 161 00:09:47 --> 00:09:51 This is by far the most, or this is one of the 162 00:09:51 --> 00:09:52 typical cases, anyway. 163 00:09:52 --> 00:09:54 I'm going to give you a couple more examples. 164 00:09:54 --> 00:09:57 Don't get too worked up over this. 165 00:09:57 --> 00:09:59 You will somehow be able to visualize it. 166 00:09:59 --> 00:10:05 I'll give you some examples to help you out with it later. 167 00:10:05 --> 00:10:06 So here's the situation. 168 00:10:06 --> 00:10:07 Here's my integral. 169 00:10:07 --> 00:10:10 And now we're faced with a trig integral. 170 00:10:10 --> 00:10:12 Which we have to remember how to do. 171 00:10:12 --> 00:10:15 Now, the trig integral here -- so first let me factor 172 00:10:15 --> 00:10:16 out the constants. 173 00:10:16 --> 00:10:22 This is 4a ^2 / 2, so it's 2a ^2 integral from - pi / 2 to pi 174 00:10:22 --> 00:10:26 / 2 of cos ^2 theta d theta. 175 00:10:26 --> 00:10:29 And now you have to remember what you're supposed 176 00:10:29 --> 00:10:32 to do at this point. 177 00:10:32 --> 00:10:35 So think, if you haven't done it yet, this is practice 178 00:10:35 --> 00:10:37 you need to do. 179 00:10:37 --> 00:10:40 This trig integral is handled by a double angle formula. 180 00:10:40 --> 00:10:43 As it happens, I'm going to be giving you these formulas 181 00:10:43 --> 00:10:44 on the review sheet. 182 00:10:44 --> 00:10:47 You'll see they're written on the review sheet. 183 00:10:47 --> 00:10:49 At least in some form. 184 00:10:49 --> 00:10:52 So for example, there's a formula, and this will 185 00:10:52 --> 00:10:54 be on the exam, too. 186 00:10:54 --> 00:10:58 So this is the correct formula to use here. 187 00:10:58 --> 00:11:04 Is that this is 1 + cos 2 theta / 2 d theta. 188 00:11:04 --> 00:11:08 So that's the substitution that you use for the cosine ^2 189 00:11:08 --> 00:11:12 in order to integrate it. 190 00:11:12 --> 00:11:15 That serves as a little review of trig integrals. 191 00:11:15 --> 00:11:18 And now, this is quite easy. 192 00:11:18 --> 00:11:27 This integral now is easy. 193 00:11:27 --> 00:11:28 Why is it easy? 194 00:11:28 --> 00:11:31 Well, because it's the antiderivative of a constant, 195 00:11:31 --> 00:11:33 and cos 2 theta its antiderivative you're supposed 196 00:11:33 --> 00:11:34 to be able to write down. 197 00:11:34 --> 00:11:36 So the antiderivative of 1 is theta. 198 00:11:36 --> 00:11:41 And the antiderivative of the cos is 1/2 sin 199 00:11:41 --> 00:11:50 when it's 2 theta. 200 00:11:50 --> 00:11:55 And that is a ^2 (pi /2 - (- pi) / 2). 201 00:11:55 --> 00:12:00 And the signs go away because they're both 0. 202 00:12:00 --> 00:12:04 So all told we get pi a ^2, which is certainly what 203 00:12:04 --> 00:12:05 we would like it to be. 204 00:12:05 --> 00:12:09 It's the area of the circle. 205 00:12:09 --> 00:12:10 Another question? 206 00:12:10 --> 00:12:29 STUDENT: [INAUDIBLE] 207 00:12:29 --> 00:12:30 PROFESSOR: The question, so I'm not sure which 208 00:12:30 --> 00:12:31 question you're asking. 209 00:12:31 --> 00:12:35 I pivoted my arm around (0, 0). 210 00:12:35 --> 00:12:37 This point, this is the point we're talking about, 211 00:12:37 --> 00:12:39 (0, 0), is a key point. 212 00:12:39 --> 00:12:43 It's where I guess you could say I stuck my elbow there. 213 00:12:43 --> 00:12:48 Now, the reason is that it's the place where r = 0. 214 00:12:48 --> 00:12:50 So it's more or less the center of the universe from the point 215 00:12:50 --> 00:12:54 of view of this problem. 216 00:12:54 --> 00:12:58 So it's the reference point and if you like, when you're doing 217 00:12:58 --> 00:13:01 this, it's a little bit like a radar screen. 218 00:13:01 --> 00:13:03 Everything is centered at the origin and you're taking 219 00:13:03 --> 00:13:07 rays coming out from it. 220 00:13:07 --> 00:13:10 And seeing where they're going to go. 221 00:13:10 --> 00:13:12 So for example, this is the theta = 0 ray, this is 222 00:13:12 --> 00:13:15 the theta = pi / 4 ray. 223 00:13:15 --> 00:13:17 This the theta = pi / 2 ray. 224 00:13:17 --> 00:13:21 And indeed, if my elbow is right at this center here, 225 00:13:21 --> 00:13:24 I'm pointing in those various directions. 226 00:13:24 --> 00:13:32 So that's what I had in mind when I did that. 227 00:13:32 --> 00:13:35 You can always get these formulas, by the way, from 228 00:13:35 --> 00:13:42 the original business, x = r cos theta, y = r sin theta. 229 00:13:42 --> 00:13:45 But it's useful to have the geometric picture as well. 230 00:13:45 --> 00:13:46 In other words, if you were a machine you'd have to 231 00:13:46 --> 00:13:48 rely on these formulas. 232 00:13:48 --> 00:13:49 And plot things using these. 233 00:13:49 --> 00:13:57 Always. 234 00:13:57 --> 00:14:00 Now, in terms of plotting I want to expand your 235 00:14:00 --> 00:14:01 brain a little bit. 236 00:14:01 --> 00:14:03 So we need just a little bit more practice with plotting. 237 00:14:03 --> 00:14:05 In polar coordinates. 238 00:14:05 --> 00:14:10 And so, the first question that I want to ask you is, 239 00:14:10 --> 00:14:15 what happens outside of this range of theta? 240 00:14:15 --> 00:14:19 In other words, what happens if theta's beyond pi / 2? 241 00:14:19 --> 00:14:22 Can somebody see what's happening to the 242 00:14:22 --> 00:14:23 formulas in that case? 243 00:14:23 --> 00:14:27 So what I'm looking at now, let's go back to it. 244 00:14:27 --> 00:14:35 What I'm looking at is this formula here. 245 00:14:35 --> 00:14:38 But to use the elbow analogy here, I'm swept 246 00:14:38 --> 00:14:39 around like this. 247 00:14:39 --> 00:14:40 But now I'm going to point this way. 248 00:14:40 --> 00:14:42 I'm going to point out over there. 249 00:14:42 --> 00:14:49 My hand is up here in the northwest direction. 250 00:14:49 --> 00:14:51 So what's going to happen? 251 00:14:51 --> 00:14:52 Somebody want to tell me? 252 00:14:52 --> 00:14:55 STUDENT: [INAUDIBLE] 253 00:14:55 --> 00:14:56 PROFESSOR: It goes around itself. 254 00:14:56 --> 00:14:57 That's right. 255 00:14:57 --> 00:15:02 What happens is that when r crosses this vertical, r = 256 00:15:02 --> 00:15:05 0, when it crosses over here it goes negative. 257 00:15:05 --> 00:15:08 So although my theta is pointing me this way, the thing 258 00:15:08 --> 00:15:10 is going to go backwards. 259 00:15:10 --> 00:15:12 And there's another clue. 260 00:15:12 --> 00:15:13 Which is very important. 261 00:15:13 --> 00:15:15 How far backwards is it going? 262 00:15:15 --> 00:15:17 Well, you don't actually need to know anything but this 263 00:15:17 --> 00:15:21 equation here, to understand that it has to be on 264 00:15:21 --> 00:15:23 the same circle. 265 00:15:23 --> 00:15:26 So when I'm pointing this way, the things points backwards 266 00:15:26 --> 00:15:29 to this point over there. 267 00:15:29 --> 00:15:32 So what happens is, it goes around once. 268 00:15:32 --> 00:15:34 And then when I point out this way, it sweeps 269 00:15:34 --> 00:15:35 around a second time. 270 00:15:35 --> 00:15:37 It just keeps on going around the same circle. 271 00:15:37 --> 00:15:39 So over here it's empty. 272 00:15:39 --> 00:15:41 Because it's pointing the other way and it's sweeping 273 00:15:41 --> 00:15:42 around the same curve. 274 00:15:42 --> 00:15:47 A second time. 275 00:15:47 --> 00:15:51 Now, if you were foolish enough to integrate, say, from 0 to 2 276 00:15:51 --> 00:15:54 pi or some wider range, what would happen is you would 277 00:15:54 --> 00:15:55 just double the area. 278 00:15:55 --> 00:16:00 Because you would have swept it out twice. 279 00:16:00 --> 00:16:02 So that's the mistake that you'll make. 280 00:16:02 --> 00:16:05 Sometimes you'll count things as negative and positive. 281 00:16:05 --> 00:16:07 But because there's a square here, it's always 282 00:16:07 --> 00:16:08 a positive quantity. 283 00:16:08 --> 00:16:13 And you'll always over-count if you go too far. 284 00:16:13 --> 00:16:15 So that's what happens. 285 00:16:15 --> 00:16:17 Again, it sweeps out the same region. 286 00:16:17 --> 00:16:20 That's because these two equations really are 287 00:16:20 --> 00:16:22 equivalent to each other. 288 00:16:22 --> 00:16:24 It's just that this one sweeps it out twice. 289 00:16:24 --> 00:16:28 And this one doesn't say how it's sweeping it out. 290 00:16:28 --> 00:16:30 Yeah, another question. 291 00:16:30 --> 00:16:32 STUDENT: Doesn't this equation also work if you just 292 00:16:32 --> 00:16:34 go from 0 to pi? 293 00:16:34 --> 00:16:36 PROFESSOR: Does the integration work if you 294 00:16:36 --> 00:16:38 just go from 0 to pi? 295 00:16:38 --> 00:16:40 The answer is yes. 296 00:16:40 --> 00:16:43 That's a very weird object, though. 297 00:16:43 --> 00:16:44 Let me just show you what that is. 298 00:16:44 --> 00:16:47 If you started from 0 to 2 pi. 299 00:16:47 --> 00:16:50 So I'll illustrate it on here. 300 00:16:50 --> 00:16:53 The first thing that you swept out between 0 and pi over 301 00:16:53 --> 00:16:54 2 is this part here. 302 00:16:54 --> 00:16:56 That was swept out. 303 00:16:56 --> 00:17:00 And then, when you're going around this next quadrant here, 304 00:17:00 --> 00:17:05 you're actually sweeping out this underside here. 305 00:17:05 --> 00:17:07 So actually, you're getting it because you're getting half of 306 00:17:07 --> 00:17:09 it on one half, and getting the other half on the 307 00:17:09 --> 00:17:10 other quadrant. 308 00:17:10 --> 00:17:14 So it's actually giving you the right answer. 309 00:17:14 --> 00:17:15 That turns out to be OK. 310 00:17:15 --> 00:17:17 It's a little weird way to chop up a circle. 311 00:17:17 --> 00:17:23 But it's legal. 312 00:17:23 --> 00:17:25 But of course, that's an accident of this 313 00:17:25 --> 00:17:26 particular figure. 314 00:17:26 --> 00:17:27 You can't count on that happening. 315 00:17:27 --> 00:17:29 It's much better to line it up exactly with 316 00:17:29 --> 00:17:32 what the figure does. 317 00:17:32 --> 00:17:35 So don't do that too often. 318 00:17:35 --> 00:17:40 You might run into troubles. 319 00:17:40 --> 00:17:44 So I'm going to give you a couple more examples of 320 00:17:44 --> 00:17:48 practice with these pictures. 321 00:17:48 --> 00:17:57 And maybe I'm going to get rid of this one up here. 322 00:17:57 --> 00:18:03 So here's another favorite. 323 00:18:03 --> 00:18:05 Here's another favorite. 324 00:18:05 --> 00:18:08 So this, if you like, is Example 2. 325 00:18:08 --> 00:18:09 I guess we had an Example 1 up there. 326 00:18:09 --> 00:18:12 And now we're really not going to try to do any 327 00:18:12 --> 00:18:13 more area examples. 328 00:18:13 --> 00:18:15 The area examples are actually straightforward. 329 00:18:15 --> 00:18:18 It's really just figuring out what the picture looks like. 330 00:18:18 --> 00:18:27 So this is examples of drawings. 331 00:18:27 --> 00:18:34 So this one is one that's kind of fun to do. 332 00:18:34 --> 00:18:37 This is r = sin 2 theta. 333 00:18:37 --> 00:18:40 Something like this is on your homework. 334 00:18:40 --> 00:18:46 And so what happens here is the following. 335 00:18:46 --> 00:18:51 What happens here is that at theta = 0, that's 336 00:18:51 --> 00:18:53 the first place. 337 00:18:53 --> 00:18:56 So let's just plot a few places here. 338 00:18:56 --> 00:18:57 I'm not going to plot very many. 339 00:18:57 --> 00:19:00 Theta's = 0, I get r as 1. 340 00:19:00 --> 00:19:02 Whoops, I get r is 0. 341 00:19:02 --> 00:19:04 Sorry. 342 00:19:04 --> 00:19:09 And then pi / 4, that's where I get sin pi / 2, I get 1 here. 343 00:19:09 --> 00:19:10 For this. 344 00:19:10 --> 00:19:14 And then again, at pi / 2, I get sin pi, which 345 00:19:14 --> 00:19:17 is back at 0 again. 346 00:19:17 --> 00:19:20 So and the other thing to say is in between 347 00:19:20 --> 00:19:21 here it's positive. 348 00:19:21 --> 00:19:22 In between. 349 00:19:22 --> 00:19:27 So what it does is, it starts out at 0 and it goes out 350 00:19:27 --> 00:19:30 to the radius 1 over here. 351 00:19:30 --> 00:19:32 And then it comes back. 352 00:19:32 --> 00:19:35 So it does something like this. 353 00:19:35 --> 00:19:39 It goes out, and it comes back. 354 00:19:39 --> 00:19:45 Now because of the symmetries of the sine function, this 355 00:19:45 --> 00:19:47 is pretty much all you need to know. 356 00:19:47 --> 00:19:51 It does something similar in all of the quadrants. 357 00:19:51 --> 00:19:55 But in order to see what it's doing, it's useful for 358 00:19:55 --> 00:19:57 you to watch me draw it. 359 00:19:57 --> 00:20:00 Because the order is very important for understanding 360 00:20:00 --> 00:20:01 what it's doing. 361 00:20:01 --> 00:20:06 It's similar to this weird business with the circle here. 362 00:20:06 --> 00:20:09 So watch me draw this guy. 363 00:20:09 --> 00:20:11 I'll draw it in red because it usually has a name. 364 00:20:11 --> 00:20:12 So here it is. 365 00:20:12 --> 00:20:15 It does this things. 366 00:20:15 --> 00:20:17 And then it does this. 367 00:20:17 --> 00:20:19 And then it does this. 368 00:20:19 --> 00:20:21 And then it does that. 369 00:20:21 --> 00:20:26 So it's called a four-leaf rose. 370 00:20:26 --> 00:20:30 I drew it in pink because it's kind of a rose here. 371 00:20:30 --> 00:20:32 So it started out over here. 372 00:20:32 --> 00:20:34 This is Step 1. 373 00:20:34 --> 00:20:40 And this is the range 0 < theta < pi / 4. 374 00:20:40 --> 00:20:42 It did this part here. 375 00:20:42 --> 00:20:46 And then it went to here. 376 00:20:46 --> 00:20:48 So I should draw these in white, because they're 377 00:20:48 --> 00:20:50 harder to read in red. 378 00:20:50 --> 00:20:52 But now look at what it did. 379 00:20:52 --> 00:20:55 It did not make a right angle turn. 380 00:20:55 --> 00:20:56 It was nice and smooth. 381 00:20:56 --> 00:20:59 It went around here and then it went down here. 382 00:20:59 --> 00:21:00 This is 3. 383 00:21:00 --> 00:21:02 Back here, that's 4. 384 00:21:02 --> 00:21:05 And then over here, that's 5. 385 00:21:05 --> 00:21:07 Back up here, that's 6. 386 00:21:07 --> 00:21:09 And then around here, that's 7. 387 00:21:09 --> 00:21:11 And down here, that's 8. 388 00:21:11 --> 00:21:14 And then back where it started and goes around again. 389 00:21:14 --> 00:21:18 And this is because actually it's switching sign when 390 00:21:18 --> 00:21:19 it crosses the origin. 391 00:21:19 --> 00:21:21 When it was over in this quadrant the first time, it 392 00:21:21 --> 00:21:28 actually was tracing what's directly behind it. 393 00:21:28 --> 00:21:29 So this is kind of amusing. 394 00:21:29 --> 00:21:32 From this little tiny formula you get this 395 00:21:32 --> 00:21:34 pretty diagram here. 396 00:21:34 --> 00:21:36 Anyway that's, as I say, an old favorite. 397 00:21:36 --> 00:21:41 And here if you want to do the area of one leaf, you've got to 398 00:21:41 --> 00:21:43 make sure you understand that it's a small piece 399 00:21:43 --> 00:21:49 of the whole. 400 00:21:49 --> 00:21:52 OK, now I have one last drawing example that I 401 00:21:52 --> 00:21:54 want to discuss with you. 402 00:21:54 --> 00:21:57 And it involves another skill that we haven't quite gotten 403 00:21:57 --> 00:21:59 enough practice with. 404 00:21:59 --> 00:22:01 So I'm going to do that one. 405 00:22:01 --> 00:22:04 And it's also preparation for an exercise. 406 00:22:04 --> 00:22:08 But one that we're going to do after the test. 407 00:22:08 --> 00:22:15 So here's my last example. 408 00:22:15 --> 00:22:17 We're going to discuss what happens with 409 00:22:17 --> 00:22:20 this function here. 410 00:22:20 --> 00:22:23 Sorry, that's not legible, is it. 411 00:22:23 --> 00:22:35 That's a cosine. r = 1 / 1 + 2 cos theta. 412 00:22:35 --> 00:22:39 Now, the first thing I want to do is just take our time a 413 00:22:39 --> 00:22:44 little bit and plot a few points. 414 00:22:44 --> 00:22:48 So here's the values of theta and here are the values of r, 415 00:22:48 --> 00:22:49 and we'll see what happens. 416 00:22:49 --> 00:22:52 And we'll try to figure out what it's doing. 417 00:22:52 --> 00:22:56 When theta = 0, cos = 1. 418 00:22:56 --> 00:23:01 So r = 1/3. 419 00:23:01 --> 00:23:07 The denominator is 1 + 2, so it's 1/3. 420 00:23:07 --> 00:23:10 If theta, I'm going to make it easy, we're not 421 00:23:10 --> 00:23:11 going to do so many. 422 00:23:11 --> 00:23:16 I'm going to do pi / 2, that's an easy value of the cosine. 423 00:23:16 --> 00:23:18 That's cos pi / 2 = 0. 424 00:23:18 --> 00:23:23 So that value of r = 1. 425 00:23:23 --> 00:23:30 And now I'm going to back up and do - pi / 2. - pi 426 00:23:30 --> 00:23:33 / 2, again, cosine = 0. 427 00:23:33 --> 00:23:39 And r = 1. 428 00:23:39 --> 00:23:42 So now I'd like to just plot those points anyway, and 429 00:23:42 --> 00:23:47 see what's going on with this expression here. 430 00:23:47 --> 00:23:49 The first one is a rectangular. 431 00:23:49 --> 00:23:53 I'm going to write the rectangular coordinates here, 432 00:23:53 --> 00:23:56 not the polar coordinates. 433 00:23:56 --> 00:24:01 The rectangular coordinates here 1/3 out at the 434 00:24:01 --> 00:24:04 horizontal, so it's (1/3, 0). 435 00:24:04 --> 00:24:08 The polar coordinates is (1/3, 0), but the rectangular 436 00:24:08 --> 00:24:10 coordinate is also that. 437 00:24:10 --> 00:24:14 And over here, at pi / 2, the distance is 1. 438 00:24:14 --> 00:24:18 So this is the point (0, 1) in x-y coordinates. 439 00:24:18 --> 00:24:26 And then down here at, - pi / 2, it's (0, - 1). 440 00:24:26 --> 00:24:28 Let me just emphasize. 441 00:24:28 --> 00:24:31 You should be able to think of this visually if you can crank 442 00:24:31 --> 00:24:33 your arm around and think it. 443 00:24:33 --> 00:24:37 Or if you're right-handed you'll bend that way now. 444 00:24:37 --> 00:24:38 Anyway. 445 00:24:38 --> 00:24:38 Or you'll have to use. 446 00:24:38 --> 00:24:44 But this also works using this formulas x = r cos 447 00:24:44 --> 00:24:48 theta, y = r sin theta. 448 00:24:48 --> 00:24:53 Notice that in this case, r was 1 but the cosine was 0. 449 00:24:53 --> 00:24:57 So you plug in theta = - pi / 2. 450 00:24:57 --> 00:24:58 And r = 1. 451 00:24:58 --> 00:25:01 And lo and behold, you get 0 here. 452 00:25:01 --> 00:25:03 And here you get - 1 here you get 1. 453 00:25:03 --> 00:25:05 So this is - 1. 454 00:25:05 --> 00:25:07 So this is an example. 455 00:25:07 --> 00:25:11 I did it purely visually or sort of organically. 456 00:25:11 --> 00:25:16 But you can also do it by plugging in the numbers. 457 00:25:16 --> 00:25:21 Now in between, the denominator is positive. 458 00:25:21 --> 00:25:22 And it's something in between. 459 00:25:22 --> 00:25:26 It's going to sweep around something like this. 460 00:25:26 --> 00:25:29 That's what happens in between. 461 00:25:29 --> 00:25:34 As theta increases from - pi / 2 to pi / 2. 462 00:25:34 --> 00:25:37 And now something interesting happens with this particular 463 00:25:37 --> 00:25:40 function, which is that we notice that the denominator 464 00:25:40 --> 00:25:43 is 0 at a certain place. 465 00:25:43 --> 00:25:49 Namely, if I solve 2 cos theta = - 1, then the denominator 466 00:25:49 --> 00:25:51 is going to be 0 there. 467 00:25:51 --> 00:25:58 That's cos theta = - 1/2, so theta is equal to, it turns 468 00:25:58 --> 00:26:01 out, plus or minus 2 pi / 3. 469 00:26:01 --> 00:26:03 Those are the values here. 470 00:26:03 --> 00:26:07 So when we're out here somewhere, in these 471 00:26:07 --> 00:26:10 directions, there's nothing. 472 00:26:10 --> 00:26:14 It's going infinitely far out. 473 00:26:14 --> 00:26:20 Those ways. 474 00:26:20 --> 00:26:23 OK that's about as much as we'll be able to figure out 475 00:26:23 --> 00:26:26 of this diagram without doing some analytic work. 476 00:26:26 --> 00:26:31 And that's the other little piece that I want to explain. 477 00:26:31 --> 00:26:34 Namely, going backwards from polar coordinates to 478 00:26:34 --> 00:26:36 rectangular coordinates. 479 00:26:36 --> 00:26:38 Which is one thing that we haven't done. 480 00:26:38 --> 00:26:40 So let's do that. 481 00:26:40 --> 00:26:48 So what is the rectangular equation? 482 00:26:48 --> 00:26:58 That means the (x, y) equation for this r = 483 00:26:58 --> 00:27:03 1 / 1 + 2 cos theta. 484 00:27:03 --> 00:27:06 And let's see what it is. 485 00:27:06 --> 00:27:08 Well, first I'm going to clear the denominator here. 486 00:27:08 --> 00:27:15 This is r + 2r cos theta = 1. 487 00:27:15 --> 00:27:23 And now I'm going to rewrite it as r = 1 - 2r cos theta. 488 00:27:23 --> 00:27:25 And the reason for that is that in a minute I'll 489 00:27:25 --> 00:27:27 explain to you why. 490 00:27:27 --> 00:27:29 This is 1 - 2x. 491 00:27:29 --> 00:27:32 And this guy, I'm going to square now. 492 00:27:32 --> 00:27:38 I'm going to make this r ^2 = (1 - 2x) ^2. 493 00:27:38 --> 00:27:49 And now, with an r^2, I can plug in x ^2 + y ^2. 494 00:27:49 --> 00:27:53 So this is a standard thing to do. 495 00:27:53 --> 00:27:55 And it's basically what you're going to do any 496 00:27:55 --> 00:27:57 time you're faced with an equation like this. 497 00:27:57 --> 00:27:59 Is try to work it out. 498 00:27:59 --> 00:28:05 And, in these situations where you have 1 / a + b cos theta, 499 00:28:05 --> 00:28:08 or sin theta, you'll always come out with some quadratic 500 00:28:08 --> 00:28:12 expression like this. 501 00:28:12 --> 00:28:14 Now, I'm going to combine terms. 502 00:28:14 --> 00:28:18 So here I have - 3x ^2 + y ^2, and put everything 503 00:28:18 --> 00:28:20 on the the left side. 504 00:28:20 --> 00:28:24 So that's this. 505 00:28:24 --> 00:28:28 And we recognize, well you're supposed to recognize, that 506 00:28:28 --> 00:28:36 this is what's known as a hyperbola. 507 00:28:36 --> 00:28:39 If the signs are the same, it's an ellipse. 508 00:28:39 --> 00:28:42 If the the signs are opposite it's a hyperbola. 509 00:28:42 --> 00:28:44 And in between, if one of the coefficients on the quadratic 510 00:28:44 --> 00:28:49 is 0, it's a parabola. 511 00:28:49 --> 00:28:54 So now we see that the picture that we drew there is actually, 512 00:28:54 --> 00:28:56 turns out it's going to have asymptotes, it's going 513 00:28:56 --> 00:29:01 to be a hyperbola. 514 00:29:01 --> 00:29:06 So now, let me ask you the last little mind-bending question 515 00:29:06 --> 00:29:08 that I want to ask. 516 00:29:08 --> 00:29:10 Which is, what happens. 517 00:29:10 --> 00:29:12 So now I'm using my right arm, I guess. 518 00:29:12 --> 00:29:14 But my elbow's at the origin here. 519 00:29:14 --> 00:29:18 What happens if I pass outside, to the range where this 520 00:29:18 --> 00:29:20 denominator is negative. 521 00:29:20 --> 00:29:24 It crossed 0 and it went to negative. 522 00:29:24 --> 00:29:28 It's sweeping out something over here. 523 00:29:28 --> 00:29:32 Is it sweeping out the same curve? 524 00:29:32 --> 00:29:34 Anybody have any idea what it's doing? 525 00:29:34 --> 00:29:34 Yeah. 526 00:29:34 --> 00:29:37 STUDENT: [INAUDIBLE] 527 00:29:37 --> 00:29:38 PROFESSOR: Yeah, exactly. 528 00:29:38 --> 00:29:38 Good answer. 529 00:29:38 --> 00:29:44 It's the other branch of the hyperbola. 530 00:29:44 --> 00:29:46 So what's actually happening is in disguise, there's another 531 00:29:46 --> 00:29:48 branch of the hyperbola which is being swept up by the 532 00:29:48 --> 00:29:50 other piece of this thing. 533 00:29:50 --> 00:29:54 Now, that is consistent with these algebraic equations. 534 00:29:54 --> 00:29:57 The algebraic equation that I got here doesn't say 535 00:29:57 --> 00:30:00 which branch of the hyperbola I've got. 536 00:30:00 --> 00:30:05 It's actually got two branches. 537 00:30:05 --> 00:30:08 And the curve really was, in disguise, capturing 538 00:30:08 --> 00:30:12 both of them. 539 00:30:12 --> 00:30:15 I want to make the connection now with the basic 540 00:30:15 --> 00:30:18 formula for area here. 541 00:30:18 --> 00:30:22 Because this is a really beautiful connection. 542 00:30:22 --> 00:30:26 And I want to make that connection in connection 543 00:30:26 --> 00:30:29 also with this example. 544 00:30:29 --> 00:30:33 The hyperbolas, as you probably know, are the 545 00:30:33 --> 00:30:37 trajectories of comets. 546 00:30:37 --> 00:30:42 And ellipses, which is what you would get if maybe you put 1/2 547 00:30:42 --> 00:30:44 here instead of a 2, would be the trajectories of 548 00:30:44 --> 00:30:47 planets or asteroids. 549 00:30:47 --> 00:30:51 But there's actually something much more important, 550 00:30:51 --> 00:30:53 physically that goes on. 551 00:30:53 --> 00:30:56 That's special about this particular representation 552 00:30:56 --> 00:30:58 of the hyperbola. 553 00:30:58 --> 00:31:01 And what happens when you get the ellipses as well. 554 00:31:01 --> 00:31:08 Which is that in this case, r = 0 is the 555 00:31:08 --> 00:31:17 focus of the hyperbola. 556 00:31:17 --> 00:31:22 And what that means is that it's actually the place 557 00:31:22 --> 00:31:28 where the sun is. 558 00:31:28 --> 00:31:31 So this is the right representation, if you want 559 00:31:31 --> 00:31:36 the center of gravity in the center of your picture. 560 00:31:36 --> 00:31:38 And pretty much any other. 561 00:31:38 --> 00:31:40 I mean, you can't tell that at all from the algebraic 562 00:31:40 --> 00:31:43 equations here. 563 00:31:43 --> 00:31:46 So this hyperbola is going to be the trajectory of some 564 00:31:46 --> 00:31:51 comet going by here. 565 00:31:51 --> 00:31:58 And this formula here is actually a rather central 566 00:31:58 --> 00:32:05 formula in astronomy. 567 00:32:05 --> 00:32:17 Namely, there's something called Kepler's Law. 568 00:32:17 --> 00:32:23 Which says that the rate of change of area which is 569 00:32:23 --> 00:32:28 swept out is constant. 570 00:32:28 --> 00:32:30 The rate of change of area relative to the center of 571 00:32:30 --> 00:32:33 mass, relative to the sun. 572 00:32:33 --> 00:32:37 So in equal areas, this is amount of area. 573 00:32:37 --> 00:32:43 So this tells you now that when a comet goes around the sun 574 00:32:43 --> 00:32:46 like this, its speed varies. 575 00:32:46 --> 00:32:49 And it's speed varies according to a very specific rule. 576 00:32:49 --> 00:32:52 Namely, this one here. 577 00:32:52 --> 00:32:54 And this rule was observed by Kepler. 578 00:32:54 --> 00:32:58 But if you have this connection here, we also 579 00:32:58 --> 00:32:59 have something else. 580 00:32:59 --> 00:33:08 We also know that dA / dt = 1/2 r ^2 d theta / dt. 581 00:33:08 --> 00:33:13 So that's this formula here, formally dividing by t. 582 00:33:13 --> 00:33:16 That's the rate of change with respect to this time parameter, 583 00:33:16 --> 00:33:18 which is the honest to goodness time. 584 00:33:18 --> 00:33:20 Real physical time. 585 00:33:20 --> 00:33:34 And that means, this quantity here is constant. 586 00:33:34 --> 00:33:40 And this is one of the key insights that physicists had, 587 00:33:40 --> 00:33:43 long after Kepler made his physical observations, they 588 00:33:43 --> 00:33:48 realized that he had managed to get the best physics experiment 589 00:33:48 --> 00:33:50 of all, because it's a frictionless setup. 590 00:33:50 --> 00:33:53 Outer space, there's no air. 591 00:33:53 --> 00:33:54 Nothing is going on. 592 00:33:54 --> 00:33:59 This is what's known nowadays as conservation 593 00:33:59 --> 00:34:10 of angular momentum. 594 00:34:10 --> 00:34:13 This is the expression for angular momentum. 595 00:34:13 --> 00:34:18 And what Kepler was observing, it turns out, is what we see 596 00:34:18 --> 00:34:19 all the time in real life. 597 00:34:19 --> 00:34:21 Which is when you start something spinning around 598 00:34:21 --> 00:34:23 it continues to spin at roughly the same rate. 599 00:34:23 --> 00:34:27 Or, if you're an ice skater and you get yourself scrunched 600 00:34:27 --> 00:34:30 together a little bit more, you can spin faster. 601 00:34:30 --> 00:34:32 And there's an exact quantitative rule 602 00:34:32 --> 00:34:33 that does that. 603 00:34:33 --> 00:34:38 And it's exactly this polar formula here. 604 00:34:38 --> 00:34:39 So that's a neat thing. 605 00:34:39 --> 00:34:41 And we will do a little exercise on this rate of 606 00:34:41 --> 00:34:45 change after the exam. 607 00:34:45 --> 00:34:52 So that's it for generalities and a little pep talk on what's 608 00:34:52 --> 00:34:57 coming up to you when you learn a little more physics. 609 00:34:57 --> 00:35:04 Right now we need to talk about the exam. 610 00:35:04 --> 00:35:14 So first of all, let me tell you what the topics are. 611 00:35:14 --> 00:35:17 They're the same as last year's test. 612 00:35:17 --> 00:35:19 Which you can take a look at. 613 00:35:19 --> 00:35:24 And let's see. 614 00:35:24 --> 00:35:25 So what did we do? 615 00:35:25 --> 00:35:30 One of the main topics of this unit were techniques 616 00:35:30 --> 00:35:35 of integration. 617 00:35:35 --> 00:35:40 And there are three, which we will test. 618 00:35:40 --> 00:35:46 One is trig substitution. 619 00:35:46 --> 00:35:53 One is integration by parts. 620 00:35:53 --> 00:36:02 And one is partial fractions. 621 00:36:02 --> 00:36:06 So that's more than half of the exam, right there. 622 00:36:06 --> 00:36:16 The other half of the exam is parametric curves. 623 00:36:16 --> 00:36:18 Arc length. 624 00:36:18 --> 00:36:20 These are all interrelated. 625 00:36:20 --> 00:36:33 And area of surfaces of revolution. 626 00:36:33 --> 00:36:35 Those are the only kind that we can handle. 627 00:36:35 --> 00:36:38 Just as we did with volume of surfaces of revolution. 628 00:36:38 --> 00:36:47 And then there's a final topic, which is polar coordinates. 629 00:36:47 --> 00:36:57 And area in polar coordinates, including area. 630 00:36:57 --> 00:36:57 That's it. 631 00:36:57 --> 00:37:00 That's what's on the test, there are six problems. 632 00:37:00 --> 00:37:02 They're very similar. 633 00:37:02 --> 00:37:04 Well, they're not actually that similar. 634 00:37:04 --> 00:37:07 But they're somewhat similar to last year's. 635 00:37:07 --> 00:37:13 I'd say the test is similar. 636 00:37:13 --> 00:37:15 Maybe a tiny bit more difficult. 637 00:37:15 --> 00:37:16 We'll see. 638 00:37:16 --> 00:37:18 We'll see. 639 00:37:18 --> 00:37:18 Yeah. 640 00:37:18 --> 00:37:23 STUDENT: [INAUDIBLE] 641 00:37:23 --> 00:37:26 PROFESSOR: The question was, we didn't do arc length in 642 00:37:26 --> 00:37:28 polar coordinates, did we? 643 00:37:28 --> 00:37:31 And the answer is no, we did not. 644 00:37:31 --> 00:37:32 We did not do arc length in polar coordinates. 645 00:37:32 --> 00:37:36 When I give you an exercise, I'm going to ask you about, if 646 00:37:36 --> 00:37:38 you know the speed of a comet here, what's the speed 647 00:37:38 --> 00:37:39 of the comet there. 648 00:37:39 --> 00:37:41 And we'll have to know about arc length for that. 649 00:37:41 --> 00:37:48 But we're not doing it on this exam. 650 00:37:48 --> 00:37:49 Other questions. 651 00:37:49 --> 00:37:59 STUDENT: [INAUDIBLE] 652 00:37:59 --> 00:38:05 PROFESSOR: The question is, will I expect you to know r 653 00:38:05 --> 00:38:10 equals, so let's see if I can formulate this question. 654 00:38:10 --> 00:38:13 It's related to this four-leaf rose here. 655 00:38:13 --> 00:38:16 So the question is, suppose I gave you something 656 00:38:16 --> 00:38:21 that looked like this. 657 00:38:21 --> 00:38:24 Would I expect you to be able to know what it is. 658 00:38:24 --> 00:38:28 I think the answer, the fair answer to give you, is if it's 659 00:38:28 --> 00:38:33 this complicated, I only have two possibilities. 660 00:38:33 --> 00:38:37 I can give you a long time to sketch this out. 661 00:38:37 --> 00:38:38 And think about what it does. 662 00:38:38 --> 00:38:40 Or I can tell you that it happens to be a 663 00:38:40 --> 00:38:42 three-leaf rose. 664 00:38:42 --> 00:38:47 And then you have some clue as to what it's doing. 665 00:38:47 --> 00:38:49 It doesn't have six. 666 00:38:49 --> 00:38:53 Because of some weird thing, having to do with repetitions. 667 00:38:53 --> 00:38:57 But the odds and evens work differently. 668 00:38:57 --> 00:39:03 So, in fact I would have to tell you what the picture 669 00:39:03 --> 00:39:07 looks like, if it's going to be this complicated. 670 00:39:07 --> 00:39:11 Similarly, so this is an important point to make, when 671 00:39:11 --> 00:39:13 we come to techniques of integration, any integral that 672 00:39:13 --> 00:39:18 you have, I'm not going to tell you which of these three 673 00:39:18 --> 00:39:20 techniques to use on the ones which are straightforward 674 00:39:20 --> 00:39:20 integrals. 675 00:39:20 --> 00:39:22 But if it's an integral that I think you're going to get stuck 676 00:39:22 --> 00:39:25 on, either I'm going to give you a hint, tell 677 00:39:25 --> 00:39:26 you how to do it. 678 00:39:26 --> 00:39:29 Or I'm going to tell you, don't do it. 679 00:39:29 --> 00:39:31 If I tell you don't do it, don't try to do it. 680 00:39:31 --> 00:39:33 It may be impossible. 681 00:39:33 --> 00:39:35 And even if it's possible, it's going to be very long. 682 00:39:35 --> 00:39:38 Like an hour. 683 00:39:38 --> 00:39:42 So don't do it unless I tell you to. 684 00:39:42 --> 00:39:46 On the other hand, all of these setups in this second half 685 00:39:46 --> 00:39:50 of this unit, they involve somehow setting something up. 686 00:39:50 --> 00:39:55 And they're basically three issues. 687 00:39:55 --> 00:39:57 One is what the integrand is. 688 00:39:57 --> 00:40:02 One is what the lower limit is, what is the upper limit. 689 00:40:02 --> 00:40:05 They're just three things, three inputs, to 690 00:40:05 --> 00:40:06 setting up an integral. 691 00:40:06 --> 00:40:10 All integrals, this is going to be the setup for all of them. 692 00:40:10 --> 00:40:13 And then the second step is evaluating. 693 00:40:13 --> 00:40:17 Which really is what we did in the first half here. 694 00:40:17 --> 00:40:20 And, unfortunately, we don't have infinitely many techniques 695 00:40:20 --> 00:40:21 and indeed there's some integrals that can't be 696 00:40:21 --> 00:40:23 evaluated and some that are too long. 697 00:40:23 --> 00:40:25 So we'll just try to avoid those. 698 00:40:25 --> 00:40:32 I'm not trying to give you ones which are hopelessly long. 699 00:40:32 --> 00:40:33 Alright, other questions. 700 00:40:33 --> 00:40:34 Yes. 701 00:40:34 --> 00:40:47 STUDENT: [INAUDIBLE] 702 00:40:47 --> 00:40:49 PROFESSOR: The question is, will the percentages 703 00:40:49 --> 00:40:50 be the same. 704 00:40:50 --> 00:40:54 And the answer is, no. 705 00:40:54 --> 00:40:55 I'll tell you exactly. 706 00:40:55 --> 00:40:57 This is 55 points. 707 00:40:57 --> 00:40:59 Unless I change the point values. 708 00:40:59 --> 00:41:07 This is 55, and this is 45. 709 00:41:07 --> 00:41:11 That's what it came out to be. 710 00:41:11 --> 00:41:14 You are going to want to know about all of the things that 711 00:41:14 --> 00:41:14 I've written down here. 712 00:41:14 --> 00:41:17 You're definitely going to want to know, for example, 713 00:41:17 --> 00:41:18 surfaces of revolution. 714 00:41:18 --> 00:41:20 How to set those up. 715 00:41:20 --> 00:41:24 Yes. there was another question I saw. 716 00:41:24 --> 00:41:24 Yes. 717 00:41:24 --> 00:41:50 STUDENT: [INAUDIBLE] 718 00:41:50 --> 00:41:55 PROFESSOR: So if you have a partial fraction with something 719 00:41:55 --> 00:42:11 like (x + 2)^2 and maybe an x and maybe an x + 1, and you're 720 00:42:11 --> 00:42:16 interested in what happens with this denominator here? 721 00:42:16 --> 00:42:24 So what's going to happen is, you're going to 722 00:42:24 --> 00:42:32 need a coefficient for each degree of this. 723 00:42:32 --> 00:42:40 So altogether, the setup is going to be this. 724 00:42:40 --> 00:42:43 Plus one for x. 725 00:42:43 --> 00:42:46 And one for x + 1. 726 00:42:46 --> 00:42:50 This is the setup. 727 00:42:50 --> 00:42:51 So you need -- 728 00:42:51 --> 00:42:58 STUDENT: [INAUDIBLE] 729 00:42:58 --> 00:43:02 PROFESSOR: So if I change this to being a 3 here, then I 730 00:43:02 --> 00:43:10 need, I guess I'll have to call it E, (x + 2) ^3. 731 00:43:10 --> 00:43:11 I need that. 732 00:43:11 --> 00:43:13 Now, it gets harder and harder. 733 00:43:13 --> 00:43:15 The more repeated roots there are, the more repeated factors 734 00:43:15 --> 00:43:17 there are, the harder it is. 735 00:43:17 --> 00:43:22 Because the ones you can pick off by the cover-up method are, 736 00:43:22 --> 00:43:24 is just the top one here. 737 00:43:24 --> 00:43:25 And these two. 738 00:43:25 --> 00:43:28 So C, D, and E you can get. 739 00:43:28 --> 00:43:31 But B and A you're going to have to do by either plugging 740 00:43:31 --> 00:43:33 in or some other, more elaborate, algebra. 741 00:43:33 --> 00:43:36 So the more of these lower terms there are, the 742 00:43:36 --> 00:43:37 worse off you are. 743 00:43:37 --> 00:43:48 STUDENT: [INAUDIBLE] 744 00:43:48 --> 00:43:56 PROFESSOR: The question is, does this x ^3 + 745 00:43:56 --> 00:43:59 21 affect this setup. 746 00:43:59 --> 00:44:03 And the answer is almost no. 747 00:44:03 --> 00:44:04 That is, not at all. 748 00:44:04 --> 00:44:06 It's the same setup exactly. 749 00:44:06 --> 00:44:08 But, there's one thing. 750 00:44:08 --> 00:44:12 If the degree gets too big, then you've got to use long 751 00:44:12 --> 00:44:18 division first to knock it down. 752 00:44:18 --> 00:44:20 I'll give you an example of this type of practice. 753 00:44:20 --> 00:44:22 Unless there are more question. 754 00:44:22 --> 00:44:22 Yes. 755 00:44:22 --> 00:44:23 STUDENT: [INAUDIBLE] 756 00:44:23 --> 00:44:26 PROFESSOR: Are you going to have to know how to 757 00:44:26 --> 00:44:29 do reduction formulas? 758 00:44:29 --> 00:44:33 Anything that's a little out of the ordinary like a reduction 759 00:44:33 --> 00:44:35 formula, I will have to coach you to do. 760 00:44:35 --> 00:44:38 So, in other words, what you'll have to be able to do in that 761 00:44:38 --> 00:44:40 situation is follow directions. 762 00:44:40 --> 00:44:42 If I tell you OK, you're faced with this, then do 763 00:44:42 --> 00:44:44 an integration by parts. 764 00:44:44 --> 00:44:48 And do that, then get the reduction formula. 765 00:44:48 --> 00:44:49 STUDENT: [INAUDIBLE] 766 00:44:49 --> 00:44:50 PROFESSOR: Yeah. 767 00:44:50 --> 00:44:56 OK, so the question had do with the partial fractions method. 768 00:44:56 --> 00:45:03 And what happens if you have a quadratic. 769 00:45:03 --> 00:45:12 So, for instance, if it were this, this one's 770 00:45:12 --> 00:45:13 too disgusting. 771 00:45:13 --> 00:45:17 I'm going to just do it with two of them. 772 00:45:17 --> 00:45:20 So the parts with x + x + 1 are the same. 773 00:45:20 --> 00:45:23 But now you have linear factors here. 774 00:45:23 --> 00:45:27 Ax + B / x^2 + 2. 775 00:45:27 --> 00:45:33 And A, maybe I'll call them 1, and A2 x + B2 / (x^2 + 776 00:45:33 --> 00:45:43 2) ^2 + C / x + D / x + 1. 777 00:45:43 --> 00:45:47 This is the way it works. 778 00:45:47 --> 00:45:52 OK, I'm going to give you one more quick example of an 779 00:45:52 --> 00:45:59 integration technique just to liven things up. 780 00:45:59 --> 00:46:02 Let's see. 781 00:46:02 --> 00:46:06 So here's a somewhat tricky example. 782 00:46:06 --> 00:46:08 This is just a little trickier than I would 783 00:46:08 --> 00:46:09 give you on a test. 784 00:46:09 --> 00:46:11 But it's the same principle, and I may do this 785 00:46:11 --> 00:46:14 on a final exam. 786 00:46:14 --> 00:46:20 So suppose you're faced with this integral. 787 00:46:20 --> 00:46:24 What are you going to do? 788 00:46:24 --> 00:46:26 Integration by parts, great. 789 00:46:26 --> 00:46:29 That's right, that's because this guy is begging to be 790 00:46:29 --> 00:46:33 differentiated, to be made simpler. 791 00:46:33 --> 00:46:37 So that means that I want this one to be u, and I 792 00:46:37 --> 00:46:40 want this one to be v '. 793 00:46:40 --> 00:46:42 And I want to use integration by parts. 794 00:46:42 --> 00:46:51 And then u ' = 1 / 1 + x ^2, and v = x ^2 / 2. 795 00:46:51 --> 00:46:58 So the answer is now, x^2 / 2 tan inverse x - the 796 00:46:58 --> 00:47:00 integral of this guy. 797 00:47:00 --> 00:47:02 Which is going to be x ^2 / 2. 798 00:47:02 --> 00:47:09 And then I have 1 / 1 + x ^2 dx. 799 00:47:09 --> 00:47:13 Now, you are not done at this point. 800 00:47:13 --> 00:47:16 You're still in slightly hot water. 801 00:47:16 --> 00:47:19 You're in tepid water, anyway. 802 00:47:19 --> 00:47:21 So what is it that you have to do here? 803 00:47:21 --> 00:47:24 You're faced with this integral, which I'll 804 00:47:24 --> 00:47:30 put on the next board. 805 00:47:30 --> 00:47:33 It's a lot simpler than the other one, but as I say you're 806 00:47:33 --> 00:47:35 not quite out of the woods. 807 00:47:35 --> 00:47:39 You're faced with the integral of 1/2, - 1/2 808 00:47:39 --> 00:47:42 x ^2 / 1 + x ^2 dx. 809 00:47:42 --> 00:47:47 STUDENT: [INAUDIBLE] 810 00:47:47 --> 00:47:50 PROFESSOR: Trig substitution actually, interestingly, 811 00:47:50 --> 00:47:53 will work. 812 00:47:53 --> 00:47:54 But that wasn't what I wanted you to do. 813 00:47:54 --> 00:47:56 I wanted you to, yeah, go ahead. 814 00:47:56 --> 00:47:57 STUDENT: [INAUDIBLE] 815 00:47:57 --> 00:47:59 PROFESSOR: Add and subtract 1 to the numerator. 816 00:47:59 --> 00:48:02 So now, that's the correct answer. 817 00:48:02 --> 00:48:04 This is the case where the numerator and the 818 00:48:04 --> 00:48:06 denominator are tied. 819 00:48:06 --> 00:48:08 And so you have to use long division. 820 00:48:08 --> 00:48:12 But a shortcut is just to observe that the result of 821 00:48:12 --> 00:48:20 long division is the same thing as doing this. 822 00:48:20 --> 00:48:26 And then noticing that this is 1 - 1 / 1 + x ^2. 823 00:48:26 --> 00:48:30 So this is the same as long division, in this case. 824 00:48:30 --> 00:48:34 Because when you divide in, it goes in with a quotient of 1. 825 00:48:34 --> 00:48:39 And so this guy turns out to be - 1/2 the integral 826 00:48:39 --> 00:48:47 of 1 - 1 / 1 + x ^2 dx. 827 00:48:47 --> 00:48:55 Which is 1/2 x - 1/2 tan inverse x + c. 828 00:48:55 --> 00:49:01 So this is one extra step that you may be faced with 829 00:49:01 --> 00:49:02 someday in your life. 830 00:49:02 --> 00:49:05 And just keep that in mind. 831 00:49:05 --> 00:49:06