1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:05,750 Your support will help MIT OpenCourseWare 4 00:00:05,750 --> 00:00:09,460 continue to offer high quality educational resources for free. 5 00:00:09,460 --> 00:00:12,550 To make a donation, or to view additional materials 6 00:00:12,550 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,150 --> 00:00:21,840 at ocw.mit.edu. 8 00:00:21,840 --> 00:00:25,190 PROFESSOR: Well, because our subject today 9 00:00:25,190 --> 00:00:28,280 is trig integrals and substitutions, 10 00:00:28,280 --> 00:00:33,640 Professor Jerison called in his substitute teacher for today. 11 00:00:33,640 --> 00:00:46,150 That's me. 12 00:00:46,150 --> 00:00:49,860 Professor Miller. 13 00:00:49,860 --> 00:00:52,790 And I'm going to try to tell you about trig substitutions 14 00:00:52,790 --> 00:00:54,610 and trig integrals. 15 00:00:54,610 --> 00:00:59,650 And I'll be here tomorrow to do more of the same, as well. 16 00:00:59,650 --> 00:01:02,320 So, this is about trigonometry, and maybe first thing I'll do 17 00:01:02,320 --> 00:01:24,180 is remind you of some basic things about trigonometry. 18 00:01:24,180 --> 00:01:27,110 So, if I have a circle, trigonometry 19 00:01:27,110 --> 00:01:29,730 is all based on the circle of radius 1 20 00:01:29,730 --> 00:01:32,000 and centered at the origin. 21 00:01:32,000 --> 00:01:34,870 And so if this is an angle of theta, up from the x-axis, 22 00:01:34,870 --> 00:01:36,350 then the coordinates of this point 23 00:01:36,350 --> 00:01:39,550 are cosine theta and sine theta. 24 00:01:39,550 --> 00:01:42,090 And so that leads right away to some trig identities, 25 00:01:42,090 --> 00:01:43,330 which you know very well. 26 00:01:43,330 --> 00:01:46,820 But I'm going to put them up here because we'll use them 27 00:01:46,820 --> 00:01:51,220 over and over again today. 28 00:01:51,220 --> 00:01:55,220 Remember the convention sin^2 theta secretly means (sin 29 00:01:55,220 --> 00:01:57,530 theta)^2. 30 00:01:57,530 --> 00:01:59,040 It would be more sensible to write 31 00:01:59,040 --> 00:02:01,600 a parenthesis around the sine of theta 32 00:02:01,600 --> 00:02:03,350 and then say you square that. 33 00:02:03,350 --> 00:02:06,380 But everybody in the world puts the 2 up there over the sin, 34 00:02:06,380 --> 00:02:09,080 and so I'll do that too. 35 00:02:09,080 --> 00:02:11,995 So that follows just because the circle has radius 1. 36 00:02:11,995 --> 00:02:13,870 But then there are some other identities too, 37 00:02:13,870 --> 00:02:15,410 which I think you remember. 38 00:02:15,410 --> 00:02:19,170 I'll write them down here. cos(2theta), 39 00:02:19,170 --> 00:02:22,490 there's this double angle formula that says cos(2theta) = 40 00:02:22,490 --> 00:02:23,840 cos^2(theta) - sin^2(theta). 41 00:02:29,090 --> 00:02:31,000 And there's also the double angle formula 42 00:02:31,000 --> 00:02:34,620 for the sin(2theta). 43 00:02:34,620 --> 00:02:38,480 Remember what that says? 44 00:02:38,480 --> 00:02:41,300 2 sin(theta) cos(theta). 45 00:02:46,437 --> 00:02:48,020 I'm going to use these trig identities 46 00:02:48,020 --> 00:02:50,370 and I'm going to use them in a slightly different way. 47 00:02:50,370 --> 00:02:53,090 And so I'd like to pay a little more attention to this one 48 00:02:53,090 --> 00:02:57,120 and get a different way of writing this one out. 49 00:02:57,120 --> 00:03:06,930 So this is actually the half angle formula. 50 00:03:06,930 --> 00:03:14,170 And that says, I'm going to try to express the cos(theta) 51 00:03:14,170 --> 00:03:16,400 in terms of the cos(2theta). 52 00:03:16,400 --> 00:03:19,590 So if I know the cos(2theta), I want 53 00:03:19,590 --> 00:03:23,320 to try to express the cos theta in terms of it. 54 00:03:23,320 --> 00:03:30,130 Well, I'll start with a cos(2theta) and play with that. 55 00:03:30,130 --> 00:03:30,630 OK. 56 00:03:30,630 --> 00:03:36,600 Well, we know what this is, it's cos^2(theta) - sin^2(theta). 57 00:03:36,600 --> 00:03:39,520 But we also know what the sin^2(theta) is in terms 58 00:03:39,520 --> 00:03:40,250 of the cosine. 59 00:03:40,250 --> 00:03:44,630 So I can eliminate the sin^2 from this picture. 60 00:03:44,630 --> 00:03:48,400 So this is equal to cos^2(theta) minus the quantity 1 - 61 00:03:48,400 --> 00:03:50,650 cos^2(theta). 62 00:03:50,650 --> 00:03:57,510 I put in what sin^2 is in terms of cos^2 And so that's 2 63 00:03:57,510 --> 00:03:59,940 cos^2(theta) - 1. 64 00:03:59,940 --> 00:04:02,240 There's this cos^2, which gets a plus sign. 65 00:04:02,240 --> 00:04:04,540 Because of these two minus signs. 66 00:04:04,540 --> 00:04:06,290 And there's the one that was there before, 67 00:04:06,290 --> 00:04:10,270 so altogether there are two of them. 68 00:04:10,270 --> 00:04:12,380 I want to isolate what cosine is. 69 00:04:12,380 --> 00:04:16,070 Or rather, what cos^2 is. 70 00:04:16,070 --> 00:04:17,680 So let's solve for that. 71 00:04:17,680 --> 00:04:20,120 So I'll put the 1 on the other side. 72 00:04:20,120 --> 00:04:24,120 And I get 1 + cos(2theta). 73 00:04:24,120 --> 00:04:27,910 And then, I want to divide by this 2, and so that puts a 2 74 00:04:27,910 --> 00:04:30,280 in this denominator here. 75 00:04:30,280 --> 00:04:33,280 So some people call that the half angle formula. 76 00:04:33,280 --> 00:04:36,730 What it really is for us is it's a way of eliminating powers 77 00:04:36,730 --> 00:04:38,800 from sines and cosines. 78 00:04:38,800 --> 00:04:41,290 I've gotten rid of this square at the expense 79 00:04:41,290 --> 00:04:43,720 of putting in a 2theta here. 80 00:04:43,720 --> 00:04:45,260 We'll use that. 81 00:04:45,260 --> 00:04:51,022 And, similarly, same calculation shows that sin^2(theta) = (1 - 82 00:04:51,022 --> 00:04:51,730 cos(2theta)) / 2. 83 00:04:55,850 --> 00:05:01,480 Same cosine, in that formula also, but it has a minus sign. 84 00:05:01,480 --> 00:05:03,640 For the sin^2. 85 00:05:03,640 --> 00:05:07,860 OK. so that's my little review of trig identities 86 00:05:07,860 --> 00:05:14,620 that we'll make use of as this lecture goes on. 87 00:05:14,620 --> 00:05:17,720 I want to talk about trig identity-- trig integrals, 88 00:05:17,720 --> 00:05:23,500 and you know some trig integrals, I'm sure, already. 89 00:05:23,500 --> 00:05:26,740 Like, well, let me write the differential form first. 90 00:05:26,740 --> 00:05:30,400 You know that d sin theta, or maybe I'll 91 00:05:30,400 --> 00:05:35,620 say d sin x, is, let's see, that's 92 00:05:35,620 --> 00:05:39,030 the derivative of sin x times dx, right. 93 00:05:39,030 --> 00:05:46,120 The derivative of sin x is cos x, dx. 94 00:05:46,120 --> 00:05:50,080 And so if I integrate both sides here, the integral form of this 95 00:05:50,080 --> 00:05:55,390 is the integral of cos x dx. 96 00:05:55,390 --> 00:05:59,850 Is sin x plus a constant. 97 00:05:59,850 --> 00:06:07,410 And in the same way, d cos x = -sin x dx. 98 00:06:07,410 --> 00:06:10,320 Right, the derivative of the cosine is minus sine. 99 00:06:10,320 --> 00:06:15,330 And when I integrate that, I find the integral of sin x dx 100 00:06:15,330 --> 00:06:21,140 is -cos x + c. 101 00:06:21,140 --> 00:06:22,820 So that's our starting point. 102 00:06:22,820 --> 00:06:27,510 And the game today, for the first half of the lecture, 103 00:06:27,510 --> 00:06:33,550 is to use that basic-- just those basic integration 104 00:06:33,550 --> 00:06:37,710 formulas, together with clever use of trig identities 105 00:06:37,710 --> 00:06:41,260 in order to compute more complicated formulas involving 106 00:06:41,260 --> 00:06:42,820 trig functions. 107 00:06:42,820 --> 00:06:47,040 So the first thing, the first topic, 108 00:06:47,040 --> 00:06:52,250 is to think about integrals of the form sin^n (x) cos^n (x) 109 00:06:52,250 --> 00:06:52,750 dx. 110 00:06:56,190 --> 00:07:04,070 Where here I have in mind m and n are non-negative integers. 111 00:07:04,070 --> 00:07:05,950 So let's try to integrate these. 112 00:07:05,950 --> 00:07:09,640 I'll show you some applications of these pretty soon. 113 00:07:09,640 --> 00:07:11,250 Looking down the road a little bit, 114 00:07:11,250 --> 00:07:14,050 integrals like this show up in Fourier series 115 00:07:14,050 --> 00:07:16,190 and many other subjects in mathematics. 116 00:07:16,190 --> 00:07:20,000 It turns out they're quite important to be able to do. 117 00:07:20,000 --> 00:07:23,860 So that's why we're doing them now. 118 00:07:23,860 --> 00:07:29,690 Well, so there are two cases to think about here. 119 00:07:29,690 --> 00:07:32,070 When you're integrating things like this. 120 00:07:32,070 --> 00:07:35,890 There's the easy case, and let's do that one first. 121 00:07:35,890 --> 00:07:49,495 The easy case is when at least one exponent is odd. 122 00:07:49,495 --> 00:07:50,370 That's the easy case. 123 00:07:50,370 --> 00:07:55,870 So, for example, suppose that I wanted to integrate, 124 00:07:55,870 --> 00:08:02,560 well, let's take the case m = 1. 125 00:08:02,560 --> 00:08:09,530 So I'm integrating sin^n (x) cos x dx. 126 00:08:09,530 --> 00:08:17,050 I'm taking-- Oh, I could do that one. 127 00:08:17,050 --> 00:08:23,600 Let's see if that's what I want to take. 128 00:08:23,600 --> 00:08:27,540 Yeah. 129 00:08:27,540 --> 00:08:30,590 My confusion is that I meant to have this a different power. 130 00:08:30,590 --> 00:08:34,750 You were thinking that. 131 00:08:34,750 --> 00:08:36,670 So let's do this case when m = 1. 132 00:08:36,670 --> 00:08:38,680 So the integral I'm trying to do is any power 133 00:08:38,680 --> 00:08:41,450 of the sine times the cosine. 134 00:08:41,450 --> 00:08:44,520 Well, here's the trick. 135 00:08:44,520 --> 00:08:48,920 Recognize, use this formula up at the top 136 00:08:48,920 --> 00:08:53,570 there to see cos x dx as something that we already 137 00:08:53,570 --> 00:08:55,670 have on the blackboard. 138 00:08:55,670 --> 00:08:59,570 So, the way to exploit that is to make a substitution. 139 00:08:59,570 --> 00:09:08,320 And substitution is going to be u = sin x. 140 00:09:08,320 --> 00:09:09,610 And here's why. 141 00:09:09,610 --> 00:09:12,820 Then this integral that I'm trying to do is the integral 142 00:09:12,820 --> 00:09:18,760 of u^n, that's already a simplification. 143 00:09:18,760 --> 00:09:22,160 And then there's that cos x dx. 144 00:09:22,160 --> 00:09:25,120 When you make a substitution, you've got to go all the way 145 00:09:25,120 --> 00:09:27,970 and replace everything in the expression 146 00:09:27,970 --> 00:09:33,000 by things involving this new variable that I've introduced. 147 00:09:33,000 --> 00:09:35,390 So I'd better get rid of the cos x dx 148 00:09:35,390 --> 00:09:39,140 and rewrite it in terms of du or in terms of u. 149 00:09:39,140 --> 00:09:45,290 And I can do that because du, according to that formula, 150 00:09:45,290 --> 00:09:50,600 is cos x dx. 151 00:09:50,600 --> 00:09:53,810 Let me put a box around that. 152 00:09:53,810 --> 00:09:55,800 That's our substitution. 153 00:09:55,800 --> 00:09:57,500 When you make a substitution, you also 154 00:09:57,500 --> 00:10:00,740 want to compute the differential of the variable 155 00:10:00,740 --> 00:10:03,100 that you substitute in. 156 00:10:03,100 --> 00:10:08,800 So the cos x dx that appears here is just, exactly, du. 157 00:10:08,800 --> 00:10:11,500 And I've replaced this trig integral with something 158 00:10:11,500 --> 00:10:13,330 that doesn't involve trig functions at all. 159 00:10:13,330 --> 00:10:14,240 This is a lot easier. 160 00:10:14,240 --> 00:10:17,420 We can just plug into what we know here. 161 00:10:17,420 --> 00:10:23,640 This is u^(n+1) / (n+1) plus a constant, 162 00:10:23,640 --> 00:10:26,620 and I've done the integral. 163 00:10:26,620 --> 00:10:29,580 But I'm not quite done with the problem yet. 164 00:10:29,580 --> 00:10:34,310 Because to be nice to your reader and to yourself, 165 00:10:34,310 --> 00:10:36,560 you should go back at this point, probably, 166 00:10:36,560 --> 00:10:40,399 go back and get rid of this new variable that you introduced. 167 00:10:40,399 --> 00:10:42,440 You're the one who introduced this variable, you. 168 00:10:42,440 --> 00:10:45,950 Nobody except you, really, knows what it is. 169 00:10:45,950 --> 00:10:48,080 But the rest of the world knows what 170 00:10:48,080 --> 00:10:51,360 they asked for the first place that involved x. 171 00:10:51,360 --> 00:10:53,460 So I have to go back and get rid of this. 172 00:10:53,460 --> 00:10:57,830 And that's not hard to do in this case, because u = sin x. 173 00:10:57,830 --> 00:11:04,390 And so I make this back substitution. 174 00:11:04,390 --> 00:11:05,820 And that's what you get. 175 00:11:05,820 --> 00:11:11,040 So there's the answer. 176 00:11:11,040 --> 00:11:15,930 OK, so the game was, I use this odd power of the cosine here, 177 00:11:15,930 --> 00:11:19,180 and I could see it appearing as the differential of the sine. 178 00:11:19,180 --> 00:11:22,470 So that's what made this substitution work. 179 00:11:22,470 --> 00:11:25,060 Let's do another example to see how that works out 180 00:11:25,060 --> 00:11:36,490 in a slightly different case. 181 00:11:36,490 --> 00:11:48,420 So here's another example. 182 00:11:48,420 --> 00:11:50,050 Now I do have an odd power. 183 00:11:50,050 --> 00:11:53,800 One of the exponents is odd, so I'm in the easy case. 184 00:11:53,800 --> 00:11:56,310 But it's not 1. 185 00:11:56,310 --> 00:12:05,590 The game now is to use this trig identity 186 00:12:05,590 --> 00:12:10,500 to get rid of the largest even power that you can, 187 00:12:10,500 --> 00:12:13,870 from this odd power here. 188 00:12:13,870 --> 00:12:24,710 So use sin^2 x = 1 - cos^2 x, to eliminate a lot of powers from 189 00:12:24,710 --> 00:12:26,220 that odd power. 190 00:12:26,220 --> 00:12:28,370 Watch what happens. 191 00:12:28,370 --> 00:12:31,490 So this is not really a substitution or anything, 192 00:12:31,490 --> 00:12:34,320 this is just a trig identity. 193 00:12:34,320 --> 00:12:38,130 This sine cubed is sine squared times the sine. 194 00:12:38,130 --> 00:12:41,360 And the sine squared is 1 - cos^2 x. 195 00:12:41,360 --> 00:12:43,770 And then I have the remaining sin x. 196 00:12:43,770 --> 00:12:48,970 And then I have cos^2 x dx. 197 00:12:48,970 --> 00:12:53,580 So let me rewrite that a little bit to see how this works out. 198 00:12:53,580 --> 00:12:59,490 This is the integral of cos^2 x minus, 199 00:12:59,490 --> 00:13:01,240 and then there's the product of these two. 200 00:13:01,240 --> 00:13:09,370 That's cos^4 x times sin x dx. 201 00:13:09,370 --> 00:13:12,200 So now I'm really exactly in the situation 202 00:13:12,200 --> 00:13:13,700 that I was in over here. 203 00:13:13,700 --> 00:13:17,570 I've got a single power of a sine or cosine. 204 00:13:17,570 --> 00:13:20,060 It happens that it's a sine here. 205 00:13:20,060 --> 00:13:21,960 But that's not going to cause any trouble, 206 00:13:21,960 --> 00:13:26,290 we can go ahead and play the same game that I did there. 207 00:13:26,290 --> 00:13:28,860 So, so far I've just been using trig identities. 208 00:13:28,860 --> 00:13:41,770 But now I'll use a trig substitution. 209 00:13:41,770 --> 00:13:45,335 And I think I want to write these as powers of a variable. 210 00:13:45,335 --> 00:13:47,960 And then this is going to be the differential of that variable. 211 00:13:47,960 --> 00:13:58,190 So I'll take u to be cos x, and that means that du = -sin x dx. 212 00:13:58,190 --> 00:14:04,950 There's the substitution. 213 00:14:04,950 --> 00:14:09,550 So when I make that substitution, what do we get. 214 00:14:09,550 --> 00:14:11,710 Cosine squared becomes u^2. 215 00:14:15,470 --> 00:14:23,150 Cosine to the 4th becomes u^4, and sin x dx becomes not quite 216 00:14:23,150 --> 00:14:27,570 du, watch for the signum, watch for this minus sign here. 217 00:14:27,570 --> 00:14:32,100 It becomes -du. 218 00:14:32,100 --> 00:14:32,820 But that's OK. 219 00:14:32,820 --> 00:14:34,720 The minus sign comes outside. 220 00:14:34,720 --> 00:14:36,860 And I can integrate both of these powers, 221 00:14:36,860 --> 00:14:43,540 so I get -u^3 / 3. 222 00:14:43,540 --> 00:14:48,950 And then this 4th power gives me a 5th power, when I integrate. 223 00:14:48,950 --> 00:14:53,390 And don't forget the constant. 224 00:14:53,390 --> 00:14:55,100 Am I done? 225 00:14:55,100 --> 00:14:55,850 Not quite done. 226 00:14:55,850 --> 00:14:57,690 I have to back substitute and get rid 227 00:14:57,690 --> 00:15:00,830 of my choice of variable, u, and replace it with yours. 228 00:15:00,830 --> 00:15:01,330 Questions? 229 00:15:01,330 --> 00:15:06,049 STUDENT: [INAUDIBLE] 230 00:15:06,049 --> 00:15:07,340 PROFESSOR: There should indeed. 231 00:15:07,340 --> 00:15:10,190 I forgot this minus sign when I came down here. 232 00:15:10,190 --> 00:15:12,690 So these two gang up to give me a plus. 233 00:15:12,690 --> 00:15:14,960 Was that what the other question was about, too? 234 00:15:14,960 --> 00:15:16,720 Thanks. 235 00:15:16,720 --> 00:15:18,030 So let's back substitute. 236 00:15:18,030 --> 00:15:23,900 And I'm going to put that over here. 237 00:15:23,900 --> 00:15:27,920 And the result is, well, I just replace the u by cosine of x. 238 00:15:27,920 --> 00:15:38,160 So this is - -cos^3(x) / 3 plus, thank you, cos^5(x) / 5 + c. 239 00:15:38,160 --> 00:15:44,670 And there's the answer. 240 00:15:44,670 --> 00:15:47,210 By the way, you can remember one of the nice things 241 00:15:47,210 --> 00:15:49,160 about doing an integral is it's fairly 242 00:15:49,160 --> 00:15:51,280 easy to check your answer. 243 00:15:51,280 --> 00:15:53,680 You can always differentiate the thing you get, 244 00:15:53,680 --> 00:15:56,890 and see whether you get the right thing when you go back. 245 00:15:56,890 --> 00:15:59,280 It's not too hard to use the power 246 00:15:59,280 --> 00:16:02,500 rules and the differentiation rule 247 00:16:02,500 --> 00:16:06,080 for the cosine to get back to this if you 248 00:16:06,080 --> 00:16:09,350 want to check the work. 249 00:16:09,350 --> 00:16:12,830 Let's do one more example, just to handle 250 00:16:12,830 --> 00:16:15,980 an example of this easy case, which you 251 00:16:15,980 --> 00:16:18,490 might have thought of at first. 252 00:16:18,490 --> 00:16:22,720 Suppose I just want to integrate a cube. sin^3 x. 253 00:16:29,320 --> 00:16:32,040 No cosine in sight. 254 00:16:32,040 --> 00:16:35,780 But I do have an odd power of a trig 255 00:16:35,780 --> 00:16:37,180 function, of a sine or cosine. 256 00:16:37,180 --> 00:16:39,020 So I'm in the easy case. 257 00:16:39,020 --> 00:16:44,740 And the procedure that I was suggesting says I want to take 258 00:16:44,740 --> 00:16:48,700 out the largest even power that I can, from the sin^3. 259 00:16:48,700 --> 00:16:52,870 So I'll take that out, that's a sin^2, and write it as 1 - 260 00:16:52,870 --> 00:16:53,370 cos^2. 261 00:16:57,020 --> 00:16:58,380 Well, now I'm very happy. 262 00:16:58,380 --> 00:17:00,470 Because it's just like the situation 263 00:17:00,470 --> 00:17:06,017 we had somewhere on the board here. 264 00:17:06,017 --> 00:17:07,850 It's just like the situation we had up here. 265 00:17:07,850 --> 00:17:11,660 I've got a power of a cosine times sin x dx. 266 00:17:11,660 --> 00:17:16,310 So exactly the same substitution steps in. 267 00:17:16,310 --> 00:17:19,070 You get, and maybe you can see what 268 00:17:19,070 --> 00:17:20,860 happens without doing the work. 269 00:17:20,860 --> 00:17:22,630 Shall I do the work here? 270 00:17:22,630 --> 00:17:24,450 I make the same substitution. 271 00:17:24,450 --> 00:17:30,680 And so this is (1 - u (1 - u^2) times -du. 272 00:17:33,540 --> 00:17:40,050 Which is u - u^3 / 3. 273 00:17:40,050 --> 00:17:42,050 But then I want to put this minus sign in place, 274 00:17:42,050 --> 00:17:47,700 and so that gives me -u + u^3 / 3 plus a constant. 275 00:17:47,700 --> 00:17:58,090 And then I back substitute and get cos x + cos^3 x / 3. 276 00:17:58,090 --> 00:17:59,350 So this is the easy case. 277 00:17:59,350 --> 00:18:02,190 If you have some odd power to play with, 278 00:18:02,190 --> 00:18:07,990 then you can make use of it and it's pretty straightforward. 279 00:18:07,990 --> 00:18:10,680 OK the harder case is when you don't have an odd power. 280 00:18:10,680 --> 00:18:11,610 So what's the program? 281 00:18:11,610 --> 00:18:13,770 I'm going to do the harder case, and then I'm 282 00:18:13,770 --> 00:18:19,340 going to show you an example of how to integrate square roots. 283 00:18:19,340 --> 00:18:26,260 And do an application, using these ideas from trigonometry. 284 00:18:26,260 --> 00:18:30,250 So I want to keep this blackboard. 285 00:18:30,250 --> 00:18:34,470 Maybe I'll come back and start here again. 286 00:18:34,470 --> 00:18:55,240 So the harder case is when they're only even exponents. 287 00:18:55,240 --> 00:18:58,190 I'm still trying to integrate the same form. 288 00:18:58,190 --> 00:19:00,370 But now all the exponents are even. 289 00:19:00,370 --> 00:19:03,510 So we have to do some game. 290 00:19:03,510 --> 00:19:10,400 And here the game is use the half angle formula. 291 00:19:10,400 --> 00:19:16,455 Which I just erased, very sadly, on the board here. 292 00:19:16,455 --> 00:19:17,830 Maybe I'll rewrite them over here 293 00:19:17,830 --> 00:19:23,550 so we have them on the board. 294 00:19:23,550 --> 00:19:44,270 I think I remember what they were. 295 00:19:44,270 --> 00:19:46,300 So the game is I'm going to use that half angle 296 00:19:46,300 --> 00:19:50,180 formula to start getting rid of those even powers. 297 00:19:50,180 --> 00:19:54,510 Half angle formula written like this, exactly, talks about-- it 298 00:19:54,510 --> 00:19:57,790 rewrites even powers of sines and cosines. 299 00:19:57,790 --> 00:20:00,950 So let's see how that works out in an example. 300 00:20:00,950 --> 00:20:08,880 How about just the cosine squared for a start. 301 00:20:08,880 --> 00:20:09,700 What to do? 302 00:20:09,700 --> 00:20:11,970 I can't pull anything out. 303 00:20:11,970 --> 00:20:15,180 I could rewrite this as 1 - sin^2, 304 00:20:15,180 --> 00:20:17,290 but then I'd be faced with integrating the sin^2, 305 00:20:17,290 --> 00:20:19,480 which is exactly as hard. 306 00:20:19,480 --> 00:20:23,100 So instead, let's use this formula here. 307 00:20:23,100 --> 00:20:29,180 This is really the same as (1+cos(2theta)) / 2. 308 00:20:29,180 --> 00:20:32,700 And now, this is easy. 309 00:20:32,700 --> 00:20:34,340 It's got two parts to it. 310 00:20:34,340 --> 00:20:38,550 Integrating one half gives me theta over-- Oh. 311 00:20:38,550 --> 00:20:42,440 Miraculously, the x turned into a theta. 312 00:20:42,440 --> 00:20:44,330 Let's put it back as x. 313 00:20:44,330 --> 00:20:47,440 I get x/2 by integrating 1/2. 314 00:20:47,440 --> 00:20:50,110 So, notice that something non-trigonometric occurs here 315 00:20:50,110 --> 00:20:52,710 when I do these even integrals. 316 00:20:52,710 --> 00:20:54,800 x/2 appears. 317 00:20:54,800 --> 00:20:57,880 And then the other one, OK, so this takes a little thought. 318 00:20:57,880 --> 00:21:01,870 The integral of the cosine is the sine, 319 00:21:01,870 --> 00:21:05,520 or is it minus the sine. 320 00:21:05,520 --> 00:21:11,390 Negative sine. 321 00:21:11,390 --> 00:21:12,634 Shall we take a vote? 322 00:21:12,634 --> 00:21:13,550 I think it's positive. 323 00:21:13,550 --> 00:21:18,310 And so you get sin(2x), but is that right? 324 00:21:18,310 --> 00:21:19,420 Over 2. 325 00:21:19,420 --> 00:21:24,150 If I differentiate the sin(2x), this 2 comes out. 326 00:21:24,150 --> 00:21:25,950 And would give me an extra 2 here. 327 00:21:25,950 --> 00:21:29,580 So there's an extra 2 that I have to put in here 328 00:21:29,580 --> 00:21:34,670 when I integrate it. 329 00:21:34,670 --> 00:21:37,230 And there's the answer. 330 00:21:37,230 --> 00:21:39,480 This is not a substitution. 331 00:21:39,480 --> 00:21:41,550 I just played with trig identities here. 332 00:21:41,550 --> 00:21:45,020 And then did a simple trig integral, 333 00:21:45,020 --> 00:21:46,900 getting your help to get the sign right. 334 00:21:46,900 --> 00:21:49,310 And thinking about what this 2 is going to do. 335 00:21:49,310 --> 00:21:52,990 It produces a 2 in the denominator. 336 00:21:52,990 --> 00:21:59,270 But it's not applying any complicated thing. 337 00:21:59,270 --> 00:22:03,100 It's just using this identity. 338 00:22:03,100 --> 00:22:05,580 Let's do another example that's a little bit harder. 339 00:22:05,580 --> 00:22:07,690 This time, sin^2 times cos^2. 340 00:22:35,940 --> 00:22:37,920 Again, no odd powers. 341 00:22:37,920 --> 00:22:40,730 I've got to work a little bit harder. 342 00:22:40,730 --> 00:22:42,930 And what I'm going to do is apply those identities 343 00:22:42,930 --> 00:22:44,210 up there. 344 00:22:44,210 --> 00:22:47,670 Now, what I recommend doing in this situation 345 00:22:47,670 --> 00:22:51,370 is going over to the side somewhere. 346 00:22:51,370 --> 00:22:55,610 And do some side work. 347 00:22:55,610 --> 00:22:58,680 Because it's all just playing with trig functions. 348 00:22:58,680 --> 00:23:06,280 It's not actually doing any integrals for a while. 349 00:23:06,280 --> 00:23:11,815 So, I guess one way to get rid of the sin^2 and the cos^2 is 350 00:23:11,815 --> 00:23:14,180 to use those identities and so let's do that. 351 00:23:14,180 --> 00:23:16,210 So the sine is (1 - cos(2x)) / 2. 352 00:23:20,350 --> 00:23:22,170 And the cosine is (1 + cos(2x)) / 2. 353 00:23:27,890 --> 00:23:29,870 So I just substitute them in. 354 00:23:29,870 --> 00:23:31,560 And now I can multiply that out. 355 00:23:31,560 --> 00:23:38,030 And what I have is a difference times a sum. 356 00:23:38,030 --> 00:23:40,771 So you know a formula for that. 357 00:23:40,771 --> 00:23:43,270 Taking the product of these two things, well there'll be a 4 358 00:23:43,270 --> 00:23:44,430 in the denominator. 359 00:23:44,430 --> 00:23:46,490 And then in the numerator, I get the square 360 00:23:46,490 --> 00:23:49,650 of this minus the square of this. 361 00:23:49,650 --> 00:23:59,840 (a-b)(a+b) = a^2 - b^2. = - So I get that. 362 00:23:59,840 --> 00:24:02,450 Well, I'm a little bit happier, because at least I 363 00:24:02,450 --> 00:24:03,710 don't have 4. 364 00:24:03,710 --> 00:24:07,430 I don't have 2 different squares. 365 00:24:07,430 --> 00:24:09,940 I still have a square, and want to integrate this. 366 00:24:09,940 --> 00:24:12,170 I'm still not in the easy case. 367 00:24:12,170 --> 00:24:16,970 I got myself back to an easier hard case. 368 00:24:16,970 --> 00:24:18,670 But we do know what to do about this. 369 00:24:18,670 --> 00:24:21,140 Because I just did it up there. 370 00:24:21,140 --> 00:24:24,430 And I could play into this formula that we got. 371 00:24:24,430 --> 00:24:29,010 But I think it's just as easy to continue to calculate here. 372 00:24:29,010 --> 00:24:32,880 Use the half angle formula again for this, 373 00:24:32,880 --> 00:24:34,900 and continue on your way. 374 00:24:34,900 --> 00:24:37,720 So I get a 1/4 from this bit. 375 00:24:37,720 --> 00:24:43,260 And then minus 1/4 of cos^2(2x). 376 00:24:45,780 --> 00:24:51,630 And when I plug in 2x in for theta, there in the top board, 377 00:24:51,630 --> 00:24:59,650 I'm going to get a 4x on the right-hand side. 378 00:24:59,650 --> 00:25:02,130 So it comes out like that. 379 00:25:02,130 --> 00:25:04,600 And I guess I could simplify that a little bit more. 380 00:25:04,600 --> 00:25:05,790 This is a 1/4. 381 00:25:05,790 --> 00:25:07,840 Oh, but then there's a 2 here. 382 00:25:07,840 --> 00:25:10,630 It's half that. 383 00:25:10,630 --> 00:25:12,200 So then I can simplify a little more. 384 00:25:12,200 --> 00:25:16,070 It's 1/4 - 1/8, which is 1/8. 385 00:25:16,070 --> 00:25:19,630 And then I have 1/8 cos(4x). 386 00:25:25,510 --> 00:25:27,560 OK, that's my side work. 387 00:25:27,560 --> 00:25:30,590 I just did some trig identities over here. 388 00:25:30,590 --> 00:25:32,840 And rewrote sine squared times cosine 389 00:25:32,840 --> 00:25:35,930 squared as something which involves just no powers 390 00:25:35,930 --> 00:25:37,940 of trig, just cosine by itself. 391 00:25:37,940 --> 00:25:41,330 And a constant. 392 00:25:41,330 --> 00:25:45,090 So I can take that and substitute it in here. 393 00:25:45,090 --> 00:25:48,810 And now the integration is pretty easy. 394 00:25:48,810 --> 00:25:57,810 1/8, cos(4x) / 8, dx, which is, OK 395 00:25:57,810 --> 00:26:01,340 the 1/8 is going to give me x/8. 396 00:26:01,340 --> 00:26:06,314 The integral or cosine is plus or minus the sine. 397 00:26:06,314 --> 00:26:08,230 The derivative of the sine is plus the cosine. 398 00:26:08,230 --> 00:26:11,440 So it's going to be plus the-- Only there's a minus here. 399 00:26:11,440 --> 00:26:17,890 So it's going to be the sine-- minus sin(4x) / 8, 400 00:26:17,890 --> 00:26:20,797 but then I have an additional factor in the denominator. 401 00:26:20,797 --> 00:26:21,880 And what's it going to be? 402 00:26:21,880 --> 00:26:28,830 I have to put a 4 there. 403 00:26:28,830 --> 00:26:32,730 So we've done that calculation, too. 404 00:26:32,730 --> 00:26:38,250 So any of these-- If you keep doing this kind of process, 405 00:26:38,250 --> 00:26:45,530 these two kinds of procedures, you 406 00:26:45,530 --> 00:26:48,840 can now integrate any expression that 407 00:26:48,840 --> 00:26:52,040 has a power of a sine times a power of a cosine in it, 408 00:26:52,040 --> 00:26:56,860 by using these ideas. 409 00:26:56,860 --> 00:27:01,730 Now, let's see. 410 00:27:01,730 --> 00:27:16,210 Oh, let me give you an alternate method for this last one here. 411 00:27:16,210 --> 00:27:26,460 I know what I'll do. 412 00:27:26,460 --> 00:27:28,850 Let me give an alternate method for doing, really 413 00:27:28,850 --> 00:27:31,670 doing the side work over there. 414 00:27:31,670 --> 00:27:35,920 I'm trying to deal with sin^2 times cos^2. 415 00:27:35,920 --> 00:27:50,080 Well that's the square of sin x cos x. 416 00:27:50,080 --> 00:27:54,240 And sin x cos x shows up right here. 417 00:27:54,240 --> 00:27:55,900 In another trig identity. 418 00:27:55,900 --> 00:27:58,410 So we can make use of that, too. 419 00:27:58,410 --> 00:28:01,330 That reduces the number of factors of sines and cosines 420 00:28:01,330 --> 00:28:01,930 by 1. 421 00:28:01,930 --> 00:28:04,190 So it's going in the right direction. 422 00:28:04,190 --> 00:28:11,270 This is equal to 1/2 sin(2x), squared. 423 00:28:11,270 --> 00:28:17,730 Sine times cosine is 1/2-- Say this right. 424 00:28:17,730 --> 00:28:21,280 It's sin(2x) / 2, and then I want to square that. 425 00:28:21,280 --> 00:28:31,030 So what I get is sin^2(2x) / 4. 426 00:28:31,030 --> 00:28:33,930 Which is, well, I'm not too happy yet, 427 00:28:33,930 --> 00:28:35,639 because I still have an even power. 428 00:28:35,639 --> 00:28:37,930 Remember I'm trying to integrate this thing in the end, 429 00:28:37,930 --> 00:28:39,060 even powers are bad. 430 00:28:39,060 --> 00:28:40,910 I try to get rid of them. 431 00:28:40,910 --> 00:28:46,360 By using that formula, the half angle formula. 432 00:28:46,360 --> 00:28:48,950 So I can apply that to sin x here again. 433 00:28:48,950 --> 00:28:52,600 I get 1/4 of (1 - cos(4x)) / 2. 434 00:28:57,480 --> 00:28:59,730 That's what the half angle formula says for sin^2(2x). 435 00:29:02,330 --> 00:29:04,790 And that's exactly the same as the expression 436 00:29:04,790 --> 00:29:08,660 that I got up here, as well. 437 00:29:08,660 --> 00:29:11,710 It's the same expression that I have there. 438 00:29:11,710 --> 00:29:16,680 So it's the same expression as I have here. 439 00:29:16,680 --> 00:29:20,000 So this is just an alternate way to play this game of using 440 00:29:20,000 --> 00:29:24,940 the half angle formula. 441 00:29:24,940 --> 00:29:27,790 OK, let's do a little application of these things 442 00:29:27,790 --> 00:29:46,060 and change the topic a little bit. 443 00:29:46,060 --> 00:29:48,210 So here's the problem. 444 00:29:48,210 --> 00:29:56,900 So this is an application and example 445 00:29:56,900 --> 00:30:07,570 of a real trig substitution. 446 00:30:07,570 --> 00:30:22,080 So here's the problem I want to look at. 447 00:30:22,080 --> 00:30:26,210 OK, so I have a circle whose radius is a. 448 00:30:26,210 --> 00:30:31,710 And I cut out from it a sort of tab, here. 449 00:30:31,710 --> 00:30:36,330 This tab here. 450 00:30:36,330 --> 00:30:38,590 And the height of this thing is b. 451 00:30:38,590 --> 00:30:42,110 So this length is a number b. 452 00:30:42,110 --> 00:30:47,219 And what I want to do is compute the area of that little tab. 453 00:30:47,219 --> 00:30:48,010 That's the problem. 454 00:30:48,010 --> 00:30:50,360 So there's an arc over here. 455 00:30:50,360 --> 00:30:55,060 And I want to find the area of this, for a and b, 456 00:30:55,060 --> 00:30:57,980 in terms of a and b. 457 00:30:57,980 --> 00:31:06,730 So the area, well, I guess one way 458 00:31:06,730 --> 00:31:12,370 to compute the area would be to take the integral of y dx. 459 00:31:12,370 --> 00:31:15,980 You've seen the idea of splitting this up 460 00:31:15,980 --> 00:31:20,600 into vertical strips whose height is given by a function 461 00:31:20,600 --> 00:31:21,100 y(x). 462 00:31:21,100 --> 00:31:22,266 And then you integrate that. 463 00:31:22,266 --> 00:31:24,040 That's an interpretation for the integral. 464 00:31:24,040 --> 00:31:27,210 The area is given by y dx. 465 00:31:27,210 --> 00:31:30,327 But that's a little bit awkward, because my formula for y 466 00:31:30,327 --> 00:31:31,660 is going to be a little strange. 467 00:31:31,660 --> 00:31:34,630 It's constant, value of b, along here, and then at this point 468 00:31:34,630 --> 00:31:37,660 it becomes this arc, of the circle. 469 00:31:37,660 --> 00:31:39,500 So working this out, I could do it 470 00:31:39,500 --> 00:31:42,100 but it's a little awkward because expressing y 471 00:31:42,100 --> 00:31:45,880 as a function of x, the top edge of this shape, 472 00:31:45,880 --> 00:31:50,090 it's a little awkward, and takes two different regions 473 00:31:50,090 --> 00:31:51,740 to express. 474 00:31:51,740 --> 00:31:58,620 So, a different way to say it is to say x dy. 475 00:31:58,620 --> 00:32:00,350 Maybe that'll work a little bit better. 476 00:32:00,350 --> 00:32:02,810 Or maybe it won't, but it's worth trying. 477 00:32:02,810 --> 00:32:05,290 I could just as well split this region up 478 00:32:05,290 --> 00:32:08,420 into horizontal strips. 479 00:32:08,420 --> 00:32:13,010 Whose width is dy, and whose length is x. 480 00:32:13,010 --> 00:32:17,250 Now I'm thinking of this as a function of y. 481 00:32:17,250 --> 00:32:20,870 This is the graph of a function of y. 482 00:32:20,870 --> 00:32:24,880 And that's much better, because the function of y is, well, 483 00:32:24,880 --> 00:32:28,010 it's the square root of a^2 - y^2, isn't it. 484 00:32:28,010 --> 00:32:34,300 That's x x^2 + y^2 = a^2. 485 00:32:34,300 --> 00:32:38,640 So that's what x is. 486 00:32:38,640 --> 00:32:41,740 And that's what I'm asked to integrate, then. 487 00:32:41,740 --> 00:32:45,580 Square root of (a^2 - y^2), dy. 488 00:32:45,580 --> 00:32:47,500 And I can even put in limits of integration. 489 00:32:47,500 --> 00:32:49,500 Maybe I should do that, because this is supposed 490 00:32:49,500 --> 00:32:50,910 to be an actual number. 491 00:32:50,910 --> 00:32:55,130 I guess I'm integrating it from y = 0, that's here. 492 00:32:55,130 --> 00:33:00,059 To y = b, dy. 493 00:33:00,059 --> 00:33:01,350 So this is what I want to find. 494 00:33:01,350 --> 00:33:07,420 This is a integral formula for the area of that region. 495 00:33:07,420 --> 00:33:08,730 And this is a new form. 496 00:33:08,730 --> 00:33:13,970 I don't think that you've thought 497 00:33:13,970 --> 00:33:20,310 about integrating expressions like this in this class before. 498 00:33:20,310 --> 00:33:23,660 So, it's a new form and I want to show you how to do it, 499 00:33:23,660 --> 00:33:30,290 how it's related to trigonometry. 500 00:33:30,290 --> 00:33:33,080 It's related to trigonometry through that exact picture 501 00:33:33,080 --> 00:33:36,710 that we have on the blackboard. 502 00:33:36,710 --> 00:33:42,610 After all, this a^2 - y^2 is the formula for this arc. 503 00:33:42,610 --> 00:33:45,460 And so, what I propose is that we 504 00:33:45,460 --> 00:33:49,700 try to exploit the connection with the circle 505 00:33:49,700 --> 00:33:52,860 and introduce polar coordinates. 506 00:33:52,860 --> 00:34:03,499 So, here if I measure this angle then there 507 00:34:03,499 --> 00:34:04,790 are various things you can say. 508 00:34:04,790 --> 00:34:08,076 Like the coordinates of this point here are a cos(theta), 509 00:34:08,076 --> 00:34:17,670 a-- Well, I'm sorry, it's not. 510 00:34:17,670 --> 00:34:20,780 That's an angle, but I want to call it theta_0. 511 00:34:20,780 --> 00:34:25,300 And, in general you know that the coordinates of this 512 00:34:25,300 --> 00:34:31,010 point are (a cos(theta), a sin(theta)). 513 00:34:31,010 --> 00:34:39,370 If the radius is a, then the angle here is theta. 514 00:34:39,370 --> 00:34:45,170 So x = a cos(theta), and y = a sin(theta), 515 00:34:45,170 --> 00:34:49,100 just from looking at the geometry of the circle. 516 00:34:49,100 --> 00:34:52,530 So let's make that substitution. y = a sin(theta). 517 00:34:56,600 --> 00:35:00,740 I'm using the picture to suggest that maybe making 518 00:35:00,740 --> 00:35:02,870 the substitution is a good thing to do. 519 00:35:02,870 --> 00:35:06,020 Let's follow along and see what happens. 520 00:35:06,020 --> 00:35:10,310 If that's true, what we're interested in is integrating, 521 00:35:10,310 --> 00:35:13,580 a^2 - y^2. 522 00:35:13,580 --> 00:35:18,900 Which is a^2-- We're interested in integrating the square root 523 00:35:18,900 --> 00:35:20,770 of a^2 - y^2. 524 00:35:20,770 --> 00:35:24,130 Which is the square root of a^2 minus this. 525 00:35:24,130 --> 00:35:27,100 a^2 sin^2(theta). 526 00:35:27,100 --> 00:35:35,870 And, well, that's equal to a cos theta. 527 00:35:35,870 --> 00:35:41,710 That's just sin^2 + cos^2 = 1, all over again. 528 00:35:41,710 --> 00:35:42,830 It's also x. 529 00:35:42,830 --> 00:35:44,560 This is x. 530 00:35:44,560 --> 00:35:46,070 And this was x. 531 00:35:46,070 --> 00:35:48,960 So there are a lot of different ways to think of this. 532 00:35:48,960 --> 00:35:51,650 But no matter how you say it, the thing we're trying 533 00:35:51,650 --> 00:35:59,270 to integrate, a^2 - y^2 is, under this substitution it is 534 00:35:59,270 --> 00:36:02,570 a cos(theta). 535 00:36:02,570 --> 00:36:06,700 So I'm interested in integrating the square root of (a^2 - y^2), 536 00:36:06,700 --> 00:36:09,260 dy. 537 00:36:09,260 --> 00:36:14,330 And I'm going to make this substitution y = a sin(theta). 538 00:36:17,330 --> 00:36:22,320 And so under that substitution, I've decided that the square 539 00:36:22,320 --> 00:36:26,170 root of a^2 - y^2 is a cos(theta). 540 00:36:31,170 --> 00:36:33,240 That's this. 541 00:36:33,240 --> 00:36:34,690 What about the dy? 542 00:36:34,690 --> 00:36:38,310 Well, I'd better compute the dy. 543 00:36:38,310 --> 00:36:40,920 So dy, just differentiating this expression, 544 00:36:40,920 --> 00:36:44,820 is a cos(theta) d theta. 545 00:36:44,820 --> 00:36:57,240 So let's put that in. dy = a cos(theta) d theta. 546 00:36:57,240 --> 00:36:58,310 OK. 547 00:36:58,310 --> 00:37:03,720 Making that trig substitution, y = a sin(theta), 548 00:37:03,720 --> 00:37:06,750 has replaced this integral that has a square root in it. 549 00:37:06,750 --> 00:37:08,640 And no trig functions. 550 00:37:08,640 --> 00:37:12,630 With an integral that involves no square roots and only trig 551 00:37:12,630 --> 00:37:15,140 functions. 552 00:37:15,140 --> 00:37:17,630 In fact, it's not too hard to integrate this now, because 553 00:37:17,630 --> 00:37:19,270 of the work that we've done. 554 00:37:19,270 --> 00:37:20,980 The a^2 comes out. 555 00:37:20,980 --> 00:37:22,630 This is cos^2(theta) d theta. 556 00:37:26,200 --> 00:37:28,580 And maybe we've done that example already today. 557 00:37:28,580 --> 00:37:35,920 I think we have. 558 00:37:35,920 --> 00:37:38,435 Maybe we can think it through, but maybe the easiest thing 559 00:37:38,435 --> 00:37:42,550 is to look back at notes and see what we got before. 560 00:37:42,550 --> 00:37:46,330 That was the first example in the hard case that I did. 561 00:37:46,330 --> 00:38:03,060 And what it came out to, I used x instead of theta at the time. 562 00:38:03,060 --> 00:38:05,570 So this is a good step forward. 563 00:38:05,570 --> 00:38:08,130 I started with this integral that I really 564 00:38:08,130 --> 00:38:12,440 didn't know how to do by any means that we've had so far. 565 00:38:12,440 --> 00:38:15,000 And I've replaced it by a trig integral 566 00:38:15,000 --> 00:38:16,250 that we do know how to do. 567 00:38:16,250 --> 00:38:19,160 And now I've done that trig integral. 568 00:38:19,160 --> 00:38:22,400 But we're still not quite done, because of the problem 569 00:38:22,400 --> 00:38:23,830 of back substituting. 570 00:38:23,830 --> 00:38:27,670 I'd like to go back and rewrite this in terms 571 00:38:27,670 --> 00:38:32,150 of the original variable, y. 572 00:38:32,150 --> 00:38:34,280 Or, I'd like to go back and rewrite it 573 00:38:34,280 --> 00:38:36,390 in terms of the original limits of integration 574 00:38:36,390 --> 00:38:40,050 that we had in the original problem. 575 00:38:40,050 --> 00:38:42,870 In doing that, it's going to be useful to rewrite 576 00:38:42,870 --> 00:38:47,150 this expression and get rid of the sin(2theta). 577 00:38:47,150 --> 00:38:51,840 After all, the original y was expressed in terms 578 00:38:51,840 --> 00:38:54,660 of sin(theta), not sin(2theta). 579 00:38:54,660 --> 00:39:04,220 So let me just do that here, and say that this, in turn, 580 00:39:04,220 --> 00:39:09,507 is equal to a^2 theta / 2 plus, well, 581 00:39:09,507 --> 00:39:11,090 sin(2theta) = 2 sin(theta) cos(theta). 582 00:39:18,080 --> 00:39:20,970 And so, when there's a 4 in the denominator, what I'll get 583 00:39:20,970 --> 00:39:25,390 is sin(theta) cos(theta) / 2. 584 00:39:32,580 --> 00:39:36,630 I did that because I'm getting closer to the original terms 585 00:39:36,630 --> 00:39:39,080 that the problem started with. 586 00:39:39,080 --> 00:39:40,130 Which was sin(theta). 587 00:40:05,480 --> 00:40:08,055 So let me write down the integral that we have now. 588 00:40:08,055 --> 00:40:14,680 The square root of a^2 - y^2, dy is, so far, 589 00:40:14,680 --> 00:40:26,810 what we know is a^2 (theta / 2 + sin(theta) cos(theta) / 2) + c. 590 00:40:26,810 --> 00:40:28,460 But I want to go back and rewrite this 591 00:40:28,460 --> 00:40:30,660 in terms of the original value. 592 00:40:30,660 --> 00:40:32,650 The original variable, y. 593 00:40:32,650 --> 00:40:37,430 Well, what is theta in terms of y? 594 00:40:37,430 --> 00:40:40,760 Let's see. y in terms of theta was given like this. 595 00:40:40,760 --> 00:40:44,120 So what is theta in terms of y? 596 00:40:44,120 --> 00:40:44,950 Ah. 597 00:40:44,950 --> 00:40:48,710 So here the fearsome arcsine rears its head, right? 598 00:40:48,710 --> 00:40:53,680 Theta is the angle so that y = a sin(theta). 599 00:40:53,680 --> 00:40:58,230 So that means that theta is the arcsine, or sine inverse, 600 00:40:58,230 --> 00:41:00,750 of y/a. 601 00:41:07,450 --> 00:41:12,630 So that's the first thing that shows up here. 602 00:41:12,630 --> 00:41:18,290 arcsin(y/a), all over 2. 603 00:41:18,290 --> 00:41:19,360 That's this term. 604 00:41:19,360 --> 00:41:24,530 Theta is arcsin(y/a) / 2. 605 00:41:24,530 --> 00:41:26,630 What about the other side, here? 606 00:41:26,630 --> 00:41:30,620 Well sine and cosine, we knew what they were in terms of y 607 00:41:30,620 --> 00:41:37,830 and in terms of x, if you like. 608 00:41:37,830 --> 00:41:40,300 Maybe I'll put the a^2 inside here. 609 00:41:40,300 --> 00:41:42,710 That makes it a little bit nicer. 610 00:41:42,710 --> 00:41:49,310 Plus, and the other term is a^2 sin(theta) cos(theta). 611 00:41:49,310 --> 00:41:52,610 So the a sin(theta) is just y. 612 00:41:52,610 --> 00:41:55,140 Maybe I'll write this (a sin(theta)) (a cos(theta)) 613 00:41:55,140 --> 00:41:57,550 / 2 + c. 614 00:42:02,330 --> 00:42:03,840 And so I get the same thing. 615 00:42:03,840 --> 00:42:06,204 And now here a sin(theta), that's y. 616 00:42:06,204 --> 00:42:07,370 And what's the a cos(theta)? 617 00:42:12,490 --> 00:42:16,440 It's x, or, if you like, it's the square root of a^2 - y^2. 618 00:42:22,010 --> 00:42:28,140 And so there I've rewritten everything, back 619 00:42:28,140 --> 00:42:31,250 in terms of the original variable, y. 620 00:42:31,250 --> 00:42:36,060 And there's an answer. 621 00:42:36,060 --> 00:42:41,070 So I've done this indefinite integration of a form-- 622 00:42:41,070 --> 00:42:44,810 of this quadratic, this square root of something which is 623 00:42:44,810 --> 00:42:46,940 a constant minus y^2. 624 00:42:46,940 --> 00:42:50,640 Whenever you see that, the thing to think of is trigonometry. 625 00:42:50,640 --> 00:42:54,310 That's going to play into the sin^2 + cos^2 identity. 626 00:42:54,310 --> 00:42:56,880 And the way to exploit it is to make the substitution 627 00:42:56,880 --> 00:43:01,100 y = a sin(theta). 628 00:43:01,100 --> 00:43:04,340 You could also make a substitution y = a cos(theta), 629 00:43:04,340 --> 00:43:05,540 if you wanted to. 630 00:43:05,540 --> 00:43:12,290 And the result would come out to exactly the same in the end. 631 00:43:12,290 --> 00:43:14,430 I'm still not quite done with the original problem 632 00:43:14,430 --> 00:43:23,990 that I had, because the original problem 633 00:43:23,990 --> 00:43:25,810 asked for a definite integral. 634 00:43:25,810 --> 00:43:33,000 So let's just go back and finish that as well. 635 00:43:33,000 --> 00:43:37,910 So the area was the integral from 0 636 00:43:37,910 --> 00:43:45,740 to b of this square root. 637 00:43:45,740 --> 00:43:48,220 So I just want to evaluate the right-hand side here. 638 00:43:48,220 --> 00:43:50,880 The answer that we came up with, this indefinite integral. 639 00:43:50,880 --> 00:43:53,374 I want to evaluate it at 0 and at b. 640 00:43:53,374 --> 00:43:54,040 Well, let's see. 641 00:43:54,040 --> 00:44:13,470 When I evaluate it at b, I get a^2 arcsin(b/a) / 2 plus y, 642 00:44:13,470 --> 00:44:18,900 which is b, times the square root of a^2 - b^2, 643 00:44:18,900 --> 00:44:23,050 putting y = b, divided by 2. 644 00:44:23,050 --> 00:44:26,300 So I've plugged in y = b into that formula, 645 00:44:26,300 --> 00:44:27,290 this is what I get. 646 00:44:27,290 --> 00:44:31,280 Then when I plug in y = 0, well the, sine of 0 is 0, 647 00:44:31,280 --> 00:44:34,040 so the arcsine of 0 is 0. 648 00:44:34,040 --> 00:44:35,720 So this term goes away. 649 00:44:35,720 --> 00:44:38,930 And when y = 0, this term is 0 also. 650 00:44:38,930 --> 00:44:43,210 And so I don't get any subtracted terms at all. 651 00:44:43,210 --> 00:44:45,600 So there's an expression for this. 652 00:44:45,600 --> 00:44:52,360 Notice that this arcsin(b/a), that's exactly this angle. 653 00:44:52,360 --> 00:45:00,550 arcsin(b/a), it's the angle that you get when y = b. 654 00:45:00,550 --> 00:45:06,760 So this theta is the arcsin(b/a). 655 00:45:09,990 --> 00:45:15,870 Put this over here. 656 00:45:15,870 --> 00:45:17,090 That is theta_0. 657 00:45:17,090 --> 00:45:21,560 That is the angle that the corner makes. 658 00:45:21,560 --> 00:45:28,460 So I could rewrite this as a a^2 theta_0 / 2 plus b times 659 00:45:28,460 --> 00:45:34,722 the square root of a^2 - b^2, over 2. 660 00:45:34,722 --> 00:45:36,430 Let's just think about this for a minute. 661 00:45:36,430 --> 00:45:40,320 I have these two terms in the sum, is that reasonable? 662 00:45:40,320 --> 00:45:44,080 The first term is a^2. 663 00:45:44,080 --> 00:45:50,370 That's the radius squared times this angle, times 1/2. 664 00:45:50,370 --> 00:45:54,280 Well, I think that is exactly the area of this sector. 665 00:45:54,280 --> 00:46:03,170 a^2 theta / 2 is the formula for the area of the sector. 666 00:46:03,170 --> 00:46:07,070 And this one, this is the vertical elevation. 667 00:46:07,070 --> 00:46:09,500 This is the horizontal. 668 00:46:09,500 --> 00:46:14,550 a^2 - b^2 is this distance. 669 00:46:14,550 --> 00:46:16,980 Square root of a^2 - b^2. 670 00:46:16,980 --> 00:46:20,420 So the right-hand term is b times the square root of a^2 - 671 00:46:20,420 --> 00:46:31,930 b^2 divided by 2, that's the area of that triangle. 672 00:46:31,930 --> 00:46:34,280 So using a little bit of geometry 673 00:46:34,280 --> 00:46:39,890 gives you the same answer as all of this elaborate calculus. 674 00:46:39,890 --> 00:46:41,610 Maybe that's enough cause for celebration 675 00:46:41,610 --> 00:46:43,620 for us to quit for today.