1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,950 Commons license. 3 00:00:03,950 --> 00:00:06,110 Your support will help MIT OpenCourseWare 4 00:00:06,110 --> 00:00:09,950 continue to offer high quality educational resources for free. 5 00:00:09,950 --> 00:00:12,640 To make a donation, or to view additional materials 6 00:00:12,640 --> 00:00:17,690 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,690 --> 00:00:21,760 at ocw.mit.edu. 8 00:00:21,760 --> 00:00:24,020 PROFESSOR: So again, welcome back. 9 00:00:24,020 --> 00:00:28,190 And today's topic is a continuation 10 00:00:28,190 --> 00:00:29,420 of what we did last time. 11 00:00:29,420 --> 00:00:31,850 We still have a little bit of work and thinking 12 00:00:31,850 --> 00:00:38,600 to do concerning polar coordinates. 13 00:00:38,600 --> 00:00:50,210 So we're going to talk about polar coordinates. 14 00:00:50,210 --> 00:00:58,320 And my first job today is to talk a little bit about area. 15 00:00:58,320 --> 00:01:01,760 That's something we didn't mention last time. 16 00:01:01,760 --> 00:01:05,320 And since we're all back from Thanksgiving, 17 00:01:05,320 --> 00:01:10,130 we can certainly talk about it in terms of a pie. 18 00:01:10,130 --> 00:01:14,650 Which is the basic idea for area in polar coordinates. 19 00:01:14,650 --> 00:01:21,840 Here's our pie, and here's a slice of the pie. 20 00:01:21,840 --> 00:01:25,700 The slice has a piece of arc length on it, which 21 00:01:25,700 --> 00:01:28,150 I'm going to call delta theta. 22 00:01:28,150 --> 00:01:31,540 And the area of that shaded-in slice, 23 00:01:31,540 --> 00:01:35,070 I'm going to call delta A. 24 00:01:35,070 --> 00:01:38,210 And let's suppose that the radius is a. 25 00:01:38,210 --> 00:01:39,350 Little a. 26 00:01:39,350 --> 00:01:45,950 So this is a pie of radius a. 27 00:01:45,950 --> 00:01:48,130 That's our picture. 28 00:01:48,130 --> 00:01:50,510 Now, it's pretty easy to figure out what 29 00:01:50,510 --> 00:01:54,390 the area that slice of pie is. 30 00:01:54,390 --> 00:01:58,650 The total area is, of course, pi a^2. 31 00:01:58,650 --> 00:02:00,030 We know that. 32 00:02:00,030 --> 00:02:05,190 And to get this fraction, delta A, all we have to do 33 00:02:05,190 --> 00:02:10,270 is take the percentage of the arc of the total circumference. 34 00:02:10,270 --> 00:02:16,720 That's (delta theta) / This is the fraction of area-- sorry, 35 00:02:16,720 --> 00:02:18,590 fraction of the total circumference, 36 00:02:18,590 --> 00:02:20,900 the total length around the rim. 37 00:02:20,900 --> 00:02:24,150 And then we multiply that by pi a^2. 38 00:02:24,150 --> 00:02:27,000 And that's giving us the total area. 39 00:02:27,000 --> 00:02:30,500 And if you work that out, that's delta A is equal 40 00:02:30,500 --> 00:02:36,860 to, the pi's cancel and we have 1/2 a^2 delta theta. 41 00:02:36,860 --> 00:02:44,990 So here's the basic formula. 42 00:02:44,990 --> 00:02:54,510 And now what we need to do is to talk about a variable pie here. 43 00:02:54,510 --> 00:02:58,790 That would be a pie with a kind of a wavy crust. 44 00:02:58,790 --> 00:03:01,520 Which is coming around like this. 45 00:03:01,520 --> 00:03:04,140 So r = r(theta). 46 00:03:04,140 --> 00:03:07,860 The distance from the center is varying 47 00:03:07,860 --> 00:03:13,420 with the place where we are, the angle where we're shooting out. 48 00:03:13,420 --> 00:03:22,070 And now I want to subdivide that into little chunks here. 49 00:03:22,070 --> 00:03:25,760 Now, the idea for adding up the area, the total area 50 00:03:25,760 --> 00:03:29,110 of this piece that's swept out, is 51 00:03:29,110 --> 00:03:33,160 to break it up into little slices whose areas 52 00:03:33,160 --> 00:03:37,580 are almost easy to calculate. 53 00:03:37,580 --> 00:03:42,900 Namely, what we're going to do is to take, 54 00:03:42,900 --> 00:03:46,000 and I'm going to label it this way. 55 00:03:46,000 --> 00:03:49,520 I'm going to take these little circular arcs, which go-- 56 00:03:49,520 --> 00:03:55,770 So I'm going to extend past where this goes. 57 00:03:55,770 --> 00:03:58,620 And then I'm going to take each circular arc here. 58 00:03:58,620 --> 00:04:00,510 So here's a circular arc. 59 00:04:00,510 --> 00:04:04,010 And then here's another circular arc. 60 00:04:04,010 --> 00:04:05,590 And here's another circular arc. 61 00:04:05,590 --> 00:04:08,250 It's just right on the nose in that case. 62 00:04:08,250 --> 00:04:12,050 Now, in these two cases, so basically 63 00:04:12,050 --> 00:04:14,910 the picture that I'm trying to draw for you is this. 64 00:04:14,910 --> 00:04:17,960 I have some sector. 65 00:04:17,960 --> 00:04:19,445 And then I have some circular arc. 66 00:04:19,445 --> 00:04:23,400 And maybe it takes a little extra. 67 00:04:23,400 --> 00:04:26,780 There's a little extra area, I'm making an error in the area. 68 00:04:26,780 --> 00:04:28,400 This is a little extra area. 69 00:04:28,400 --> 00:04:32,410 And maybe to draw it the other way. 70 00:04:32,410 --> 00:04:34,530 I'm a little short on this one. 71 00:04:34,530 --> 00:04:37,190 And let's say on this one I'm right on the nose. 72 00:04:37,190 --> 00:04:42,950 I have the same arc as the curve of the surface. 73 00:04:42,950 --> 00:04:45,580 Now this is a little bit like the step functions 74 00:04:45,580 --> 00:04:47,240 that we used in Riemann sums. 75 00:04:47,240 --> 00:04:48,940 It's practically the same. 76 00:04:48,940 --> 00:04:52,490 Eventually, this little band of stuff that we're missing by, 77 00:04:52,490 --> 00:04:56,110 if we take very, very narrow little slices here, 78 00:04:56,110 --> 00:04:58,610 is going to be negligible. 79 00:04:58,610 --> 00:05:01,360 It'll get closer and closer to the curve itself. 80 00:05:01,360 --> 00:05:04,290 So that area will tend to 0 in the limit. 81 00:05:04,290 --> 00:05:05,940 So we don't have to worry about it. 82 00:05:05,940 --> 00:05:09,970 And the approximate relationship is sitting here. 83 00:05:09,970 --> 00:05:12,110 Where this distance now is r. 84 00:05:12,110 --> 00:05:14,340 So this radius is r. 85 00:05:14,340 --> 00:05:18,330 And this is this delta theta. 86 00:05:18,330 --> 00:05:23,080 And so in the approximate case, what we have is that delta A is 87 00:05:23,080 --> 00:05:28,830 approximately 1/2 r^2 delta theta. 88 00:05:28,830 --> 00:05:30,840 Which is practically the same thing we had here. 89 00:05:30,840 --> 00:05:34,985 Except that that r is replacing the constant there. 90 00:05:34,985 --> 00:05:39,290 And it's approximately true, because r is varying. 91 00:05:39,290 --> 00:05:42,230 And then in the limit, we have the exact formula 92 00:05:42,230 --> 00:05:43,870 for the differential. 93 00:05:43,870 --> 00:05:46,200 Which is this one. 94 00:05:46,200 --> 00:05:49,190 So this is the main formula for area. 95 00:05:49,190 --> 00:05:52,000 And if you like, the total area then is going to be 96 00:05:52,000 --> 00:05:56,290 the integral from some starting place to some end place of 1/2 97 00:05:56,290 --> 00:05:57,070 r^2 d theta. 98 00:06:00,020 --> 00:06:04,210 Now, this is only useful in the situation that we're in. 99 00:06:04,210 --> 00:06:07,120 Namely-- So this is the other important formula. 100 00:06:07,120 --> 00:06:11,440 And this is only useful when r is a function of theta. 101 00:06:11,440 --> 00:06:17,930 When this is the way in which the region is presented to us. 102 00:06:17,930 --> 00:06:20,430 So that's the setup. 103 00:06:20,430 --> 00:06:23,310 And that's our main formula. 104 00:06:23,310 --> 00:06:27,884 Let's do what example. 105 00:06:27,884 --> 00:06:29,300 The example that I'm going to take 106 00:06:29,300 --> 00:06:31,750 is the one that we did at the end of last time, 107 00:06:31,750 --> 00:06:33,790 or near the end of last time. 108 00:06:33,790 --> 00:06:39,260 Which was this formula here. r = 2a cos(theta). 109 00:06:39,260 --> 00:06:46,920 Remember, that was the same as (x-a)^2 + y^2 = a^2. 110 00:06:46,920 --> 00:06:49,120 So this is what we did last time. 111 00:06:49,120 --> 00:06:53,550 We connected this rectangular representation 112 00:06:53,550 --> 00:06:54,920 to that polar representation. 113 00:06:54,920 --> 00:07:02,130 And the picture is of a circle. 114 00:07:02,130 --> 00:07:11,580 Where this is the point (2a, 0). 115 00:07:11,580 --> 00:07:16,140 So let's figure out what the area is. 116 00:07:16,140 --> 00:07:19,480 Well, first of all, we have to figure out 117 00:07:19,480 --> 00:07:22,040 when we sweep out the area, we have 118 00:07:22,040 --> 00:07:27,160 to realize that we only go from -pi/2 to pi/2. 119 00:07:27,160 --> 00:07:30,620 So that's something we can get from the picture. 120 00:07:30,620 --> 00:07:32,900 You can also get it directly from this formula 121 00:07:32,900 --> 00:07:38,030 if you realize that cosine is positive in this range here. 122 00:07:38,030 --> 00:07:40,430 And at the ends, it's 0. 123 00:07:40,430 --> 00:07:46,870 So the thing encloses a region at these ends. 124 00:07:46,870 --> 00:07:54,550 So at the ends, cosine of plus or minus pi/2 is equal to 0. 125 00:07:54,550 --> 00:08:02,320 That's what cinches this up like a little sack, if you like. 126 00:08:02,320 --> 00:08:07,530 So the area is now going to be the integral from -pi/2 to pi/2 127 00:08:07,530 --> 00:08:13,410 of 1/2 times the square of r, that's (2a cos(theta))^2, 128 00:08:13,410 --> 00:08:18,111 d theta. 129 00:08:18,111 --> 00:08:18,610 Question. 130 00:08:18,610 --> 00:08:25,434 STUDENT: [INAUDIBLE] 131 00:08:25,434 --> 00:08:27,600 PROFESSOR: How do I know from looking at the picture 132 00:08:27,600 --> 00:08:33,400 that I'm going from -pi/2 to pi/2, is the question. 133 00:08:33,400 --> 00:08:36,820 I do it with my whole body. 134 00:08:36,820 --> 00:08:39,860 I say, here I am pointing down. 135 00:08:39,860 --> 00:08:41,120 That's -pi/2. 136 00:08:41,120 --> 00:08:43,890 I sweep up, that's 0. 137 00:08:43,890 --> 00:08:47,710 And I get all the way up to here. that's pi/2. 138 00:08:47,710 --> 00:08:48,910 So that's the way I do it. 139 00:08:48,910 --> 00:08:51,450 That's really the way I do it, I'm being honest. 140 00:08:51,450 --> 00:08:54,520 Now if you're a machine, you can't actually look. 141 00:08:54,520 --> 00:08:57,800 And you don't have a body, so you can't point your arms. 142 00:08:57,800 --> 00:08:59,650 Then you would have to go by the formulas. 143 00:08:59,650 --> 00:09:02,950 And you'd have to actually use something like this formula 144 00:09:02,950 --> 00:09:03,900 here. 145 00:09:03,900 --> 00:09:07,830 The fact that this is where the loop cinches up. 146 00:09:07,830 --> 00:09:10,750 This is where the radius comes into 0. 147 00:09:10,750 --> 00:09:11,270 At pi/2. 148 00:09:11,270 --> 00:09:17,230 So you need to know that in order to understand the range. 149 00:09:17,230 --> 00:09:17,970 Another question. 150 00:09:17,970 --> 00:09:23,089 STUDENT: [INAUDIBLE] 151 00:09:23,089 --> 00:09:24,630 PROFESSOR: So when we're doing these, 152 00:09:24,630 --> 00:09:28,000 should we just guess that it's going to be a loop? 153 00:09:28,000 --> 00:09:30,590 I'm probably going to give you some clues as to what's 154 00:09:30,590 --> 00:09:31,090 going on. 155 00:09:31,090 --> 00:09:33,990 Because it's very hard to figure these things out. 156 00:09:33,990 --> 00:09:36,920 Sometimes it'll be bounded by one curve and another curve, 157 00:09:36,920 --> 00:09:39,780 and I'll say it's the thing in between those two curves. 158 00:09:39,780 --> 00:09:42,710 That's the kind of thing that I could do. 159 00:09:42,710 --> 00:09:47,590 Here, you really should know this one in advance. 160 00:09:47,590 --> 00:09:51,270 This is by far the most-- or this is one 161 00:09:51,270 --> 00:09:52,540 of the typical cases, anyway. 162 00:09:52,540 --> 00:09:54,690 I'm going to give you a couple more examples. 163 00:09:54,690 --> 00:09:57,130 Don't get too worked up over this. 164 00:09:57,130 --> 00:09:59,400 You will somehow be able to visualize it. 165 00:09:59,400 --> 00:10:05,070 I'll give you some examples to help you out with it later. 166 00:10:05,070 --> 00:10:06,360 So here's the situation. 167 00:10:06,360 --> 00:10:07,620 Here's my integral. 168 00:10:07,620 --> 00:10:10,450 And now we're faced with a trig integral. 169 00:10:10,450 --> 00:10:12,970 Which we have to remember how to do. 170 00:10:12,970 --> 00:10:14,770 Now, the trig integral here-- so first 171 00:10:14,770 --> 00:10:16,760 let me factor out the constants. 172 00:10:16,760 --> 00:10:21,740 This is 4a 4a^2 / 2, so it's 2a^2 integral from -pi/2 173 00:10:21,740 --> 00:10:26,870 to pi/2 of cos^2(theta) d theta. 174 00:10:26,870 --> 00:10:29,510 And now you have to remember what you're 175 00:10:29,510 --> 00:10:32,570 supposed to do at this point. 176 00:10:32,570 --> 00:10:34,310 So think, if you haven't done it yet, 177 00:10:34,310 --> 00:10:37,830 this is practice you need to do. 178 00:10:37,830 --> 00:10:40,660 This trig integral is handled by a double angle formula. 179 00:10:40,660 --> 00:10:42,520 As it happens, I'm going to be giving you 180 00:10:42,520 --> 00:10:44,670 these formulas on the review sheet. 181 00:10:44,670 --> 00:10:47,480 You'll see they're written on the review sheet. 182 00:10:47,480 --> 00:10:49,940 At least in some form. 183 00:10:49,940 --> 00:10:51,940 So for example, there's a formula, 184 00:10:51,940 --> 00:10:54,480 and this will be on the exam, too. 185 00:10:54,480 --> 00:10:58,440 So this is the correct formula to use here. 186 00:10:58,440 --> 00:11:04,240 Is that this is + (1 + cos(2theta)) / 2 d theta. 187 00:11:04,240 --> 00:11:08,230 So that's the substitution that you use for the cosine squared 188 00:11:08,230 --> 00:11:12,490 in order to integrate it. 189 00:11:12,490 --> 00:11:15,440 That serves as a little review of trig integrals. 190 00:11:15,440 --> 00:11:18,830 And now, this is quite easy. 191 00:11:18,830 --> 00:11:27,560 This integral now is easy. 192 00:11:27,560 --> 00:11:28,380 Why is it easy? 193 00:11:28,380 --> 00:11:30,640 Well, because it's the antiderivative 194 00:11:30,640 --> 00:11:32,650 of a constant, cos(2theta), its antiderivative 195 00:11:32,650 --> 00:11:34,640 you're supposed to be able to write down. 196 00:11:34,640 --> 00:11:36,820 So the antiderivative of 1 is theta. 197 00:11:36,820 --> 00:11:39,510 And the antiderivative of the cosine 198 00:11:39,510 --> 00:11:50,120 is 1/2 the sine when it's 2 theta. 199 00:11:50,120 --> 00:11:52,930 And that is a ^2 a^2 (pi/2 - (-pi/2)). 200 00:11:55,920 --> 00:12:00,070 And the signs go away because they're both 0. 201 00:12:00,070 --> 00:12:04,900 So all told we get pi a^2, which is certainly what we would like 202 00:12:04,900 --> 00:12:05,400 it to be. 203 00:12:05,400 --> 00:12:09,482 It's the area of the circle. 204 00:12:09,482 --> 00:12:10,190 Another question? 205 00:12:10,190 --> 00:12:28,982 STUDENT: [INAUDIBLE] 206 00:12:28,982 --> 00:12:30,440 PROFESSOR: The question, so I'm not 207 00:12:30,440 --> 00:12:31,990 sure which question you're asking. 208 00:12:31,990 --> 00:12:35,360 I pivoted my arm around (0, 0). 209 00:12:35,360 --> 00:12:38,550 This point, this is the point we're talking about, (0, 0), 210 00:12:38,550 --> 00:12:39,290 is a key point. 211 00:12:39,290 --> 00:12:43,310 It's where I guess you could say I stuck my elbow there. 212 00:12:43,310 --> 00:12:48,367 Now, the reason is that it's the place where r = 0. 213 00:12:48,367 --> 00:12:50,950 So it's more or less the center of the universe from the point 214 00:12:50,950 --> 00:12:54,630 of view of this problem. 215 00:12:54,630 --> 00:12:57,730 So it's the reference point and if you like, 216 00:12:57,730 --> 00:13:01,500 when you're doing this, it's a little bit like a radar screen. 217 00:13:01,500 --> 00:13:03,050 Everything is centered at the origin 218 00:13:03,050 --> 00:13:07,135 and you're taking rays coming out from it. 219 00:13:07,135 --> 00:13:10,390 And seeing where they're going to go. 220 00:13:10,390 --> 00:13:12,230 So for example, this is the theta = 0 ray, 221 00:13:12,230 --> 00:13:15,240 this is the theta = pi/4 ray. 222 00:13:15,240 --> 00:13:17,930 This the theta = pi/2 ray. 223 00:13:17,930 --> 00:13:21,250 And indeed, if my elbow is right at this center here, 224 00:13:21,250 --> 00:13:24,610 I'm pointing in those various directions. 225 00:13:24,610 --> 00:13:32,650 So that's what I had in mind when I did that. 226 00:13:32,650 --> 00:13:35,140 You can always get these formulas, by the way, 227 00:13:35,140 --> 00:13:42,120 from the original business, x = r cos(theta), y = r sin(theta). 228 00:13:42,120 --> 00:13:44,689 But it's useful to have the geometric picture as well. 229 00:13:44,689 --> 00:13:46,230 In other words, if you were a machine 230 00:13:46,230 --> 00:13:48,150 you'd have to rely on these formulas. 231 00:13:48,150 --> 00:13:49,480 And plot things using these. 232 00:13:49,480 --> 00:13:57,090 Always. 233 00:13:57,090 --> 00:14:00,509 Now, in terms of plotting I want to expand your brain 234 00:14:00,509 --> 00:14:01,050 a little bit. 235 00:14:01,050 --> 00:14:03,810 So we need just a little bit more practice with plotting. 236 00:14:03,810 --> 00:14:05,870 In polar coordinates. 237 00:14:05,870 --> 00:14:09,580 And so, the first question that I want to ask you 238 00:14:09,580 --> 00:14:15,370 is, what happens outside of this range of theta? 239 00:14:15,370 --> 00:14:19,960 In other words, what happens if theta's beyond pi/2? 240 00:14:19,960 --> 00:14:22,080 Can somebody see what's happening 241 00:14:22,080 --> 00:14:23,340 to the formulas in that case? 242 00:14:23,340 --> 00:14:27,480 So what I'm looking at now, let's go back to it. 243 00:14:27,480 --> 00:14:35,600 What I'm looking at is this formula here. 244 00:14:35,600 --> 00:14:37,690 But to use the elbow analogy here, 245 00:14:37,690 --> 00:14:39,280 I'm swept around like this. 246 00:14:39,280 --> 00:14:40,810 But now I'm going to point this way. 247 00:14:40,810 --> 00:14:42,340 I'm going to point out over there. 248 00:14:42,340 --> 00:14:49,450 My hand is up here in the northwest direction. 249 00:14:49,450 --> 00:14:51,519 So what's going to happen? 250 00:14:51,519 --> 00:14:52,560 Somebody want to tell me? 251 00:14:52,560 --> 00:14:55,210 STUDENT: [INAUDIBLE] 252 00:14:55,210 --> 00:14:56,920 PROFESSOR: It goes around itself. 253 00:14:56,920 --> 00:14:57,640 That's right. 254 00:14:57,640 --> 00:15:02,400 What happens is that when r crosses this vertical, r = 0, 255 00:15:02,400 --> 00:15:05,210 when it crosses over here it goes negative. 256 00:15:05,210 --> 00:15:08,540 So although my theta is pointing me this way, 257 00:15:08,540 --> 00:15:10,490 the thing is going to go backwards. 258 00:15:10,490 --> 00:15:12,590 And there's another clue. 259 00:15:12,590 --> 00:15:13,590 Which is very important. 260 00:15:13,590 --> 00:15:15,290 How far backwards is it going? 261 00:15:15,290 --> 00:15:18,370 Well, you don't actually need to know anything but this equation 262 00:15:18,370 --> 00:15:23,530 here, to understand that it has to be on the same circle. 263 00:15:23,530 --> 00:15:26,040 So when I'm pointing this way, the things 264 00:15:26,040 --> 00:15:29,550 points backwards to this point over there. 265 00:15:29,550 --> 00:15:32,012 So what happens is, it goes around once. 266 00:15:32,012 --> 00:15:33,470 And then when I point out this way, 267 00:15:33,470 --> 00:15:35,614 it sweeps around a second time. 268 00:15:35,614 --> 00:15:37,530 It just keeps on going around the same circle. 269 00:15:37,530 --> 00:15:38,867 So over here it's empty. 270 00:15:38,867 --> 00:15:40,700 Because it's pointing the other way and it's 271 00:15:40,700 --> 00:15:42,570 sweeping around the same curve. 272 00:15:42,570 --> 00:15:47,600 A second time. 273 00:15:47,600 --> 00:15:50,960 Now, if you were foolish enough to integrate, say, 274 00:15:50,960 --> 00:15:54,060 from 0 to 2pi or some wider range, what would happen 275 00:15:54,060 --> 00:15:55,810 is you would just double the area. 276 00:15:55,810 --> 00:16:00,970 Because you would have swept it out twice. 277 00:16:00,970 --> 00:16:02,890 So that's the mistake that you'll make. 278 00:16:02,890 --> 00:16:05,390 Sometimes you'll count things as negative and positive. 279 00:16:05,390 --> 00:16:07,392 But because there's a square here, 280 00:16:07,392 --> 00:16:08,725 it's always a positive quantity. 281 00:16:08,725 --> 00:16:13,890 And you'll always over-count if you go too far. 282 00:16:13,890 --> 00:16:15,930 So that's what happens. 283 00:16:15,930 --> 00:16:17,930 Again, it sweeps out the same region. 284 00:16:17,930 --> 00:16:19,890 That's because these two equations really 285 00:16:19,890 --> 00:16:22,050 are equivalent to each other. 286 00:16:22,050 --> 00:16:24,080 It's just that this one sweeps it out twice. 287 00:16:24,080 --> 00:16:28,930 And this one doesn't say how it's sweeping it out. 288 00:16:28,930 --> 00:16:30,380 Yeah, another question. 289 00:16:30,380 --> 00:16:32,046 STUDENT: Doesn't this equation also work 290 00:16:32,046 --> 00:16:34,320 if you just go from 0 to pi? 291 00:16:34,320 --> 00:16:36,630 PROFESSOR: Does the integration work 292 00:16:36,630 --> 00:16:38,900 if you just go from 0 to pi? 293 00:16:38,900 --> 00:16:40,730 The answer is yes. 294 00:16:40,730 --> 00:16:42,884 That's a very weird object, though. 295 00:16:42,884 --> 00:16:44,300 Let me just show you what that is. 296 00:16:44,300 --> 00:16:47,360 If you started from 0 to 2pi. 297 00:16:47,360 --> 00:16:50,120 So I'll illustrate it on here. 298 00:16:50,120 --> 00:16:53,290 The first thing that you swept out between 0 and pi/2 299 00:16:53,290 --> 00:16:54,750 is this part here. 300 00:16:54,750 --> 00:16:56,160 That was swept out. 301 00:16:56,160 --> 00:17:00,190 And then, when you're going around this next quadrant here, 302 00:17:00,190 --> 00:17:05,026 you're actually sweeping out this underside here. 303 00:17:05,026 --> 00:17:06,900 So actually, you're getting it because you're 304 00:17:06,900 --> 00:17:09,710 getting half of it on one half, and getting the other half 305 00:17:09,710 --> 00:17:10,990 on the other quadrant. 306 00:17:10,990 --> 00:17:14,120 So it's actually giving you the right answer. 307 00:17:14,120 --> 00:17:15,590 That turns out to be OK. 308 00:17:15,590 --> 00:17:17,740 It's a little weird way to chop up a circle. 309 00:17:17,740 --> 00:17:23,706 But it's legal. 310 00:17:23,706 --> 00:17:25,080 But of course, that's an accident 311 00:17:25,080 --> 00:17:26,260 of this particular figure. 312 00:17:26,260 --> 00:17:27,830 You can't count on that happening. 313 00:17:27,830 --> 00:17:29,580 It's much better to line it up exactly 314 00:17:29,580 --> 00:17:32,240 with what the figure does. 315 00:17:32,240 --> 00:17:35,630 So don't do that too often. 316 00:17:35,630 --> 00:17:40,640 You might run into troubles. 317 00:17:40,640 --> 00:17:44,110 So I'm going to give you a couple more examples 318 00:17:44,110 --> 00:17:48,970 of practice with these pictures. 319 00:17:48,970 --> 00:17:57,080 And maybe I'm going to get rid of this one up here. 320 00:17:57,080 --> 00:18:03,850 So here's another favorite. 321 00:18:03,850 --> 00:18:05,140 Here's another favorite. 322 00:18:05,140 --> 00:18:08,000 So this, if you like, is Example 2. 323 00:18:08,000 --> 00:18:09,720 I guess we had an Example 1 up there. 324 00:18:09,720 --> 00:18:12,160 And now we're really not going to try 325 00:18:12,160 --> 00:18:13,690 to do any more area examples. 326 00:18:13,690 --> 00:18:15,900 The area examples are actually straightforward. 327 00:18:15,900 --> 00:18:18,830 It's really just figuring out what the picture looks like. 328 00:18:18,830 --> 00:18:27,200 So this is examples of drawings. 329 00:18:27,200 --> 00:18:34,890 So this one is one that's kind of fun to do. 330 00:18:34,890 --> 00:18:37,500 This is r = sin(2theta). 331 00:18:37,500 --> 00:18:40,120 Something like this is on your homework. 332 00:18:40,120 --> 00:18:46,830 And so what happens here is the following. 333 00:18:46,830 --> 00:18:50,850 What happens here is that at theta = 0, 334 00:18:50,850 --> 00:18:53,100 that's the first place. 335 00:18:53,100 --> 00:18:56,470 So let's just plot a few places here. 336 00:18:56,470 --> 00:18:57,960 I'm not going to plot very many. 337 00:18:57,960 --> 00:19:00,590 Theta = 0, I get r is 1. 338 00:19:00,590 --> 00:19:02,940 Whoops, I get r is 0. 339 00:19:02,940 --> 00:19:04,020 Sorry. 340 00:19:04,020 --> 00:19:09,900 And then pi/4, that's where I get sin(pi/2), I get 1 here. 341 00:19:09,900 --> 00:19:10,570 For this. 342 00:19:10,570 --> 00:19:14,400 And then again, at pi/2 I get sin(pi), 343 00:19:14,400 --> 00:19:17,090 which is back at 0 again. 344 00:19:17,090 --> 00:19:21,010 So it's-- And the other thing to say is in between here 345 00:19:21,010 --> 00:19:21,810 it's positive. 346 00:19:21,810 --> 00:19:22,860 In between. 347 00:19:22,860 --> 00:19:26,120 So what it does is, it starts out at 0 348 00:19:26,120 --> 00:19:30,290 and it goes out to the radius 1 over here. 349 00:19:30,290 --> 00:19:32,640 And then it comes back. 350 00:19:32,640 --> 00:19:35,350 So it does something like this. 351 00:19:35,350 --> 00:19:39,790 It goes out, and it comes back. 352 00:19:39,790 --> 00:19:44,900 Now because of the symmetries of the sine function, 353 00:19:44,900 --> 00:19:47,070 this is pretty much all you need to know. 354 00:19:47,070 --> 00:19:51,550 It does something similar in all of the quadrants. 355 00:19:51,550 --> 00:19:56,000 But in order to see what it's doing, it's useful for you 356 00:19:56,000 --> 00:19:57,450 to watch me draw it. 357 00:19:57,450 --> 00:20:00,930 Because the order is very important for understanding 358 00:20:00,930 --> 00:20:01,970 what it's doing. 359 00:20:01,970 --> 00:20:06,170 It's similar to this weird business with the circle here. 360 00:20:06,170 --> 00:20:09,110 So watch me draw this guy. 361 00:20:09,110 --> 00:20:11,700 I'll draw it in red because it usually has a name. 362 00:20:11,700 --> 00:20:12,870 So here it is. 363 00:20:12,870 --> 00:20:15,450 It does this thing. 364 00:20:15,450 --> 00:20:17,280 And then it does this. 365 00:20:17,280 --> 00:20:19,330 And then it does this. 366 00:20:19,330 --> 00:20:21,770 And then it does that. 367 00:20:21,770 --> 00:20:26,130 So it's called a four-leaf rose. 368 00:20:26,130 --> 00:20:30,380 I drew it in pink because it's kind of a rose here. 369 00:20:30,380 --> 00:20:32,050 So it started out over here. 370 00:20:32,050 --> 00:20:34,800 This is Step 1. 371 00:20:34,800 --> 00:20:40,370 And this is the range 0 < theta < pi/4. 372 00:20:40,370 --> 00:20:42,200 It did this part here. 373 00:20:42,200 --> 00:20:46,180 And then it went 2 here. 374 00:20:46,180 --> 00:20:48,510 So I should draw these in white, because they're 375 00:20:48,510 --> 00:20:50,580 harder to read in red. 376 00:20:50,580 --> 00:20:52,800 But now look at what it did. 377 00:20:52,800 --> 00:20:55,160 It did not make a right angle turn. 378 00:20:55,160 --> 00:20:56,930 It was nice and smooth. 379 00:20:56,930 --> 00:20:59,500 It went around here and then it went down here. 380 00:20:59,500 --> 00:21:00,900 This is 3. 381 00:21:00,900 --> 00:21:02,840 Back here, that's 4. 382 00:21:02,840 --> 00:21:05,370 And then over here, that's 5. 383 00:21:05,370 --> 00:21:07,540 Back up here, that's 6. 384 00:21:07,540 --> 00:21:09,650 And then around here, that's 7. 385 00:21:09,650 --> 00:21:11,030 And down here, that's 8. 386 00:21:11,030 --> 00:21:14,750 And then back where it started and goes around again. 387 00:21:14,750 --> 00:21:17,940 And this is because actually it's switching sign 388 00:21:17,940 --> 00:21:19,470 when it crosses the origin. 389 00:21:19,470 --> 00:21:21,620 When it was over in this quadrant the first time, 390 00:21:21,620 --> 00:21:28,050 it actually was tracing what's directly behind it. 391 00:21:28,050 --> 00:21:29,470 So this is kind of amusing. 392 00:21:29,470 --> 00:21:33,681 From this little tiny formula you get this pretty diagram 393 00:21:33,681 --> 00:21:34,180 here. 394 00:21:34,180 --> 00:21:36,490 Anyway that's, as I say, an old favorite. 395 00:21:36,490 --> 00:21:40,490 And here if you want to do the area of one leaf, 396 00:21:40,490 --> 00:21:42,810 you've got to make sure you understand that it's 397 00:21:42,810 --> 00:21:49,650 a small piece of the whole. 398 00:21:49,650 --> 00:21:52,230 OK, now I have one last drawing example 399 00:21:52,230 --> 00:21:54,280 that I want to discuss with you. 400 00:21:54,280 --> 00:21:57,150 And it involves another skill that we haven't quite 401 00:21:57,150 --> 00:21:59,550 gotten enough practice with. 402 00:21:59,550 --> 00:22:01,320 So I'm going to do that one. 403 00:22:01,320 --> 00:22:04,210 And it's also preparation for an exercise. 404 00:22:04,210 --> 00:22:08,360 But one that we're going to do after the test. 405 00:22:08,360 --> 00:22:15,110 So here's my last example. 406 00:22:15,110 --> 00:22:18,300 We're going to discuss what happens with this function 407 00:22:18,300 --> 00:22:20,510 here. 408 00:22:20,510 --> 00:22:23,440 Sorry, that's not legible, is it. 409 00:22:23,440 --> 00:22:29,790 That's a cosine. r = r = 1/(1 + 2cos(theta)). 410 00:22:35,310 --> 00:22:37,470 Now, the first thing I want to do 411 00:22:37,470 --> 00:22:44,400 is just take our time a little bit and plot a few points. 412 00:22:44,400 --> 00:22:48,260 So here's the values of theta and here are the values of r, 413 00:22:48,260 --> 00:22:49,770 and we'll see what happens. 414 00:22:49,770 --> 00:22:52,450 And we'll try to figure out what it's doing. 415 00:22:52,450 --> 00:22:56,440 When theta = 0, cosine is 1. 416 00:22:56,440 --> 00:23:01,230 So r = 1/3. 417 00:23:01,230 --> 00:23:07,950 The denominator is 1 + 2, so it's 1/3. 418 00:23:07,950 --> 00:23:10,060 If theta-- I'm going to make it easy, 419 00:23:10,060 --> 00:23:11,540 we're not going to do so many. 420 00:23:11,540 --> 00:23:16,120 I'm going to do pi/2, that's an easy value of the cosine. 421 00:23:16,120 --> 00:23:18,200 That's cos(pi/2) = 0. 422 00:23:18,200 --> 00:23:23,070 So that value of r = 1. 423 00:23:23,070 --> 00:23:30,060 And now I'm going to back up and do -pi/2. 424 00:23:30,060 --> 00:23:33,390 -pi/2, again, cosine is 0. 425 00:23:33,390 --> 00:23:39,500 And r = 1. 426 00:23:39,500 --> 00:23:42,930 So now I'd like to just plot those points anyway, and see 427 00:23:42,930 --> 00:23:47,410 what's going on with this expression here. 428 00:23:47,410 --> 00:23:49,910 The first one is a rectangular-- I'm 429 00:23:49,910 --> 00:23:53,450 going to write the rectangular coordinates here, 430 00:23:53,450 --> 00:23:56,010 not the polar coordinates. 431 00:23:56,010 --> 00:24:01,670 The rectangular coordinates here are 1/3 out at the horizontal, 432 00:24:01,670 --> 00:24:04,220 so it's (1/3, 0). 433 00:24:04,220 --> 00:24:07,300 The polar coordinates is (1/3, 0), 434 00:24:07,300 --> 00:24:10,890 but the rectangular coordinate is also that. 435 00:24:10,890 --> 00:24:14,610 And over here, at pi/2, the distance is 1. 436 00:24:14,610 --> 00:24:18,940 So this is the point (0, 1) in x-y coordinates. 437 00:24:18,940 --> 00:24:26,940 And then down here at, -pi/2, it's (0, -1). 438 00:24:26,940 --> 00:24:28,817 Let me just emphasize. 439 00:24:28,817 --> 00:24:30,650 You should be able to think of this visually 440 00:24:30,650 --> 00:24:33,590 if you can crank your arm around and think it. 441 00:24:33,590 --> 00:24:37,630 Or if you're right-handed you'll bend that way, no. 442 00:24:37,630 --> 00:24:38,270 Anyway. 443 00:24:38,270 --> 00:24:40,250 Or you'll have to use-- But this also 444 00:24:40,250 --> 00:24:48,890 works using this formulas x = r cos(theta), y = r sin(theta). 445 00:24:48,890 --> 00:24:53,010 Notice that in this case, r was 1 but the cosine was 0. 446 00:24:53,010 --> 00:24:57,100 So you plug in theta = -pi/2. 447 00:24:57,100 --> 00:24:58,990 And r = 1. 448 00:24:58,990 --> 00:25:01,520 And lo and behold, you get 0 here. 449 00:25:01,520 --> 00:25:03,640 And here you get -1 here you get 1. 450 00:25:03,640 --> 00:25:05,830 So this is -1. 451 00:25:05,830 --> 00:25:07,750 So this is an example. 452 00:25:07,750 --> 00:25:11,960 I did it purely visually or sort of organically. 453 00:25:11,960 --> 00:25:16,190 But you can also do it by plugging in the numbers. 454 00:25:16,190 --> 00:25:21,130 Now in between, the denominator is positive. 455 00:25:21,130 --> 00:25:22,810 And it's something in between. 456 00:25:22,810 --> 00:25:26,600 It's going to sweep around something like this. 457 00:25:26,600 --> 00:25:29,200 That's what happens in between. 458 00:25:29,200 --> 00:25:32,090 As theta increases from -pi/2 to pi/2. 459 00:25:34,600 --> 00:25:36,430 And now something interesting happens 460 00:25:36,430 --> 00:25:38,260 with this particular function, which 461 00:25:38,260 --> 00:25:40,450 is that we notice that the denominator is 462 00:25:40,450 --> 00:25:43,280 0 at a certain place. 463 00:25:43,280 --> 00:25:48,610 Namely, if I solve 2 cos(theta) = -1, 464 00:25:48,610 --> 00:25:51,290 then the denominator is going to be 0 there. 465 00:25:51,290 --> 00:25:57,470 That's cos(theta) = -1/2, so theta is equal 466 00:25:57,470 --> 00:26:01,240 to, it turns out, plus or minus 2pi/3. 467 00:26:01,240 --> 00:26:03,800 Those are the values here. 468 00:26:03,800 --> 00:26:09,050 So when we're out here somewhere, in these directions, 469 00:26:09,050 --> 00:26:10,120 there's nothing. 470 00:26:10,120 --> 00:26:14,220 It's going infinitely far out. 471 00:26:14,220 --> 00:26:20,120 Those ways. 472 00:26:20,120 --> 00:26:22,500 OK that's about as much as we'll be 473 00:26:22,500 --> 00:26:24,740 able to figure out of this diagram 474 00:26:24,740 --> 00:26:26,980 without doing some analytic work. 475 00:26:26,980 --> 00:26:31,970 And that's the other little piece that I want to explain. 476 00:26:31,970 --> 00:26:34,700 Namely, going backwards from polar coordinates 477 00:26:34,700 --> 00:26:36,740 to rectangular coordinates. 478 00:26:36,740 --> 00:26:38,930 Which is one thing that we haven't done. 479 00:26:38,930 --> 00:26:40,870 So let's do that. 480 00:26:40,870 --> 00:26:48,680 So what is the rectangular equation? 481 00:26:48,680 --> 00:26:59,690 That means the (x, y) equation for this r = 1 / 482 00:26:59,690 --> 00:27:00,440 (1 + 2cos(theta)). 483 00:27:03,400 --> 00:27:06,320 And let's see what it is. 484 00:27:06,320 --> 00:27:08,850 Well, first I'm going to clear the denominator here. 485 00:27:08,850 --> 00:27:15,200 This is r + 2r cos(theta) = 1. 486 00:27:15,200 --> 00:27:23,020 And now I'm going to rewrite it as r = 1 - 2r cos(theta). 487 00:27:23,020 --> 00:27:25,280 And the reason for that is that in a minute 488 00:27:25,280 --> 00:27:27,300 I'll explain to you why. 489 00:27:27,300 --> 00:27:29,620 This is 1 - 2x. 490 00:27:29,620 --> 00:27:32,810 And this guy, I'm going to square now. 491 00:27:32,810 --> 00:27:38,010 I'm going to make this r^2 = (1 - 2x)^2. 492 00:27:38,010 --> 00:27:41,030 And now, with an r^2, I can plug in x^2 + y^2. 493 00:27:49,680 --> 00:27:53,381 So this is a standard thing to do. 494 00:27:53,381 --> 00:27:54,880 And it's basically what you're going 495 00:27:54,880 --> 00:27:57,870 to do any time you're faced with an equation like this. 496 00:27:57,870 --> 00:27:59,470 Is try to work it out. 497 00:27:59,470 --> 00:28:04,180 And, in these situations where you have 1 / 498 00:28:04,180 --> 00:28:06,990 (a + b cos(theta)), or sin(theta), 499 00:28:06,990 --> 00:28:12,400 you'll always come out with some quadratic expression like this. 500 00:28:12,400 --> 00:28:14,590 Now, I'm going to combine terms. 501 00:28:14,590 --> 00:28:19,921 So here I have -3x^2 + y^2, and put everything on the the left 502 00:28:19,921 --> 00:28:20,420 side. 503 00:28:20,420 --> 00:28:24,390 So that's this. 504 00:28:24,390 --> 00:28:28,140 And we recognize, well you're supposed to recognize, 505 00:28:28,140 --> 00:28:36,580 that this is what's known as a hyperbola. 506 00:28:36,580 --> 00:28:39,620 If the signs are the same, it's an ellipse. 507 00:28:39,620 --> 00:28:42,390 If the the signs are opposite it's a hyperbola. 508 00:28:42,390 --> 00:28:44,920 And in between, if one of the coefficients on the quadratic 509 00:28:44,920 --> 00:28:49,210 is 0, it's a parabola. 510 00:28:49,210 --> 00:28:54,844 So now we see that the picture that we drew there is actually, 511 00:28:54,844 --> 00:28:56,510 turns out it's going to have asymptotes, 512 00:28:56,510 --> 00:29:01,920 it's going to be a hyperbola. 513 00:29:01,920 --> 00:29:06,650 So now, let me ask you the last little mind-bending question 514 00:29:06,650 --> 00:29:08,020 that I want to ask. 515 00:29:08,020 --> 00:29:12,060 Which is, what happens-- So now I'm using my right arm, 516 00:29:12,060 --> 00:29:12,770 I guess. 517 00:29:12,770 --> 00:29:14,190 But my elbow's at the origin here. 518 00:29:14,190 --> 00:29:18,290 What happens if I pass outside, to the range where 519 00:29:18,290 --> 00:29:20,280 this denominator is negative. 520 00:29:20,280 --> 00:29:24,210 It crossed 0 and it went to negative. 521 00:29:24,210 --> 00:29:28,050 It's sweeping out something over here. 522 00:29:28,050 --> 00:29:32,420 Is it sweeping out the same curve? 523 00:29:32,420 --> 00:29:34,320 Anybody have any idea what it's doing? 524 00:29:34,320 --> 00:29:34,820 Yeah. 525 00:29:34,820 --> 00:29:37,460 STUDENT: [INAUDIBLE] 526 00:29:37,460 --> 00:29:38,590 PROFESSOR: Yeah, exactly. 527 00:29:38,590 --> 00:29:39,090 Good answer. 528 00:29:39,090 --> 00:29:44,220 It's the other branch of the hyperbola. 529 00:29:44,220 --> 00:29:46,390 So what's actually happening is in disguise, 530 00:29:46,390 --> 00:29:48,015 there's another branch of the hyperbola 531 00:29:48,015 --> 00:29:50,700 which is being swept up by the other piece of this thing. 532 00:29:50,700 --> 00:29:54,220 Now, that is consistent with these algebraic equations. 533 00:29:54,220 --> 00:29:56,500 The algebraic equation that I got here 534 00:29:56,500 --> 00:30:00,900 doesn't say which branch of the hyperbola I've got. 535 00:30:00,900 --> 00:30:05,930 It's actually got two branches. 536 00:30:05,930 --> 00:30:07,960 And the curve really was, in disguise, 537 00:30:07,960 --> 00:30:12,130 capturing both of them. 538 00:30:12,130 --> 00:30:14,250 I want to make the connection now 539 00:30:14,250 --> 00:30:18,770 with the basic formula for area here. 540 00:30:18,770 --> 00:30:22,920 Because this is a really beautiful connection. 541 00:30:22,920 --> 00:30:26,520 And I want to make that connection in connection 542 00:30:26,520 --> 00:30:29,110 also with this example. 543 00:30:29,110 --> 00:30:32,750 The hyperbolas, as you probably know, 544 00:30:32,750 --> 00:30:37,770 are the trajectories of comets. 545 00:30:37,770 --> 00:30:42,290 And ellipses, which is what you would get if maybe you put 1/2 546 00:30:42,290 --> 00:30:44,570 here instead of a 2, would be the trajectories 547 00:30:44,570 --> 00:30:47,450 of planets or asteroids. 548 00:30:47,450 --> 00:30:51,770 But there's actually something much more important, 549 00:30:51,770 --> 00:30:53,410 physically that goes on. 550 00:30:53,410 --> 00:30:56,820 That's special about this particular representation 551 00:30:56,820 --> 00:30:58,900 of the hyperbola. 552 00:30:58,900 --> 00:31:01,610 And what happens when you get the ellipses as well. 553 00:31:01,610 --> 00:31:06,860 Which is that in this case, r = 0 554 00:31:06,860 --> 00:31:17,520 is the focus of the hyperbola. 555 00:31:17,520 --> 00:31:21,390 And what that means is that it's actually 556 00:31:21,390 --> 00:31:28,910 the place where the sun is. 557 00:31:28,910 --> 00:31:31,220 So this is the right representation, 558 00:31:31,220 --> 00:31:32,670 if you want the center of gravity 559 00:31:32,670 --> 00:31:36,720 in the center of your picture. 560 00:31:36,720 --> 00:31:38,540 And pretty much any other. 561 00:31:38,540 --> 00:31:40,050 I mean, you can't tell that at all 562 00:31:40,050 --> 00:31:43,150 from the algebraic equations here. 563 00:31:43,150 --> 00:31:46,340 So this hyperbola is going to be the trajectory 564 00:31:46,340 --> 00:31:51,900 of some comet going by here. 565 00:31:51,900 --> 00:31:57,240 And this formula here is actually 566 00:31:57,240 --> 00:32:05,040 a rather central formula in astronomy. 567 00:32:05,040 --> 00:32:17,100 Namely, there's something called Kepler's Law. 568 00:32:17,100 --> 00:32:23,820 Which says that the rate of change of area which 569 00:32:23,820 --> 00:32:28,674 is swept out is constant. 570 00:32:28,674 --> 00:32:31,090 The rate of change of area relative to the center of mass, 571 00:32:31,090 --> 00:32:33,640 relative to the sun. 572 00:32:33,640 --> 00:32:37,000 So in equal areas, this is amount of area. 573 00:32:37,000 --> 00:32:43,570 So this tells you now that when a comet goes around the sun 574 00:32:43,570 --> 00:32:46,500 like this, its speed varies. 575 00:32:46,500 --> 00:32:49,180 And it's speed varies according to a very specific rule. 576 00:32:49,180 --> 00:32:52,220 Namely, this one here. 577 00:32:52,220 --> 00:32:54,960 And this rule was observed by Kepler. 578 00:32:54,960 --> 00:32:57,910 But if you have this connection here, 579 00:32:57,910 --> 00:32:59,400 we also have something else. 580 00:32:59,400 --> 00:33:08,390 We also know that dA/dt = 1/2 r^2 d theta / dt. 581 00:33:08,390 --> 00:33:13,387 So that's this formula here, formally dividing by t. 582 00:33:13,387 --> 00:33:14,970 That's the rate of change with respect 583 00:33:14,970 --> 00:33:18,040 to this time parameter, which is the honest to goodness time. 584 00:33:18,040 --> 00:33:20,820 Real physical time. 585 00:33:20,820 --> 00:33:34,920 And that means, this quantity here is constant. 586 00:33:34,920 --> 00:33:37,810 And this is one of the key insights 587 00:33:37,810 --> 00:33:41,540 that physicists had, long after Kepler 588 00:33:41,540 --> 00:33:43,760 made his physical observations, they realized 589 00:33:43,760 --> 00:33:48,280 that he had managed to get the best physics experiment of all, 590 00:33:48,280 --> 00:33:50,790 because it's a frictionless setup. 591 00:33:50,790 --> 00:33:53,320 Outer space, there's no air. 592 00:33:53,320 --> 00:33:54,710 Nothing is going on. 593 00:33:54,710 --> 00:33:59,260 This is what's known nowadays as conservation 594 00:33:59,260 --> 00:34:10,030 of angular momentum. 595 00:34:10,030 --> 00:34:13,720 This is the expression for angular momentum. 596 00:34:13,720 --> 00:34:16,810 And what Kepler was observing, it turns out, 597 00:34:16,810 --> 00:34:18,924 is what we see all the time in real life. 598 00:34:18,924 --> 00:34:21,090 Which is when you start something spinning around it 599 00:34:21,090 --> 00:34:23,570 continues to spin at roughly the same rate. 600 00:34:23,570 --> 00:34:26,517 Or, if you're an ice skater and you 601 00:34:26,517 --> 00:34:28,600 get yourself scrunched together a little bit more, 602 00:34:28,600 --> 00:34:30,330 you can spin faster. 603 00:34:30,330 --> 00:34:32,850 And there's an exact quantitative rule 604 00:34:32,850 --> 00:34:33,740 that does that. 605 00:34:33,740 --> 00:34:38,240 And it's exactly this polar formula here. 606 00:34:38,240 --> 00:34:39,730 So that's a neat thing. 607 00:34:39,730 --> 00:34:43,280 And we will do a little exercise on this rate of change 608 00:34:43,280 --> 00:34:45,780 after the exam. 609 00:34:45,780 --> 00:34:52,030 So that's it for generalities and a little pep talk 610 00:34:52,030 --> 00:34:54,940 on what's coming up to you when you 611 00:34:54,940 --> 00:34:57,820 learn a little more physics. 612 00:34:57,820 --> 00:35:04,660 Right now we need to talk about the exam. 613 00:35:04,660 --> 00:35:14,010 So first of all, let me tell you what the topics are. 614 00:35:14,010 --> 00:35:17,500 They're the same as last year's test. 615 00:35:17,500 --> 00:35:19,990 Which you can take a look at. 616 00:35:19,990 --> 00:35:24,060 And let's see. 617 00:35:24,060 --> 00:35:25,920 So what did we do? 618 00:35:25,920 --> 00:35:29,960 One of the main topics of this unit 619 00:35:29,960 --> 00:35:35,660 were techniques of integration. 620 00:35:35,660 --> 00:35:40,390 And there are three, which we will test. 621 00:35:40,390 --> 00:35:46,660 One is trig substitution. 622 00:35:46,660 --> 00:35:53,950 One is integration by parts. 623 00:35:53,950 --> 00:36:02,000 And one is partial fractions. 624 00:36:02,000 --> 00:36:06,020 So that's more than half of the exam, right there. 625 00:36:06,020 --> 00:36:16,210 The other half of the exam is parametric curves. 626 00:36:16,210 --> 00:36:18,420 Arc length. 627 00:36:18,420 --> 00:36:20,520 These are all interrelated. 628 00:36:20,520 --> 00:36:33,140 And area of surfaces of revolution. 629 00:36:33,140 --> 00:36:35,150 Those are the only kind that we can handle. 630 00:36:35,150 --> 00:36:38,730 Just as we did with volume of surfaces of revolution. 631 00:36:38,730 --> 00:36:47,590 And then there's a final topic, which is polar coordinates. 632 00:36:47,590 --> 00:36:57,260 And area in polar coordinates, including area. 633 00:36:57,260 --> 00:36:57,810 That's it. 634 00:36:57,810 --> 00:37:00,270 That's what's on the test, there are six problems. 635 00:37:00,270 --> 00:37:02,200 They're very similar. 636 00:37:02,200 --> 00:37:04,040 Well, they're not actually that similar. 637 00:37:04,040 --> 00:37:07,610 But they're somewhat similar to last year's. 638 00:37:07,610 --> 00:37:13,860 I'd say the test is similar. 639 00:37:13,860 --> 00:37:16,960 Maybe a tiny bit more difficult. We'll see. 640 00:37:16,960 --> 00:37:18,241 We'll see. 641 00:37:18,241 --> 00:37:18,740 Yeah. 642 00:37:18,740 --> 00:37:23,420 STUDENT: [INAUDIBLE] 643 00:37:23,420 --> 00:37:26,410 PROFESSOR: The question was, we didn't do arc length 644 00:37:26,410 --> 00:37:28,310 in polar coordinates, did we? 645 00:37:28,310 --> 00:37:30,954 And the answer is no, we did not. 646 00:37:30,954 --> 00:37:32,870 We did not do arc length in polar coordinates. 647 00:37:32,870 --> 00:37:35,840 When I give you an exercise, I'm going to ask you about, 648 00:37:35,840 --> 00:37:37,850 if you know the speed of a comet here, what's 649 00:37:37,850 --> 00:37:39,540 the speed of the comet there. 650 00:37:39,540 --> 00:37:41,710 And we'll have to know about arc length for that. 651 00:37:41,710 --> 00:37:48,534 But we're not doing it on this exam. 652 00:37:48,534 --> 00:37:49,200 Other questions. 653 00:37:49,200 --> 00:37:59,880 STUDENT: [INAUDIBLE] 654 00:37:59,880 --> 00:38:05,650 PROFESSOR: The question is, will I expect you to know r 655 00:38:05,650 --> 00:38:10,220 equals--- so let's see if I can formulate this question. 656 00:38:10,220 --> 00:38:13,430 It's related to this four-leaf rose here. 657 00:38:13,430 --> 00:38:16,020 So the question is, suppose I gave you something 658 00:38:16,020 --> 00:38:21,220 that looked like this. 659 00:38:21,220 --> 00:38:24,170 Would I expect you to be able to know what it is. 660 00:38:24,170 --> 00:38:27,300 I think the answer, the fair answer to give you, 661 00:38:27,300 --> 00:38:33,440 is if it's this complicated, I only have two possibilities. 662 00:38:33,440 --> 00:38:37,190 I can give you a long time to sketch this out. 663 00:38:37,190 --> 00:38:38,510 And think about what it does. 664 00:38:38,510 --> 00:38:40,580 Or I can tell you that it happens 665 00:38:40,580 --> 00:38:42,600 to be a three-leaf rose. 666 00:38:42,600 --> 00:38:47,530 And then you have some clue as to what it's doing. 667 00:38:47,530 --> 00:38:49,620 It doesn't have six. 668 00:38:49,620 --> 00:38:53,770 Because of some weird thing, having to do with repetitions. 669 00:38:53,770 --> 00:38:57,790 But the odds and evens work differently. 670 00:38:57,790 --> 00:39:02,540 So, in fact I would have to tell you 671 00:39:02,540 --> 00:39:04,100 what the picture looks like, if it's 672 00:39:04,100 --> 00:39:07,070 going to be this complicated. 673 00:39:07,070 --> 00:39:10,860 Similarly, so this is an important point to make, 674 00:39:10,860 --> 00:39:13,500 when we come to techniques of integration, any integral 675 00:39:13,500 --> 00:39:17,982 that you have, I'm not going to tell you which of these three 676 00:39:17,982 --> 00:39:19,440 techniques to use on the ones which 677 00:39:19,440 --> 00:39:20,810 are straightforward integrals. 678 00:39:20,810 --> 00:39:22,050 But if it's an integral that I think 679 00:39:22,050 --> 00:39:23,760 you're going to get stuck on, either I'm 680 00:39:23,760 --> 00:39:26,330 going to give you a hint, tell you how to do it. 681 00:39:26,330 --> 00:39:29,280 Or I'm going to tell you, don't do it. 682 00:39:29,280 --> 00:39:31,680 If I tell you don't do it, don't try to do it. 683 00:39:31,680 --> 00:39:33,050 It may be impossible. 684 00:39:33,050 --> 00:39:35,950 And even if it's possible, it's going to be very long. 685 00:39:35,950 --> 00:39:38,010 Like an hour. 686 00:39:38,010 --> 00:39:42,750 So don't do it unless I tell you to. 687 00:39:42,750 --> 00:39:44,990 On the other hand, all of these setups 688 00:39:44,990 --> 00:39:47,720 in this second half of this unit, 689 00:39:47,720 --> 00:39:50,290 they involve somehow setting something up. 690 00:39:50,290 --> 00:39:55,230 And they're basically three issues. 691 00:39:55,230 --> 00:39:57,310 One is what the integrand is. 692 00:39:57,310 --> 00:40:02,440 One is what the lower limit is, what is the upper limit. 693 00:40:02,440 --> 00:40:05,510 They're just three things, three inputs, 694 00:40:05,510 --> 00:40:06,710 to setting up an integral. 695 00:40:06,710 --> 00:40:10,590 All integrals, this is going to be the setup for all of them. 696 00:40:10,590 --> 00:40:13,260 And then the second step is evaluating. 697 00:40:13,260 --> 00:40:17,070 Which really is what we did in the first half here. 698 00:40:17,070 --> 00:40:19,856 And, unfortunately, we don't have infinitely many techniques 699 00:40:19,856 --> 00:40:21,230 and indeed there's some integrals 700 00:40:21,230 --> 00:40:23,790 that can't be evaluated and some that are too long. 701 00:40:23,790 --> 00:40:25,575 So we'll just try to avoid those. 702 00:40:25,575 --> 00:40:32,400 I'm not trying to give you ones which are hopelessly long. 703 00:40:32,400 --> 00:40:33,680 Alright, other questions. 704 00:40:33,680 --> 00:40:34,180 Yes. 705 00:40:34,180 --> 00:40:47,620 STUDENT: [INAUDIBLE] 706 00:40:47,620 --> 00:40:50,610 PROFESSOR: The question is, will the percentages be the same. 707 00:40:50,610 --> 00:40:54,834 And the answer is, no. 708 00:40:54,834 --> 00:40:55,750 I'll tell you exactly. 709 00:40:55,750 --> 00:40:57,200 This is 55 points. 710 00:40:57,200 --> 00:40:59,340 Unless I change the point values. 711 00:40:59,340 --> 00:41:07,780 This is 55, and this is 45. 712 00:41:07,780 --> 00:41:11,020 That's what it came out to be. 713 00:41:11,020 --> 00:41:13,554 You are going to want to know about all of the things 714 00:41:13,554 --> 00:41:14,720 that I've written down here. 715 00:41:14,720 --> 00:41:16,386 You're definitely going to want to know, 716 00:41:16,386 --> 00:41:18,650 for example, surfaces of revolution. 717 00:41:18,650 --> 00:41:20,710 How to set those up. 718 00:41:20,710 --> 00:41:24,341 Yes. there was another question I saw. 719 00:41:24,341 --> 00:41:24,840 Yes. 720 00:41:24,840 --> 00:41:50,770 STUDENT: [INAUDIBLE] 721 00:41:50,770 --> 00:41:55,010 PROFESSOR: So if you have a partial fraction with something 722 00:41:55,010 --> 00:42:04,110 like (x+2)^2 and maybe an x and maybe an x+1, 723 00:42:04,110 --> 00:42:15,460 and you're interested in what happens with this denominator 724 00:42:15,460 --> 00:42:16,120 here? 725 00:42:16,120 --> 00:42:23,600 So what's going to happen is, you're 726 00:42:23,600 --> 00:42:32,180 going to need a coefficient for each degree of this. 727 00:42:32,180 --> 00:42:40,400 So altogether, the setup is going to be this. 728 00:42:40,400 --> 00:42:43,080 Plus one for x. 729 00:42:43,080 --> 00:42:46,440 And one for x+1. 730 00:42:46,440 --> 00:42:50,770 This is the setup. 731 00:42:50,770 --> 00:42:51,850 So you need-- 732 00:42:51,850 --> 00:42:58,890 STUDENT: [INAUDIBLE] 733 00:42:58,890 --> 00:43:05,310 PROFESSOR: So if I change this to being a 3 here, then I need, 734 00:43:05,310 --> 00:43:10,490 I guess I'll have to call it E, (x+2)^3. 735 00:43:10,490 --> 00:43:11,460 I need that. 736 00:43:11,460 --> 00:43:13,000 Now, it gets harder and harder. 737 00:43:13,000 --> 00:43:14,500 The more repeated roots there are, 738 00:43:14,500 --> 00:43:17,440 the more repeated factors there are, the harder it is. 739 00:43:17,440 --> 00:43:22,080 Because the ones you can pick off by the cover-up method are, 740 00:43:22,080 --> 00:43:24,020 is just the top one here. 741 00:43:24,020 --> 00:43:25,240 And these two. 742 00:43:25,240 --> 00:43:28,340 So C, D, and E you can get. 743 00:43:28,340 --> 00:43:30,170 But B and A you're going to have to do 744 00:43:30,170 --> 00:43:33,740 by either plugging in or some other, more elaborate, algebra. 745 00:43:33,740 --> 00:43:36,650 So the more of these lower terms there are, the worse off 746 00:43:36,650 --> 00:43:37,180 you are. 747 00:43:37,180 --> 00:43:48,170 STUDENT: [INAUDIBLE] 748 00:43:48,170 --> 00:43:57,990 PROFESSOR: The question is, does this x^3 + 21 affect this 749 00:43:57,990 --> 00:43:59,220 setup. 750 00:43:59,220 --> 00:44:03,290 And the answer is almost no. 751 00:44:03,290 --> 00:44:04,790 That is, not at all. 752 00:44:04,790 --> 00:44:06,510 It's the same setup exactly. 753 00:44:06,510 --> 00:44:08,260 But, there's one thing. 754 00:44:08,260 --> 00:44:11,510 If the degree gets too big, then you've 755 00:44:11,510 --> 00:44:18,270 got to use long division first to knock it down. 756 00:44:18,270 --> 00:44:20,910 I'll give you an example of this type of practice. 757 00:44:20,910 --> 00:44:22,220 Unless there are more question. 758 00:44:22,220 --> 00:44:22,440 Yes. 759 00:44:22,440 --> 00:44:23,410 STUDENT: [INAUDIBLE] 760 00:44:23,410 --> 00:44:26,944 PROFESSOR: Are you going to have to know 761 00:44:26,944 --> 00:44:29,580 how to do reduction formulas? 762 00:44:29,580 --> 00:44:33,160 Anything that's a little out of the ordinary like a reduction 763 00:44:33,160 --> 00:44:35,581 formula, I will have to coach you to do. 764 00:44:35,581 --> 00:44:37,080 So, in other words, what you'll have 765 00:44:37,080 --> 00:44:40,280 to be able to do in that situation is follow directions. 766 00:44:40,280 --> 00:44:42,230 If I tell you OK, you're faced with this, 767 00:44:42,230 --> 00:44:44,770 then do an integration by parts. 768 00:44:44,770 --> 00:44:48,530 And do that, then get the reduction formula. 769 00:44:48,530 --> 00:44:49,410 STUDENT: [INAUDIBLE] 770 00:44:49,410 --> 00:44:50,810 PROFESSOR: Yeah. 771 00:44:50,810 --> 00:44:56,460 OK, so the question had do with the partial fractions method. 772 00:44:56,460 --> 00:45:03,250 And what happens if you have a quadratic. 773 00:45:03,250 --> 00:45:11,970 So, for instance, if it were this, 774 00:45:11,970 --> 00:45:13,200 this one's too disgusting. 775 00:45:13,200 --> 00:45:17,220 I'm going to just do it with two of them. 776 00:45:17,220 --> 00:45:20,990 So the parts with x and x+1 are the same. 777 00:45:20,990 --> 00:45:23,760 But now you have linear factors here. 778 00:45:23,760 --> 00:45:27,510 (Ax + B) / (x^2 + 2). 779 00:45:27,510 --> 00:45:33,310 And A-- maybe I'll call them 1, and A_2 x + (A_2 x + B_2) / 780 00:45:33,310 --> 00:45:43,790 (x^2 + 2)^2 + C / x + D / (x+1). 781 00:45:43,790 --> 00:45:47,060 This is the way it works. 782 00:45:47,060 --> 00:45:52,100 OK, I'm going to give you one more quick example 783 00:45:52,100 --> 00:45:59,720 of an integration technique just to liven things up. 784 00:45:59,720 --> 00:46:02,700 Let's see. 785 00:46:02,700 --> 00:46:06,330 So here's a somewhat tricky example. 786 00:46:06,330 --> 00:46:08,080 This is just a little trickier than I 787 00:46:08,080 --> 00:46:09,640 would give you on a test. 788 00:46:09,640 --> 00:46:14,040 But it's the same principle, and I may do this on a final exam. 789 00:46:14,040 --> 00:46:20,300 So suppose you're faced with this integral. 790 00:46:20,300 --> 00:46:24,250 What are you going to do? 791 00:46:24,250 --> 00:46:26,140 Integration by parts, great. 792 00:46:26,140 --> 00:46:29,360 That's right, that's because this guy is 793 00:46:29,360 --> 00:46:33,820 begging to be differentiated, to be made simpler. 794 00:46:33,820 --> 00:46:37,290 So that means that I want this one to be u, 795 00:46:37,290 --> 00:46:40,190 and I want this one to be v'. 796 00:46:40,190 --> 00:46:42,120 And I want to use integration by parts. 797 00:46:42,120 --> 00:46:48,400 And then u ' = 1 / (1+x^2), and v = x^2 / 2. 798 00:46:51,360 --> 00:46:58,200 So the answer is now, x^2 / x^2/2 tan^(-1) x minus 799 00:46:58,200 --> 00:47:00,160 the integral of this guy. 800 00:47:00,160 --> 00:47:02,530 Which is going to be x^2 / 2. 801 00:47:02,530 --> 00:47:04,040 And then I have 1 / (1 + x^2) dx. 802 00:47:09,200 --> 00:47:13,300 Now, you are not done at this point. 803 00:47:13,300 --> 00:47:16,780 You're still in slightly hot water. 804 00:47:16,780 --> 00:47:19,010 You're in tepid water, anyway. 805 00:47:19,010 --> 00:47:21,470 So what is it that you have to do here? 806 00:47:21,470 --> 00:47:24,280 You're faced with this integral, which 807 00:47:24,280 --> 00:47:30,777 I'll put on the next board. 808 00:47:30,777 --> 00:47:32,360 It's a lot simpler than the other one, 809 00:47:32,360 --> 00:47:35,970 but as I say you're not quite out of the woods. 810 00:47:35,970 --> 00:47:38,570 You're faced with the integral of 1/2-- 811 00:47:38,570 --> 00:47:42,410 -1/2 x^2 / (1 + x^2) dx. dx. 812 00:47:42,410 --> 00:47:50,350 STUDENT: [INAUDIBLE] PROFESSOR: Trig substitution actually, 813 00:47:50,350 --> 00:47:52,914 interestingly, will work. 814 00:47:52,914 --> 00:47:54,580 But that wasn't what I wanted you to do. 815 00:47:54,580 --> 00:47:56,200 I wanted you to, yeah, go ahead. 816 00:47:56,200 --> 00:47:57,430 STUDENT: [INAUDIBLE] 817 00:47:57,430 --> 00:47:59,450 PROFESSOR: Add and subtract 1 to the numerator. 818 00:47:59,450 --> 00:48:02,330 So now, that's the correct answer. 819 00:48:02,330 --> 00:48:04,850 This is the case where the numerator and the denominator 820 00:48:04,850 --> 00:48:06,200 are tied. 821 00:48:06,200 --> 00:48:08,270 And so you have to use long division. 822 00:48:08,270 --> 00:48:10,690 But a shortcut is just to observe 823 00:48:10,690 --> 00:48:13,300 that the result of long division is 824 00:48:13,300 --> 00:48:20,920 the same thing as doing this. 825 00:48:20,920 --> 00:48:26,360 And then noticing that this is 1 - 1 / (1 + x^2). 826 00:48:26,360 --> 00:48:30,040 So this is the same as long division, in this case. 827 00:48:30,040 --> 00:48:34,970 Because when you divide in, it goes in with a quotient of 1. 828 00:48:34,970 --> 00:48:43,509 And so this guy turns out to be -1/2 the integral of 1 - 829 00:48:43,509 --> 00:48:44,050 1/(1+x^2) dx. 830 00:48:47,100 --> 00:48:55,000 Which is 1/2 x - 1/2 tan^(-1) x + c. 831 00:48:55,000 --> 00:49:02,230 So this is one extra step that you may be faced with someday 832 00:49:02,230 --> 00:49:02,900 in your life. 833 00:49:02,900 --> 00:49:05,620 And just keep that in mind.