WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.750 Your support will help MIT OpenCourseWare 00:00:05.750 --> 00:00:09.460 continue to offer high quality educational resources for free. 00:00:09.460 --> 00:00:12.550 To make a donation, or to view additional materials 00:00:12.550 --> 00:00:16.150 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.150 --> 00:00:21.840 at ocw.mit.edu. 00:00:21.840 --> 00:00:25.190 PROFESSOR: Well, because our subject today 00:00:25.190 --> 00:00:28.280 is trig integrals and substitutions, 00:00:28.280 --> 00:00:33.640 Professor Jerison called in his substitute teacher for today. 00:00:33.640 --> 00:00:46.150 That's me. 00:00:46.150 --> 00:00:49.860 Professor Miller. 00:00:49.860 --> 00:00:52.790 And I'm going to try to tell you about trig substitutions 00:00:52.790 --> 00:00:54.610 and trig integrals. 00:00:54.610 --> 00:00:59.650 And I'll be here tomorrow to do more of the same, as well. 00:00:59.650 --> 00:01:02.320 So, this is about trigonometry, and maybe first thing I'll do 00:01:02.320 --> 00:01:24.180 is remind you of some basic things about trigonometry. 00:01:24.180 --> 00:01:27.110 So, if I have a circle, trigonometry 00:01:27.110 --> 00:01:29.730 is all based on the circle of radius 1 00:01:29.730 --> 00:01:32.000 and centered at the origin. 00:01:32.000 --> 00:01:34.870 And so if this is an angle of theta, up from the x-axis, 00:01:34.870 --> 00:01:36.350 then the coordinates of this point 00:01:36.350 --> 00:01:39.550 are cosine theta and sine theta. 00:01:39.550 --> 00:01:42.090 And so that leads right away to some trig identities, 00:01:42.090 --> 00:01:43.330 which you know very well. 00:01:43.330 --> 00:01:46.820 But I'm going to put them up here because we'll use them 00:01:46.820 --> 00:01:51.220 over and over again today. 00:01:51.220 --> 00:01:55.220 Remember the convention sin^2 theta secretly means (sin 00:01:55.220 --> 00:01:57.530 theta)^2. 00:01:57.530 --> 00:01:59.040 It would be more sensible to write 00:01:59.040 --> 00:02:01.600 a parenthesis around the sine of theta 00:02:01.600 --> 00:02:03.350 and then say you square that. 00:02:03.350 --> 00:02:06.380 But everybody in the world puts the 2 up there over the sin, 00:02:06.380 --> 00:02:09.080 and so I'll do that too. 00:02:09.080 --> 00:02:11.995 So that follows just because the circle has radius 1. 00:02:11.995 --> 00:02:13.870 But then there are some other identities too, 00:02:13.870 --> 00:02:15.410 which I think you remember. 00:02:15.410 --> 00:02:19.170 I'll write them down here. cos(2theta), 00:02:19.170 --> 00:02:22.490 there's this double angle formula that says cos(2theta) = 00:02:22.490 --> 00:02:23.840 cos^2(theta) - sin^2(theta). 00:02:29.090 --> 00:02:31.000 And there's also the double angle formula 00:02:31.000 --> 00:02:34.620 for the sin(2theta). 00:02:34.620 --> 00:02:38.480 Remember what that says? 00:02:38.480 --> 00:02:41.300 2 sin(theta) cos(theta). 00:02:46.437 --> 00:02:48.020 I'm going to use these trig identities 00:02:48.020 --> 00:02:50.370 and I'm going to use them in a slightly different way. 00:02:50.370 --> 00:02:53.090 And so I'd like to pay a little more attention to this one 00:02:53.090 --> 00:02:57.120 and get a different way of writing this one out. 00:02:57.120 --> 00:03:06.930 So this is actually the half angle formula. 00:03:06.930 --> 00:03:14.170 And that says, I'm going to try to express the cos(theta) 00:03:14.170 --> 00:03:16.400 in terms of the cos(2theta). 00:03:16.400 --> 00:03:19.590 So if I know the cos(2theta), I want 00:03:19.590 --> 00:03:23.320 to try to express the cos theta in terms of it. 00:03:23.320 --> 00:03:30.130 Well, I'll start with a cos(2theta) and play with that. 00:03:30.130 --> 00:03:30.630 OK. 00:03:30.630 --> 00:03:36.600 Well, we know what this is, it's cos^2(theta) - sin^2(theta). 00:03:36.600 --> 00:03:39.520 But we also know what the sin^2(theta) is in terms 00:03:39.520 --> 00:03:40.250 of the cosine. 00:03:40.250 --> 00:03:44.630 So I can eliminate the sin^2 from this picture. 00:03:44.630 --> 00:03:48.400 So this is equal to cos^2(theta) minus the quantity 1 - 00:03:48.400 --> 00:03:50.650 cos^2(theta). 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 00:03:57.510 --> 00:03:59.940 cos^2(theta) - 1. 00:03:59.940 --> 00:04:02.240 There's this cos^2, which gets a plus sign. 00:04:02.240 --> 00:04:04.540 Because of these two minus signs. 00:04:04.540 --> 00:04:06.290 And there's the one that was there before, 00:04:06.290 --> 00:04:10.270 so altogether there are two of them. 00:04:10.270 --> 00:04:12.380 I want to isolate what cosine is. 00:04:12.380 --> 00:04:16.070 Or rather, what cos^2 is. 00:04:16.070 --> 00:04:17.680 So let's solve for that. 00:04:17.680 --> 00:04:20.120 So I'll put the 1 on the other side. 00:04:20.120 --> 00:04:24.120 And I get 1 + cos(2theta). 00:04:24.120 --> 00:04:27.910 And then, I want to divide by this 2, and so that puts a 2 00:04:27.910 --> 00:04:30.280 in this denominator here. 00:04:30.280 --> 00:04:33.280 So some people call that the half angle formula. 00:04:33.280 --> 00:04:36.730 What it really is for us is it's a way of eliminating powers 00:04:36.730 --> 00:04:38.800 from sines and cosines. 00:04:38.800 --> 00:04:41.290 I've gotten rid of this square at the expense 00:04:41.290 --> 00:04:43.720 of putting in a 2theta here. 00:04:43.720 --> 00:04:45.260 We'll use that. 00:04:45.260 --> 00:04:51.022 And, similarly, same calculation shows that sin^2(theta) = (1 - 00:04:51.022 --> 00:04:51.730 cos(2theta)) / 2. 00:04:55.850 --> 00:05:01.480 Same cosine, in that formula also, but it has a minus sign. 00:05:01.480 --> 00:05:03.640 For the sin^2. 00:05:03.640 --> 00:05:07.860 OK. so that's my little review of trig identities 00:05:07.860 --> 00:05:14.620 that we'll make use of as this lecture goes on. 00:05:14.620 --> 00:05:17.720 I want to talk about trig identity-- trig integrals, 00:05:17.720 --> 00:05:23.500 and you know some trig integrals, I'm sure, already. 00:05:23.500 --> 00:05:26.740 Like, well, let me write the differential form first. 00:05:26.740 --> 00:05:30.400 You know that d sin theta, or maybe I'll 00:05:30.400 --> 00:05:35.620 say d sin x, is, let's see, that's 00:05:35.620 --> 00:05:39.030 the derivative of sin x times dx, right. 00:05:39.030 --> 00:05:46.120 The derivative of sin x is cos x, dx. 00:05:46.120 --> 00:05:50.080 And so if I integrate both sides here, the integral form of this 00:05:50.080 --> 00:05:55.390 is the integral of cos x dx. 00:05:55.390 --> 00:05:59.850 Is sin x plus a constant. 00:05:59.850 --> 00:06:07.410 And in the same way, d cos x = -sin x dx. 00:06:07.410 --> 00:06:10.320 Right, the derivative of the cosine is minus sine. 00:06:10.320 --> 00:06:15.330 And when I integrate that, I find the integral of sin x dx 00:06:15.330 --> 00:06:21.140 is -cos x + c. 00:06:21.140 --> 00:06:22.820 So that's our starting point. 00:06:22.820 --> 00:06:27.510 And the game today, for the first half of the lecture, 00:06:27.510 --> 00:06:33.550 is to use that basic-- just those basic integration 00:06:33.550 --> 00:06:37.710 formulas, together with clever use of trig identities 00:06:37.710 --> 00:06:41.260 in order to compute more complicated formulas involving 00:06:41.260 --> 00:06:42.820 trig functions. 00:06:42.820 --> 00:06:47.040 So the first thing, the first topic, 00:06:47.040 --> 00:06:52.250 is to think about integrals of the form sin^n (x) cos^n (x) 00:06:52.250 --> 00:06:52.750 dx. 00:06:56.190 --> 00:07:04.070 Where here I have in mind m and n are non-negative integers. 00:07:04.070 --> 00:07:05.950 So let's try to integrate these. 00:07:05.950 --> 00:07:09.640 I'll show you some applications of these pretty soon. 00:07:09.640 --> 00:07:11.250 Looking down the road a little bit, 00:07:11.250 --> 00:07:14.050 integrals like this show up in Fourier series 00:07:14.050 --> 00:07:16.190 and many other subjects in mathematics. 00:07:16.190 --> 00:07:20.000 It turns out they're quite important to be able to do. 00:07:20.000 --> 00:07:23.860 So that's why we're doing them now. 00:07:23.860 --> 00:07:29.690 Well, so there are two cases to think about here. 00:07:29.690 --> 00:07:32.070 When you're integrating things like this. 00:07:32.070 --> 00:07:35.890 There's the easy case, and let's do that one first. 00:07:35.890 --> 00:07:49.495 The easy case is when at least one exponent is odd. 00:07:49.495 --> 00:07:50.370 That's the easy case. 00:07:50.370 --> 00:07:55.870 So, for example, suppose that I wanted to integrate, 00:07:55.870 --> 00:08:02.560 well, let's take the case m = 1. 00:08:02.560 --> 00:08:09.530 So I'm integrating sin^n (x) cos x dx. 00:08:09.530 --> 00:08:17.050 I'm taking-- Oh, I could do that one. 00:08:17.050 --> 00:08:23.600 Let's see if that's what I want to take. 00:08:23.600 --> 00:08:27.540 Yeah. 00:08:27.540 --> 00:08:30.590 My confusion is that I meant to have this a different power. 00:08:30.590 --> 00:08:34.750 You were thinking that. 00:08:34.750 --> 00:08:36.670 So let's do this case when m = 1. 00:08:36.670 --> 00:08:38.680 So the integral I'm trying to do is any power 00:08:38.680 --> 00:08:41.450 of the sine times the cosine. 00:08:41.450 --> 00:08:44.520 Well, here's the trick. 00:08:44.520 --> 00:08:48.920 Recognize, use this formula up at the top 00:08:48.920 --> 00:08:53.570 there to see cos x dx as something that we already 00:08:53.570 --> 00:08:55.670 have on the blackboard. 00:08:55.670 --> 00:08:59.570 So, the way to exploit that is to make a substitution. 00:08:59.570 --> 00:09:08.320 And substitution is going to be u = sin x. 00:09:08.320 --> 00:09:09.610 And here's why. 00:09:09.610 --> 00:09:12.820 Then this integral that I'm trying to do is the integral 00:09:12.820 --> 00:09:18.760 of u^n, that's already a simplification. 00:09:18.760 --> 00:09:22.160 And then there's that cos x dx. 00:09:22.160 --> 00:09:25.120 When you make a substitution, you've got to go all the way 00:09:25.120 --> 00:09:27.970 and replace everything in the expression 00:09:27.970 --> 00:09:33.000 by things involving this new variable that I've introduced. 00:09:33.000 --> 00:09:35.390 So I'd better get rid of the cos x dx 00:09:35.390 --> 00:09:39.140 and rewrite it in terms of du or in terms of u. 00:09:39.140 --> 00:09:45.290 And I can do that because du, according to that formula, 00:09:45.290 --> 00:09:50.600 is cos x dx. 00:09:50.600 --> 00:09:53.810 Let me put a box around that. 00:09:53.810 --> 00:09:55.800 That's our substitution. 00:09:55.800 --> 00:09:57.500 When you make a substitution, you also 00:09:57.500 --> 00:10:00.740 want to compute the differential of the variable 00:10:00.740 --> 00:10:03.100 that you substitute in. 00:10:03.100 --> 00:10:08.800 So the cos x dx that appears here is just, exactly, du. 00:10:08.800 --> 00:10:11.500 And I've replaced this trig integral with something 00:10:11.500 --> 00:10:13.330 that doesn't involve trig functions at all. 00:10:13.330 --> 00:10:14.240 This is a lot easier. 00:10:14.240 --> 00:10:17.420 We can just plug into what we know here. 00:10:17.420 --> 00:10:23.640 This is u^(n+1) / (n+1) plus a constant, 00:10:23.640 --> 00:10:26.620 and I've done the integral. 00:10:26.620 --> 00:10:29.580 But I'm not quite done with the problem yet. 00:10:29.580 --> 00:10:34.310 Because to be nice to your reader and to yourself, 00:10:34.310 --> 00:10:36.560 you should go back at this point, probably, 00:10:36.560 --> 00:10:40.399 go back and get rid of this new variable that you introduced. 00:10:40.399 --> 00:10:42.440 You're the one who introduced this variable, you. 00:10:42.440 --> 00:10:45.950 Nobody except you, really, knows what it is. 00:10:45.950 --> 00:10:48.080 But the rest of the world knows what 00:10:48.080 --> 00:10:51.360 they asked for the first place that involved x. 00:10:51.360 --> 00:10:53.460 So I have to go back and get rid of this. 00:10:53.460 --> 00:10:57.830 And that's not hard to do in this case, because u = sin x. 00:10:57.830 --> 00:11:04.390 And so I make this back substitution. 00:11:04.390 --> 00:11:05.820 And that's what you get. 00:11:05.820 --> 00:11:11.040 So there's the answer. 00:11:11.040 --> 00:11:15.930 OK, so the game was, I use this odd power of the cosine here, 00:11:15.930 --> 00:11:19.180 and I could see it appearing as the differential of the sine. 00:11:19.180 --> 00:11:22.470 So that's what made this substitution work. 00:11:22.470 --> 00:11:25.060 Let's do another example to see how that works out 00:11:25.060 --> 00:11:36.490 in a slightly different case. 00:11:36.490 --> 00:11:48.420 So here's another example. 00:11:48.420 --> 00:11:50.050 Now I do have an odd power. 00:11:50.050 --> 00:11:53.800 One of the exponents is odd, so I'm in the easy case. 00:11:53.800 --> 00:11:56.310 But it's not 1. 00:11:56.310 --> 00:12:05.590 The game now is to use this trig identity 00:12:05.590 --> 00:12:10.500 to get rid of the largest even power that you can, 00:12:10.500 --> 00:12:13.870 from this odd power here. 00:12:13.870 --> 00:12:24.710 So use sin^2 x = 1 - cos^2 x, to eliminate a lot of powers from 00:12:24.710 --> 00:12:26.220 that odd power. 00:12:26.220 --> 00:12:28.370 Watch what happens. 00:12:28.370 --> 00:12:31.490 So this is not really a substitution or anything, 00:12:31.490 --> 00:12:34.320 this is just a trig identity. 00:12:34.320 --> 00:12:38.130 This sine cubed is sine squared times the sine. 00:12:38.130 --> 00:12:41.360 And the sine squared is 1 - cos^2 x. 00:12:41.360 --> 00:12:43.770 And then I have the remaining sin x. 00:12:43.770 --> 00:12:48.970 And then I have cos^2 x dx. 00:12:48.970 --> 00:12:53.580 So let me rewrite that a little bit to see how this works out. 00:12:53.580 --> 00:12:59.490 This is the integral of cos^2 x minus, 00:12:59.490 --> 00:13:01.240 and then there's the product of these two. 00:13:01.240 --> 00:13:09.370 That's cos^4 x times sin x dx. 00:13:09.370 --> 00:13:12.200 So now I'm really exactly in the situation 00:13:12.200 --> 00:13:13.700 that I was in over here. 00:13:13.700 --> 00:13:17.570 I've got a single power of a sine or cosine. 00:13:17.570 --> 00:13:20.060 It happens that it's a sine here. 00:13:20.060 --> 00:13:21.960 But that's not going to cause any trouble, 00:13:21.960 --> 00:13:26.290 we can go ahead and play the same game that I did there. 00:13:26.290 --> 00:13:28.860 So, so far I've just been using trig identities. 00:13:28.860 --> 00:13:41.770 But now I'll use a trig substitution. 00:13:41.770 --> 00:13:45.335 And I think I want to write these as powers of a variable. 00:13:45.335 --> 00:13:47.960 And then this is going to be the differential of that variable. 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. 00:13:58.190 --> 00:14:04.950 There's the substitution. 00:14:04.950 --> 00:14:09.550 So when I make that substitution, what do we get. 00:14:09.550 --> 00:14:11.710 Cosine squared becomes u^2. 00:14:15.470 --> 00:14:23.150 Cosine to the 4th becomes u^4, and sin x dx becomes not quite 00:14:23.150 --> 00:14:27.570 du, watch for the signum, watch for this minus sign here. 00:14:27.570 --> 00:14:32.100 It becomes -du. 00:14:32.100 --> 00:14:32.820 But that's OK. 00:14:32.820 --> 00:14:34.720 The minus sign comes outside. 00:14:34.720 --> 00:14:36.860 And I can integrate both of these powers, 00:14:36.860 --> 00:14:43.540 so I get -u^3 / 3. 00:14:43.540 --> 00:14:48.950 And then this 4th power gives me a 5th power, when I integrate. 00:14:48.950 --> 00:14:53.390 And don't forget the constant. 00:14:53.390 --> 00:14:55.100 Am I done? 00:14:55.100 --> 00:14:55.850 Not quite done. 00:14:55.850 --> 00:14:57.690 I have to back substitute and get rid 00:14:57.690 --> 00:15:00.830 of my choice of variable, u, and replace it with yours. 00:15:00.830 --> 00:15:01.330 Questions? 00:15:01.330 --> 00:15:06.049 STUDENT: [INAUDIBLE] 00:15:06.049 --> 00:15:07.340 PROFESSOR: There should indeed. 00:15:07.340 --> 00:15:10.190 I forgot this minus sign when I came down here. 00:15:10.190 --> 00:15:12.690 So these two gang up to give me a plus. 00:15:12.690 --> 00:15:14.960 Was that what the other question was about, too? 00:15:14.960 --> 00:15:16.720 Thanks. 00:15:16.720 --> 00:15:18.030 So let's back substitute. 00:15:18.030 --> 00:15:23.900 And I'm going to put that over here. 00:15:23.900 --> 00:15:27.920 And the result is, well, I just replace the u by cosine of x. 00:15:27.920 --> 00:15:38.160 So this is - -cos^3(x) / 3 plus, thank you, cos^5(x) / 5 + c. 00:15:38.160 --> 00:15:44.670 And there's the answer. 00:15:44.670 --> 00:15:47.210 By the way, you can remember one of the nice things 00:15:47.210 --> 00:15:49.160 about doing an integral is it's fairly 00:15:49.160 --> 00:15:51.280 easy to check your answer. 00:15:51.280 --> 00:15:53.680 You can always differentiate the thing you get, 00:15:53.680 --> 00:15:56.890 and see whether you get the right thing when you go back. 00:15:56.890 --> 00:15:59.280 It's not too hard to use the power 00:15:59.280 --> 00:16:02.500 rules and the differentiation rule 00:16:02.500 --> 00:16:06.080 for the cosine to get back to this if you 00:16:06.080 --> 00:16:09.350 want to check the work. 00:16:09.350 --> 00:16:12.830 Let's do one more example, just to handle 00:16:12.830 --> 00:16:15.980 an example of this easy case, which you 00:16:15.980 --> 00:16:18.490 might have thought of at first. 00:16:18.490 --> 00:16:22.720 Suppose I just want to integrate a cube. sin^3 x. 00:16:29.320 --> 00:16:32.040 No cosine in sight. 00:16:32.040 --> 00:16:35.780 But I do have an odd power of a trig 00:16:35.780 --> 00:16:37.180 function, of a sine or cosine. 00:16:37.180 --> 00:16:39.020 So I'm in the easy case. 00:16:39.020 --> 00:16:44.740 And the procedure that I was suggesting says I want to take 00:16:44.740 --> 00:16:48.700 out the largest even power that I can, from the sin^3. 00:16:48.700 --> 00:16:52.870 So I'll take that out, that's a sin^2, and write it as 1 - 00:16:52.870 --> 00:16:53.370 cos^2. 00:16:57.020 --> 00:16:58.380 Well, now I'm very happy. 00:16:58.380 --> 00:17:00.470 Because it's just like the situation 00:17:00.470 --> 00:17:06.017 we had somewhere on the board here. 00:17:06.017 --> 00:17:07.850 It's just like the situation we had up here. 00:17:07.850 --> 00:17:11.660 I've got a power of a cosine times sin x dx. 00:17:11.660 --> 00:17:16.310 So exactly the same substitution steps in. 00:17:16.310 --> 00:17:19.070 You get, and maybe you can see what 00:17:19.070 --> 00:17:20.860 happens without doing the work. 00:17:20.860 --> 00:17:22.630 Shall I do the work here? 00:17:22.630 --> 00:17:24.450 I make the same substitution. 00:17:24.450 --> 00:17:30.680 And so this is (1 - u (1 - u^2) times -du. 00:17:33.540 --> 00:17:40.050 Which is u - u^3 / 3. 00:17:40.050 --> 00:17:42.050 But then I want to put this minus sign in place, 00:17:42.050 --> 00:17:47.700 and so that gives me -u + u^3 / 3 plus a constant. 00:17:47.700 --> 00:17:58.090 And then I back substitute and get cos x + cos^3 x / 3. 00:17:58.090 --> 00:17:59.350 So this is the easy case. 00:17:59.350 --> 00:18:02.190 If you have some odd power to play with, 00:18:02.190 --> 00:18:07.990 then you can make use of it and it's pretty straightforward. 00:18:07.990 --> 00:18:10.680 OK the harder case is when you don't have an odd power. 00:18:10.680 --> 00:18:11.610 So what's the program? 00:18:11.610 --> 00:18:13.770 I'm going to do the harder case, and then I'm 00:18:13.770 --> 00:18:19.340 going to show you an example of how to integrate square roots. 00:18:19.340 --> 00:18:26.260 And do an application, using these ideas from trigonometry. 00:18:26.260 --> 00:18:30.250 So I want to keep this blackboard. 00:18:30.250 --> 00:18:34.470 Maybe I'll come back and start here again. 00:18:34.470 --> 00:18:55.240 So the harder case is when they're only even exponents. 00:18:55.240 --> 00:18:58.190 I'm still trying to integrate the same form. 00:18:58.190 --> 00:19:00.370 But now all the exponents are even. 00:19:00.370 --> 00:19:03.510 So we have to do some game. 00:19:03.510 --> 00:19:10.400 And here the game is use the half angle formula. 00:19:10.400 --> 00:19:16.455 Which I just erased, very sadly, on the board here. 00:19:16.455 --> 00:19:17.830 Maybe I'll rewrite them over here 00:19:17.830 --> 00:19:23.550 so we have them on the board. 00:19:23.550 --> 00:19:44.270 I think I remember what they were. 00:19:44.270 --> 00:19:46.300 So the game is I'm going to use that half angle 00:19:46.300 --> 00:19:50.180 formula to start getting rid of those even powers. 00:19:50.180 --> 00:19:54.510 Half angle formula written like this, exactly, talks about-- it 00:19:54.510 --> 00:19:57.790 rewrites even powers of sines and cosines. 00:19:57.790 --> 00:20:00.950 So let's see how that works out in an example. 00:20:00.950 --> 00:20:08.880 How about just the cosine squared for a start. 00:20:08.880 --> 00:20:09.700 What to do? 00:20:09.700 --> 00:20:11.970 I can't pull anything out. 00:20:11.970 --> 00:20:15.180 I could rewrite this as 1 - sin^2, 00:20:15.180 --> 00:20:17.290 but then I'd be faced with integrating the sin^2, 00:20:17.290 --> 00:20:19.480 which is exactly as hard. 00:20:19.480 --> 00:20:23.100 So instead, let's use this formula here. 00:20:23.100 --> 00:20:29.180 This is really the same as (1+cos(2theta)) / 2. 00:20:29.180 --> 00:20:32.700 And now, this is easy. 00:20:32.700 --> 00:20:34.340 It's got two parts to it. 00:20:34.340 --> 00:20:38.550 Integrating one half gives me theta over-- Oh. 00:20:38.550 --> 00:20:42.440 Miraculously, the x turned into a theta. 00:20:42.440 --> 00:20:44.330 Let's put it back as x. 00:20:44.330 --> 00:20:47.440 I get x/2 by integrating 1/2. 00:20:47.440 --> 00:20:50.110 So, notice that something non-trigonometric occurs here 00:20:50.110 --> 00:20:52.710 when I do these even integrals. 00:20:52.710 --> 00:20:54.800 x/2 appears. 00:20:54.800 --> 00:20:57.880 And then the other one, OK, so this takes a little thought. 00:20:57.880 --> 00:21:01.870 The integral of the cosine is the sine, 00:21:01.870 --> 00:21:05.520 or is it minus the sine. 00:21:05.520 --> 00:21:11.390 Negative sine. 00:21:11.390 --> 00:21:12.634 Shall we take a vote? 00:21:12.634 --> 00:21:13.550 I think it's positive. 00:21:13.550 --> 00:21:18.310 And so you get sin(2x), but is that right? 00:21:18.310 --> 00:21:19.420 Over 2. 00:21:19.420 --> 00:21:24.150 If I differentiate the sin(2x), this 2 comes out. 00:21:24.150 --> 00:21:25.950 And would give me an extra 2 here. 00:21:25.950 --> 00:21:29.580 So there's an extra 2 that I have to put in here 00:21:29.580 --> 00:21:34.670 when I integrate it. 00:21:34.670 --> 00:21:37.230 And there's the answer. 00:21:37.230 --> 00:21:39.480 This is not a substitution. 00:21:39.480 --> 00:21:41.550 I just played with trig identities here. 00:21:41.550 --> 00:21:45.020 And then did a simple trig integral, 00:21:45.020 --> 00:21:46.900 getting your help to get the sign right. 00:21:46.900 --> 00:21:49.310 And thinking about what this 2 is going to do. 00:21:49.310 --> 00:21:52.990 It produces a 2 in the denominator. 00:21:52.990 --> 00:21:59.270 But it's not applying any complicated thing. 00:21:59.270 --> 00:22:03.100 It's just using this identity. 00:22:03.100 --> 00:22:05.580 Let's do another example that's a little bit harder. 00:22:05.580 --> 00:22:07.690 This time, sin^2 times cos^2. 00:22:35.940 --> 00:22:37.920 Again, no odd powers. 00:22:37.920 --> 00:22:40.730 I've got to work a little bit harder. 00:22:40.730 --> 00:22:42.930 And what I'm going to do is apply those identities 00:22:42.930 --> 00:22:44.210 up there. 00:22:44.210 --> 00:22:47.670 Now, what I recommend doing in this situation 00:22:47.670 --> 00:22:51.370 is going over to the side somewhere. 00:22:51.370 --> 00:22:55.610 And do some side work. 00:22:55.610 --> 00:22:58.680 Because it's all just playing with trig functions. 00:22:58.680 --> 00:23:06.280 It's not actually doing any integrals for a while. 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 00:23:11.815 --> 00:23:14.180 to use those identities and so let's do that. 00:23:14.180 --> 00:23:16.210 So the sine is (1 - cos(2x)) / 2. 00:23:20.350 --> 00:23:22.170 And the cosine is (1 + cos(2x)) / 2. 00:23:27.890 --> 00:23:29.870 So I just substitute them in. 00:23:29.870 --> 00:23:31.560 And now I can multiply that out. 00:23:31.560 --> 00:23:38.030 And what I have is a difference times a sum. 00:23:38.030 --> 00:23:40.771 So you know a formula for that. 00:23:40.771 --> 00:23:43.270 Taking the product of these two things, well there'll be a 4 00:23:43.270 --> 00:23:44.430 in the denominator. 00:23:44.430 --> 00:23:46.490 And then in the numerator, I get the square 00:23:46.490 --> 00:23:49.650 of this minus the square of this. 00:23:49.650 --> 00:23:59.840 (a-b)(a+b) = a^2 - b^2. = - So I get that. 00:23:59.840 --> 00:24:02.450 Well, I'm a little bit happier, because at least I 00:24:02.450 --> 00:24:03.710 don't have 4. 00:24:03.710 --> 00:24:07.430 I don't have 2 different squares. 00:24:07.430 --> 00:24:09.940 I still have a square, and want to integrate this. 00:24:09.940 --> 00:24:12.170 I'm still not in the easy case. 00:24:12.170 --> 00:24:16.970 I got myself back to an easier hard case. 00:24:16.970 --> 00:24:18.670 But we do know what to do about this. 00:24:18.670 --> 00:24:21.140 Because I just did it up there. 00:24:21.140 --> 00:24:24.430 And I could play into this formula that we got. 00:24:24.430 --> 00:24:29.010 But I think it's just as easy to continue to calculate here. 00:24:29.010 --> 00:24:32.880 Use the half angle formula again for this, 00:24:32.880 --> 00:24:34.900 and continue on your way. 00:24:34.900 --> 00:24:37.720 So I get a 1/4 from this bit. 00:24:37.720 --> 00:24:43.260 And then minus 1/4 of cos^2(2x). 00:24:45.780 --> 00:24:51.630 And when I plug in 2x in for theta, there in the top board, 00:24:51.630 --> 00:24:59.650 I'm going to get a 4x on the right-hand side. 00:24:59.650 --> 00:25:02.130 So it comes out like that. 00:25:02.130 --> 00:25:04.600 And I guess I could simplify that a little bit more. 00:25:04.600 --> 00:25:05.790 This is a 1/4. 00:25:05.790 --> 00:25:07.840 Oh, but then there's a 2 here. 00:25:07.840 --> 00:25:10.630 It's half that. 00:25:10.630 --> 00:25:12.200 So then I can simplify a little more. 00:25:12.200 --> 00:25:16.070 It's 1/4 - 1/8, which is 1/8. 00:25:16.070 --> 00:25:19.630 And then I have 1/8 cos(4x). 00:25:25.510 --> 00:25:27.560 OK, that's my side work. 00:25:27.560 --> 00:25:30.590 I just did some trig identities over here. 00:25:30.590 --> 00:25:32.840 And rewrote sine squared times cosine 00:25:32.840 --> 00:25:35.930 squared as something which involves just no powers 00:25:35.930 --> 00:25:37.940 of trig, just cosine by itself. 00:25:37.940 --> 00:25:41.330 And a constant. 00:25:41.330 --> 00:25:45.090 So I can take that and substitute it in here. 00:25:45.090 --> 00:25:48.810 And now the integration is pretty easy. 00:25:48.810 --> 00:25:57.810 1/8, cos(4x) / 8, dx, which is, OK 00:25:57.810 --> 00:26:01.340 the 1/8 is going to give me x/8. 00:26:01.340 --> 00:26:06.314 The integral or cosine is plus or minus the sine. 00:26:06.314 --> 00:26:08.230 The derivative of the sine is plus the cosine. 00:26:08.230 --> 00:26:11.440 So it's going to be plus the-- Only there's a minus here. 00:26:11.440 --> 00:26:17.890 So it's going to be the sine-- minus sin(4x) / 8, 00:26:17.890 --> 00:26:20.797 but then I have an additional factor in the denominator. 00:26:20.797 --> 00:26:21.880 And what's it going to be? 00:26:21.880 --> 00:26:28.830 I have to put a 4 there. 00:26:28.830 --> 00:26:32.730 So we've done that calculation, too. 00:26:32.730 --> 00:26:38.250 So any of these-- If you keep doing this kind of process, 00:26:38.250 --> 00:26:45.530 these two kinds of procedures, you 00:26:45.530 --> 00:26:48.840 can now integrate any expression that 00:26:48.840 --> 00:26:52.040 has a power of a sine times a power of a cosine in it, 00:26:52.040 --> 00:26:56.860 by using these ideas. 00:26:56.860 --> 00:27:01.730 Now, let's see. 00:27:01.730 --> 00:27:16.210 Oh, let me give you an alternate method for this last one here. 00:27:16.210 --> 00:27:26.460 I know what I'll do. 00:27:26.460 --> 00:27:28.850 Let me give an alternate method for doing, really 00:27:28.850 --> 00:27:31.670 doing the side work over there. 00:27:31.670 --> 00:27:35.920 I'm trying to deal with sin^2 times cos^2. 00:27:35.920 --> 00:27:50.080 Well that's the square of sin x cos x. 00:27:50.080 --> 00:27:54.240 And sin x cos x shows up right here. 00:27:54.240 --> 00:27:55.900 In another trig identity. 00:27:55.900 --> 00:27:58.410 So we can make use of that, too. 00:27:58.410 --> 00:28:01.330 That reduces the number of factors of sines and cosines 00:28:01.330 --> 00:28:01.930 by 1. 00:28:01.930 --> 00:28:04.190 So it's going in the right direction. 00:28:04.190 --> 00:28:11.270 This is equal to 1/2 sin(2x), squared. 00:28:11.270 --> 00:28:17.730 Sine times cosine is 1/2-- Say this right. 00:28:17.730 --> 00:28:21.280 It's sin(2x) / 2, and then I want to square that. 00:28:21.280 --> 00:28:31.030 So what I get is sin^2(2x) / 4. 00:28:31.030 --> 00:28:33.930 Which is, well, I'm not too happy yet, 00:28:33.930 --> 00:28:35.639 because I still have an even power. 00:28:35.639 --> 00:28:37.930 Remember I'm trying to integrate this thing in the end, 00:28:37.930 --> 00:28:39.060 even powers are bad. 00:28:39.060 --> 00:28:40.910 I try to get rid of them. 00:28:40.910 --> 00:28:46.360 By using that formula, the half angle formula. 00:28:46.360 --> 00:28:48.950 So I can apply that to sin x here again. 00:28:48.950 --> 00:28:52.600 I get 1/4 of (1 - cos(4x)) / 2. 00:28:57.480 --> 00:28:59.730 That's what the half angle formula says for sin^2(2x). 00:29:02.330 --> 00:29:04.790 And that's exactly the same as the expression 00:29:04.790 --> 00:29:08.660 that I got up here, as well. 00:29:08.660 --> 00:29:11.710 It's the same expression that I have there. 00:29:11.710 --> 00:29:16.680 So it's the same expression as I have here. 00:29:16.680 --> 00:29:20.000 So this is just an alternate way to play this game of using 00:29:20.000 --> 00:29:24.940 the half angle formula. 00:29:24.940 --> 00:29:27.790 OK, let's do a little application of these things 00:29:27.790 --> 00:29:46.060 and change the topic a little bit. 00:29:46.060 --> 00:29:48.210 So here's the problem. 00:29:48.210 --> 00:29:56.900 So this is an application and example 00:29:56.900 --> 00:30:07.570 of a real trig substitution. 00:30:07.570 --> 00:30:22.080 So here's the problem I want to look at. 00:30:22.080 --> 00:30:26.210 OK, so I have a circle whose radius is a. 00:30:26.210 --> 00:30:31.710 And I cut out from it a sort of tab, here. 00:30:31.710 --> 00:30:36.330 This tab here. 00:30:36.330 --> 00:30:38.590 And the height of this thing is b. 00:30:38.590 --> 00:30:42.110 So this length is a number b. 00:30:42.110 --> 00:30:47.219 And what I want to do is compute the area of that little tab. 00:30:47.219 --> 00:30:48.010 That's the problem. 00:30:48.010 --> 00:30:50.360 So there's an arc over here. 00:30:50.360 --> 00:30:55.060 And I want to find the area of this, for a and b, 00:30:55.060 --> 00:30:57.980 in terms of a and b. 00:30:57.980 --> 00:31:06.730 So the area, well, I guess one way 00:31:06.730 --> 00:31:12.370 to compute the area would be to take the integral of y dx. 00:31:12.370 --> 00:31:15.980 You've seen the idea of splitting this up 00:31:15.980 --> 00:31:20.600 into vertical strips whose height is given by a function 00:31:20.600 --> 00:31:21.100 y(x). 00:31:21.100 --> 00:31:22.266 And then you integrate that. 00:31:22.266 --> 00:31:24.040 That's an interpretation for the integral. 00:31:24.040 --> 00:31:27.210 The area is given by y dx. 00:31:27.210 --> 00:31:30.327 But that's a little bit awkward, because my formula for y 00:31:30.327 --> 00:31:31.660 is going to be a little strange. 00:31:31.660 --> 00:31:34.630 It's constant, value of b, along here, and then at this point 00:31:34.630 --> 00:31:37.660 it becomes this arc, of the circle. 00:31:37.660 --> 00:31:39.500 So working this out, I could do it 00:31:39.500 --> 00:31:42.100 but it's a little awkward because expressing y 00:31:42.100 --> 00:31:45.880 as a function of x, the top edge of this shape, 00:31:45.880 --> 00:31:50.090 it's a little awkward, and takes two different regions 00:31:50.090 --> 00:31:51.740 to express. 00:31:51.740 --> 00:31:58.620 So, a different way to say it is to say x dy. 00:31:58.620 --> 00:32:00.350 Maybe that'll work a little bit better. 00:32:00.350 --> 00:32:02.810 Or maybe it won't, but it's worth trying. 00:32:02.810 --> 00:32:05.290 I could just as well split this region up 00:32:05.290 --> 00:32:08.420 into horizontal strips. 00:32:08.420 --> 00:32:13.010 Whose width is dy, and whose length is x. 00:32:13.010 --> 00:32:17.250 Now I'm thinking of this as a function of y. 00:32:17.250 --> 00:32:20.870 This is the graph of a function of y. 00:32:20.870 --> 00:32:24.880 And that's much better, because the function of y is, well, 00:32:24.880 --> 00:32:28.010 it's the square root of a^2 - y^2, isn't it. 00:32:28.010 --> 00:32:34.300 That's x x^2 + y^2 = a^2. 00:32:34.300 --> 00:32:38.640 So that's what x is. 00:32:38.640 --> 00:32:41.740 And that's what I'm asked to integrate, then. 00:32:41.740 --> 00:32:45.580 Square root of (a^2 - y^2), dy. 00:32:45.580 --> 00:32:47.500 And I can even put in limits of integration. 00:32:47.500 --> 00:32:49.500 Maybe I should do that, because this is supposed 00:32:49.500 --> 00:32:50.910 to be an actual number. 00:32:50.910 --> 00:32:55.130 I guess I'm integrating it from y = 0, that's here. 00:32:55.130 --> 00:33:00.059 To y = b, dy. 00:33:00.059 --> 00:33:01.350 So this is what I want to find. 00:33:01.350 --> 00:33:07.420 This is a integral formula for the area of that region. 00:33:07.420 --> 00:33:08.730 And this is a new form. 00:33:08.730 --> 00:33:13.970 I don't think that you've thought 00:33:13.970 --> 00:33:20.310 about integrating expressions like this in this class before. 00:33:20.310 --> 00:33:23.660 So, it's a new form and I want to show you how to do it, 00:33:23.660 --> 00:33:30.290 how it's related to trigonometry. 00:33:30.290 --> 00:33:33.080 It's related to trigonometry through that exact picture 00:33:33.080 --> 00:33:36.710 that we have on the blackboard. 00:33:36.710 --> 00:33:42.610 After all, this a^2 - y^2 is the formula for this arc. 00:33:42.610 --> 00:33:45.460 And so, what I propose is that we 00:33:45.460 --> 00:33:49.700 try to exploit the connection with the circle 00:33:49.700 --> 00:33:52.860 and introduce polar coordinates. 00:33:52.860 --> 00:34:03.499 So, here if I measure this angle then there 00:34:03.499 --> 00:34:04.790 are various things you can say. 00:34:04.790 --> 00:34:08.076 Like the coordinates of this point here are a cos(theta), 00:34:08.076 --> 00:34:17.670 a-- Well, I'm sorry, it's not. 00:34:17.670 --> 00:34:20.780 That's an angle, but I want to call it theta_0. 00:34:20.780 --> 00:34:25.300 And, in general you know that the coordinates of this 00:34:25.300 --> 00:34:31.010 point are (a cos(theta), a sin(theta)). 00:34:31.010 --> 00:34:39.370 If the radius is a, then the angle here is theta. 00:34:39.370 --> 00:34:45.170 So x = a cos(theta), and y = a sin(theta), 00:34:45.170 --> 00:34:49.100 just from looking at the geometry of the circle. 00:34:49.100 --> 00:34:52.530 So let's make that substitution. y = a sin(theta). 00:34:56.600 --> 00:35:00.740 I'm using the picture to suggest that maybe making 00:35:00.740 --> 00:35:02.870 the substitution is a good thing to do. 00:35:02.870 --> 00:35:06.020 Let's follow along and see what happens. 00:35:06.020 --> 00:35:10.310 If that's true, what we're interested in is integrating, 00:35:10.310 --> 00:35:13.580 a^2 - y^2. 00:35:13.580 --> 00:35:18.900 Which is a^2-- We're interested in integrating the square root 00:35:18.900 --> 00:35:20.770 of a^2 - y^2. 00:35:20.770 --> 00:35:24.130 Which is the square root of a^2 minus this. 00:35:24.130 --> 00:35:27.100 a^2 sin^2(theta). 00:35:27.100 --> 00:35:35.870 And, well, that's equal to a cos theta. 00:35:35.870 --> 00:35:41.710 That's just sin^2 + cos^2 = 1, all over again. 00:35:41.710 --> 00:35:42.830 It's also x. 00:35:42.830 --> 00:35:44.560 This is x. 00:35:44.560 --> 00:35:46.070 And this was x. 00:35:46.070 --> 00:35:48.960 So there are a lot of different ways to think of this. 00:35:48.960 --> 00:35:51.650 But no matter how you say it, the thing we're trying 00:35:51.650 --> 00:35:59.270 to integrate, a^2 - y^2 is, under this substitution it is 00:35:59.270 --> 00:36:02.570 a cos(theta). 00:36:02.570 --> 00:36:06.700 So I'm interested in integrating the square root of (a^2 - y^2), 00:36:06.700 --> 00:36:09.260 dy. 00:36:09.260 --> 00:36:14.330 And I'm going to make this substitution y = a sin(theta). 00:36:17.330 --> 00:36:22.320 And so under that substitution, I've decided that the square 00:36:22.320 --> 00:36:26.170 root of a^2 - y^2 is a cos(theta). 00:36:31.170 --> 00:36:33.240 That's this. 00:36:33.240 --> 00:36:34.690 What about the dy? 00:36:34.690 --> 00:36:38.310 Well, I'd better compute the dy. 00:36:38.310 --> 00:36:40.920 So dy, just differentiating this expression, 00:36:40.920 --> 00:36:44.820 is a cos(theta) d theta. 00:36:44.820 --> 00:36:57.240 So let's put that in. dy = a cos(theta) d theta. 00:36:57.240 --> 00:36:58.310 OK. 00:36:58.310 --> 00:37:03.720 Making that trig substitution, y = a sin(theta), 00:37:03.720 --> 00:37:06.750 has replaced this integral that has a square root in it. 00:37:06.750 --> 00:37:08.640 And no trig functions. 00:37:08.640 --> 00:37:12.630 With an integral that involves no square roots and only trig 00:37:12.630 --> 00:37:15.140 functions. 00:37:15.140 --> 00:37:17.630 In fact, it's not too hard to integrate this now, because 00:37:17.630 --> 00:37:19.270 of the work that we've done. 00:37:19.270 --> 00:37:20.980 The a^2 comes out. 00:37:20.980 --> 00:37:22.630 This is cos^2(theta) d theta. 00:37:26.200 --> 00:37:28.580 And maybe we've done that example already today. 00:37:28.580 --> 00:37:35.920 I think we have. 00:37:35.920 --> 00:37:38.435 Maybe we can think it through, but maybe the easiest thing 00:37:38.435 --> 00:37:42.550 is to look back at notes and see what we got before. 00:37:42.550 --> 00:37:46.330 That was the first example in the hard case that I did. 00:37:46.330 --> 00:38:03.060 And what it came out to, I used x instead of theta at the time. 00:38:03.060 --> 00:38:05.570 So this is a good step forward. 00:38:05.570 --> 00:38:08.130 I started with this integral that I really 00:38:08.130 --> 00:38:12.440 didn't know how to do by any means that we've had so far. 00:38:12.440 --> 00:38:15.000 And I've replaced it by a trig integral 00:38:15.000 --> 00:38:16.250 that we do know how to do. 00:38:16.250 --> 00:38:19.160 And now I've done that trig integral. 00:38:19.160 --> 00:38:22.400 But we're still not quite done, because of the problem 00:38:22.400 --> 00:38:23.830 of back substituting. 00:38:23.830 --> 00:38:27.670 I'd like to go back and rewrite this in terms 00:38:27.670 --> 00:38:32.150 of the original variable, y. 00:38:32.150 --> 00:38:34.280 Or, I'd like to go back and rewrite it 00:38:34.280 --> 00:38:36.390 in terms of the original limits of integration 00:38:36.390 --> 00:38:40.050 that we had in the original problem. 00:38:40.050 --> 00:38:42.870 In doing that, it's going to be useful to rewrite 00:38:42.870 --> 00:38:47.150 this expression and get rid of the sin(2theta). 00:38:47.150 --> 00:38:51.840 After all, the original y was expressed in terms 00:38:51.840 --> 00:38:54.660 of sin(theta), not sin(2theta). 00:38:54.660 --> 00:39:04.220 So let me just do that here, and say that this, in turn, 00:39:04.220 --> 00:39:09.507 is equal to a^2 theta / 2 plus, well, 00:39:09.507 --> 00:39:11.090 sin(2theta) = 2 sin(theta) cos(theta). 00:39:18.080 --> 00:39:20.970 And so, when there's a 4 in the denominator, what I'll get 00:39:20.970 --> 00:39:25.390 is sin(theta) cos(theta) / 2. 00:39:32.580 --> 00:39:36.630 I did that because I'm getting closer to the original terms 00:39:36.630 --> 00:39:39.080 that the problem started with. 00:39:39.080 --> 00:39:40.130 Which was sin(theta). 00:40:05.480 --> 00:40:08.055 So let me write down the integral that we have now. 00:40:08.055 --> 00:40:14.680 The square root of a^2 - y^2, dy is, so far, 00:40:14.680 --> 00:40:26.810 what we know is a^2 (theta / 2 + sin(theta) cos(theta) / 2) + c. 00:40:26.810 --> 00:40:28.460 But I want to go back and rewrite this 00:40:28.460 --> 00:40:30.660 in terms of the original value. 00:40:30.660 --> 00:40:32.650 The original variable, y. 00:40:32.650 --> 00:40:37.430 Well, what is theta in terms of y? 00:40:37.430 --> 00:40:40.760 Let's see. y in terms of theta was given like this. 00:40:40.760 --> 00:40:44.120 So what is theta in terms of y? 00:40:44.120 --> 00:40:44.950 Ah. 00:40:44.950 --> 00:40:48.710 So here the fearsome arcsine rears its head, right? 00:40:48.710 --> 00:40:53.680 Theta is the angle so that y = a sin(theta). 00:40:53.680 --> 00:40:58.230 So that means that theta is the arcsine, or sine inverse, 00:40:58.230 --> 00:41:00.750 of y/a. 00:41:07.450 --> 00:41:12.630 So that's the first thing that shows up here. 00:41:12.630 --> 00:41:18.290 arcsin(y/a), all over 2. 00:41:18.290 --> 00:41:19.360 That's this term. 00:41:19.360 --> 00:41:24.530 Theta is arcsin(y/a) / 2. 00:41:24.530 --> 00:41:26.630 What about the other side, here? 00:41:26.630 --> 00:41:30.620 Well sine and cosine, we knew what they were in terms of y 00:41:30.620 --> 00:41:37.830 and in terms of x, if you like. 00:41:37.830 --> 00:41:40.300 Maybe I'll put the a^2 inside here. 00:41:40.300 --> 00:41:42.710 That makes it a little bit nicer. 00:41:42.710 --> 00:41:49.310 Plus, and the other term is a^2 sin(theta) cos(theta). 00:41:49.310 --> 00:41:52.610 So the a sin(theta) is just y. 00:41:52.610 --> 00:41:55.140 Maybe I'll write this (a sin(theta)) (a cos(theta)) 00:41:55.140 --> 00:41:57.550 / 2 + c. 00:42:02.330 --> 00:42:03.840 And so I get the same thing. 00:42:03.840 --> 00:42:06.204 And now here a sin(theta), that's y. 00:42:06.204 --> 00:42:07.370 And what's the a cos(theta)? 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. 00:42:22.010 --> 00:42:28.140 And so there I've rewritten everything, back 00:42:28.140 --> 00:42:31.250 in terms of the original variable, y. 00:42:31.250 --> 00:42:36.060 And there's an answer. 00:42:36.060 --> 00:42:41.070 So I've done this indefinite integration of a form-- 00:42:41.070 --> 00:42:44.810 of this quadratic, this square root of something which is 00:42:44.810 --> 00:42:46.940 a constant minus y^2. 00:42:46.940 --> 00:42:50.640 Whenever you see that, the thing to think of is trigonometry. 00:42:50.640 --> 00:42:54.310 That's going to play into the sin^2 + cos^2 identity. 00:42:54.310 --> 00:42:56.880 And the way to exploit it is to make the substitution 00:42:56.880 --> 00:43:01.100 y = a sin(theta). 00:43:01.100 --> 00:43:04.340 You could also make a substitution y = a cos(theta), 00:43:04.340 --> 00:43:05.540 if you wanted to. 00:43:05.540 --> 00:43:12.290 And the result would come out to exactly the same in the end. 00:43:12.290 --> 00:43:14.430 I'm still not quite done with the original problem 00:43:14.430 --> 00:43:23.990 that I had, because the original problem 00:43:23.990 --> 00:43:25.810 asked for a definite integral. 00:43:25.810 --> 00:43:33.000 So let's just go back and finish that as well. 00:43:33.000 --> 00:43:37.910 So the area was the integral from 0 00:43:37.910 --> 00:43:45.740 to b of this square root. 00:43:45.740 --> 00:43:48.220 So I just want to evaluate the right-hand side here. 00:43:48.220 --> 00:43:50.880 The answer that we came up with, this indefinite integral. 00:43:50.880 --> 00:43:53.374 I want to evaluate it at 0 and at b. 00:43:53.374 --> 00:43:54.040 Well, let's see. 00:43:54.040 --> 00:44:13.470 When I evaluate it at b, I get a^2 arcsin(b/a) / 2 plus y, 00:44:13.470 --> 00:44:18.900 which is b, times the square root of a^2 - b^2, 00:44:18.900 --> 00:44:23.050 putting y = b, divided by 2. 00:44:23.050 --> 00:44:26.300 So I've plugged in y = b into that formula, 00:44:26.300 --> 00:44:27.290 this is what I get. 00:44:27.290 --> 00:44:31.280 Then when I plug in y = 0, well the, sine of 0 is 0, 00:44:31.280 --> 00:44:34.040 so the arcsine of 0 is 0. 00:44:34.040 --> 00:44:35.720 So this term goes away. 00:44:35.720 --> 00:44:38.930 And when y = 0, this term is 0 also. 00:44:38.930 --> 00:44:43.210 And so I don't get any subtracted terms at all. 00:44:43.210 --> 00:44:45.600 So there's an expression for this. 00:44:45.600 --> 00:44:52.360 Notice that this arcsin(b/a), that's exactly this angle. 00:44:52.360 --> 00:45:00.550 arcsin(b/a), it's the angle that you get when y = b. 00:45:00.550 --> 00:45:06.760 So this theta is the arcsin(b/a). 00:45:09.990 --> 00:45:15.870 Put this over here. 00:45:15.870 --> 00:45:17.090 That is theta_0. 00:45:17.090 --> 00:45:21.560 That is the angle that the corner makes. 00:45:21.560 --> 00:45:28.460 So I could rewrite this as a a^2 theta_0 / 2 plus b times 00:45:28.460 --> 00:45:34.722 the square root of a^2 - b^2, over 2. 00:45:34.722 --> 00:45:36.430 Let's just think about this for a minute. 00:45:36.430 --> 00:45:40.320 I have these two terms in the sum, is that reasonable? 00:45:40.320 --> 00:45:44.080 The first term is a^2. 00:45:44.080 --> 00:45:50.370 That's the radius squared times this angle, times 1/2. 00:45:50.370 --> 00:45:54.280 Well, I think that is exactly the area of this sector. 00:45:54.280 --> 00:46:03.170 a^2 theta / 2 is the formula for the area of the sector. 00:46:03.170 --> 00:46:07.070 And this one, this is the vertical elevation. 00:46:07.070 --> 00:46:09.500 This is the horizontal. 00:46:09.500 --> 00:46:14.550 a^2 - b^2 is this distance. 00:46:14.550 --> 00:46:16.980 Square root of a^2 - b^2. 00:46:16.980 --> 00:46:20.420 So the right-hand term is b times the square root of a^2 - 00:46:20.420 --> 00:46:31.930 b^2 divided by 2, that's the area of that triangle. 00:46:31.930 --> 00:46:34.280 So using a little bit of geometry 00:46:34.280 --> 00:46:39.890 gives you the same answer as all of this elaborate calculus. 00:46:39.890 --> 00:46:41.610 Maybe that's enough cause for celebration 00:46:41.610 --> 00:46:43.620 for us to quit for today.