WEBVTT 00:00:00.000 --> 00:00:08.500 CHRISTINE BREINER: Welcome back to recitation. 00:00:08.500 --> 00:00:11.010 In this video I'd like us to do two things. 00:00:11.010 --> 00:00:14.010 The first thing we're going to do is we're going to graph the 00:00:14.010 --> 00:00:18.570 curve, r equals 1 plus cosine theta over 2 for theta between 00:00:18.570 --> 00:00:22.400 0 and 4 pi and we're going to graph it in the x, y plane. 00:00:22.400 --> 00:00:25.040 And then after we've done that, we're going to take a 00:00:25.040 --> 00:00:28.420 look at some components of that curve and we're going to 00:00:28.420 --> 00:00:32.310 calculate the area of some components that close up. 00:00:32.310 --> 00:00:37.190 So what I'd like you to do first is get a good picture of 00:00:37.190 --> 00:00:39.550 this curve in the x, y plane. 00:00:39.550 --> 00:00:40.910 I'll give you a little while to do that. 00:00:40.910 --> 00:00:43.320 So why don't you pause the video, get a good picture of 00:00:43.320 --> 00:00:46.230 that curve, then come back when you're ready and I'll 00:00:46.230 --> 00:00:49.440 show you how I graph it, and then we'll get into these area 00:00:49.440 --> 00:00:58.520 problems. 00:00:58.520 --> 00:00:58.760 OK. 00:00:58.760 --> 00:00:59.720 Welcome back. 00:00:59.720 --> 00:01:03.190 So the goal, again, was to graph a certain curve 00:01:03.190 --> 00:01:06.690 described by r and theta, but in the x, y plane. 00:01:06.690 --> 00:01:09.880 And for theta between 0 and pi over 4. 00:01:09.880 --> 00:01:15.600 And when I do these problems, we want to make sure that we 00:01:15.600 --> 00:01:18.460 understand how r depends on theta. 00:01:18.460 --> 00:01:21.700 That's sort of the main goal to graph this curve. 00:01:21.700 --> 00:01:25.590 So what I do, actually, is I look at this not in the x, y 00:01:25.590 --> 00:01:28.720 plane, which was what I said to do in the problem, but I 00:01:28.720 --> 00:01:32.070 first look at it in what we see as the r, theta plane. 00:01:32.070 --> 00:01:35.170 And what we mean by that is we're going to graph this just 00:01:35.170 --> 00:01:38.430 like we would if this variable was x and this variable was y. 00:01:38.430 --> 00:01:43.440 So we move out of what we know about how r relates to x and 00:01:43.440 --> 00:01:46.040 y, and how theta relates to x and y, and we just look at how 00:01:46.040 --> 00:01:47.660 r relates to theta. 00:01:47.660 --> 00:01:50.430 So let me draw that first and we'll see if we can sort of 00:01:50.430 --> 00:01:54.620 understand what I mean by that, and then we'll put that, 00:01:54.620 --> 00:01:58.030 use that curve to put that into the x, y plane. 00:01:58.030 --> 00:01:58.290 OK. 00:01:58.290 --> 00:02:03.500 So, the first thing we do is I'm going to let this be the 00:02:03.500 --> 00:02:07.570 theta-axis and this be the r-axis. 00:02:07.570 --> 00:02:10.350 And let's look at--I want to write down over here what the 00:02:10.350 --> 00:02:12.520 equation is so I don't have to keep turning around. 00:02:15.550 --> 00:02:17.800 So again, this is not, I don't want to think about this as 00:02:17.800 --> 00:02:18.880 the x, y plane. 00:02:18.880 --> 00:02:21.820 Because in the x, y plane, theta has certain values at 00:02:21.820 --> 00:02:24.680 each point that are fixed based on the angle. 00:02:24.680 --> 00:02:28.815 But now I'm letting theta vary in this direction, and r is 00:02:28.815 --> 00:02:30.290 varying in this direction. 00:02:30.290 --> 00:02:34.620 And my theta, I said, was between 0 and pi over 4-- 00:02:34.620 --> 00:02:37.000 I'm sorry, not pi over 4. 00:02:37.000 --> 00:02:38.350 4 pi. 00:02:38.350 --> 00:02:40.200 0 and 4 pi. 00:02:40.200 --> 00:02:42.730 pi over 4 would be a very small component of this that 00:02:42.730 --> 00:02:43.920 I'm interested in. 00:02:43.920 --> 00:02:47.290 And let's think about what happens, what kind of 00:02:47.290 --> 00:02:50.780 transformations have been done to the normal cosine function. 00:02:50.780 --> 00:02:55.860 So if I took the normal cosine function and I take it instead 00:02:55.860 --> 00:02:59.260 of cosine theta, I look it cosine theta over 2, what 00:02:59.260 --> 00:03:01.250 that's doing is that's stretching it 00:03:01.250 --> 00:03:02.900 horizontally out. 00:03:02.900 --> 00:03:05.820 So think about the period of the cosine function 00:03:05.820 --> 00:03:07.910 usually is 2 pi. 00:03:07.910 --> 00:03:10.630 But notice what happens when I put in 2 pi for theta, I'm 00:03:10.630 --> 00:03:12.700 getting cosine of pi. 00:03:12.700 --> 00:03:15.200 If I want to get cosine of 2 pi, I have to let theta go to 00:03:15.200 --> 00:03:19.080 4 pi, which is why I'm letting theta be between 0 and 4 pi. 00:03:19.080 --> 00:03:23.870 So dividing the input value by 2 doubles your period. 00:03:23.870 --> 00:03:26.280 So the period is now 4 pi. 00:03:26.280 --> 00:03:28.190 So to get all the way through, I'm going to have 00:03:28.190 --> 00:03:29.900 to go up to 4 pi. 00:03:29.900 --> 00:03:34.440 So let me just make a 2 pi here, and this is 00:03:34.440 --> 00:03:36.910 about a 4 pi here. 00:03:36.910 --> 00:03:40.620 So here's around 3 pi, and here's around pi. 00:03:40.620 --> 00:03:44.010 So that now, instead of the usual cosine function, it's 00:03:44.010 --> 00:03:46.380 going to take twice as long to get all the way through. 00:03:46.380 --> 00:03:47.660 That's one thing we know. 00:03:47.660 --> 00:03:49.730 What else have we done to the usual cosine function? 00:03:49.730 --> 00:03:51.420 We've moved it up by 1. 00:03:51.420 --> 00:03:56.550 And so instead of starting out when your input is 0, starting 00:03:56.550 --> 00:03:59.910 out at height 1, when you're input is 0, you start out at 00:03:59.910 --> 00:04:00.920 height 1 plus 1. 00:04:00.920 --> 00:04:03.180 You start out at height 2. 00:04:03.180 --> 00:04:09.250 So in fact, this function, let me point out, it's going to 00:04:09.250 --> 00:04:12.030 start at 2, which means it also is 00:04:12.030 --> 00:04:13.520 going to end over here. 00:04:13.520 --> 00:04:16.680 Because it's periodic it's going to end at 2. 00:04:16.680 --> 00:04:18.870 And let's think about what else we know. 00:04:18.870 --> 00:04:21.070 We know that the usual cosine function goes 00:04:21.070 --> 00:04:22.800 down to negative 1. 00:04:22.800 --> 00:04:25.257 But I've added 1 to it, so now it only goes down to 0. 00:04:25.257 --> 00:04:26.880 OK? 00:04:26.880 --> 00:04:27.790 Hopefully that makes sense. 00:04:27.790 --> 00:04:31.090 Maybe I should even-- 00:04:31.090 --> 00:04:32.010 hmm. 00:04:32.010 --> 00:04:34.170 I don't want to draw the actual cosine function again 00:04:34.170 --> 00:04:35.100 right on here. 00:04:35.100 --> 00:04:38.080 But let me draw the regular cosine function here. 00:04:38.080 --> 00:04:42.710 So we have it, the regular cosine function-- 00:04:42.710 --> 00:04:45.620 because I keep talking about it-- 00:04:45.620 --> 00:04:46.720 does something like this. 00:04:46.720 --> 00:04:50.380 It goes at 1 here, and it's at 1 again here. 00:04:50.380 --> 00:04:53.530 And so it's at minus 1 at pi. 00:04:53.530 --> 00:04:56.290 And so, very roughly, it looks something like this. 00:04:56.290 --> 00:04:57.680 Right? 00:04:57.680 --> 00:04:59.970 So I keep referencing the cosine function, so this is 00:04:59.970 --> 00:05:01.730 the part I'm referencing. 00:05:01.730 --> 00:05:05.620 So we have to stretch it by 2, and then shift it up by 1. 00:05:05.620 --> 00:05:09.680 And so we see what was at pi, negative 1, I'm now going to 00:05:09.680 --> 00:05:13.110 be at 2 pi, 0. 00:05:13.110 --> 00:05:14.500 And then where do these points go? 00:05:14.500 --> 00:05:16.540 This was pi over 2. 00:05:16.540 --> 00:05:19.160 The x value is going to be doubled. 00:05:19.160 --> 00:05:20.950 I'm going to be at pi, and the y value is 00:05:20.950 --> 00:05:22.420 going to go up by 1. 00:05:22.420 --> 00:05:26.050 So I'm at pi, 1 and 3, pi 1. 00:05:26.050 --> 00:05:30.410 And so the curve will look something like this. 00:05:30.410 --> 00:05:35.500 I'm not an expert here, but hopefully that looks something 00:05:35.500 --> 00:05:36.780 like the cosine time function. 00:05:36.780 --> 00:05:43.230 But with a stretch and a shift. 00:05:43.230 --> 00:05:48.700 So that is the curve, r equals 1 plus cosine theta over 2 in 00:05:48.700 --> 00:05:51.510 what we consider the r, theta plane. 00:05:51.510 --> 00:05:54.560 So theta is varying in this direction and r is varying in 00:05:54.560 --> 00:05:55.840 this direction. 00:05:55.840 --> 00:05:58.790 But how do I transfer that to the x, y plane? 00:05:58.790 --> 00:06:00.630 That's the real point that I want to 00:06:00.630 --> 00:06:02.830 make about this problem. 00:06:02.830 --> 00:06:06.980 So let's look at what's happening in the x, y plane. 00:06:06.980 --> 00:06:10.160 So this will be x and this will be y. 00:06:10.160 --> 00:06:11.560 Let's pick some points and try to 00:06:11.560 --> 00:06:13.120 figure out what's happening. 00:06:13.120 --> 00:06:15.150 So where is this point? 00:06:15.150 --> 00:06:17.360 This is the point zero comma two. 00:06:17.360 --> 00:06:20.820 When theta is 0, r equals 2. 00:06:20.820 --> 00:06:23.350 Where is theta equal to 0 in this picture? 00:06:23.350 --> 00:06:24.525 It's in the x direction. 00:06:24.525 --> 00:06:25.310 Right? 00:06:25.310 --> 00:06:31.270 That's theta equals 0, and also 2 pi and 4 pi. 00:06:31.270 --> 00:06:36.360 Any multiple of 2 pi, theta is pointing in this direction. 00:06:36.360 --> 00:06:38.470 And r equals 2 there. 00:06:38.470 --> 00:06:41.830 So in a strange twist, this is the point, 2, 00:06:41.830 --> 00:06:43.540 0 on the x, y plane. 00:06:43.540 --> 00:06:46.210 But that's no going to, that's just a coincidence, OK? 00:06:46.210 --> 00:06:48.560 Don't think, oh, I'm just going to flip the values 00:06:48.560 --> 00:06:49.020 everywhere. 00:06:49.020 --> 00:06:49.795 That's not going to happen. 00:06:49.795 --> 00:06:51.540 OK? 00:06:51.540 --> 00:06:53.520 Actually, and also, before I make a mistake, I'm going to 00:06:53.520 --> 00:06:55.190 make this a little bit bigger. 00:06:55.190 --> 00:06:57.360 I want it to be bigger. 00:06:57.360 --> 00:06:59.020 So I don't want this to be 2. 00:06:59.020 --> 00:07:00.350 I want this to be 1. 00:07:00.350 --> 00:07:04.380 I'm going to make this 2, 0. 00:07:04.380 --> 00:07:06.980 I want to have a little bigger picture. 00:07:06.980 --> 00:07:07.290 OK. 00:07:07.290 --> 00:07:14.010 So it's going to be 2, 0 at theta equals 0 and r equal 2. 00:07:14.010 --> 00:07:16.070 Is it ever hit this point again? 00:07:16.070 --> 00:07:16.930 Well, it is. 00:07:16.930 --> 00:07:18.610 And it's going to hit that point again because it's 00:07:18.610 --> 00:07:20.840 periodic and I've gone out to 4 pi. 00:07:20.840 --> 00:07:23.870 If I rotate all the way around, I'm at 2 pi for theta. 00:07:23.870 --> 00:07:26.330 If I rotate all the way around again I'm at 4 pi for theta, 00:07:26.330 --> 00:07:28.740 and my radius there is 2, also. 00:07:28.740 --> 00:07:32.120 So this 2, 0 happens again when I have this point. 00:07:32.120 --> 00:07:34.160 So it's going to close up. 00:07:34.160 --> 00:07:34.430 OK. 00:07:34.430 --> 00:07:35.430 And then what else happens? 00:07:35.430 --> 00:07:38.320 Well, as theta is rotating, let's take theta 00:07:38.320 --> 00:07:40.110 between 0 and pi. 00:07:40.110 --> 00:07:43.560 theta rotates from 0 to pi going like this. 00:07:43.560 --> 00:07:46.270 Notice what's happening to the r value. 00:07:46.270 --> 00:07:49.910 The r value is going from 2 down to 1. 00:07:49.910 --> 00:07:53.780 Now, I'm not going to be totally exact, but-- 00:07:53.780 --> 00:07:58.450 here's minus 1, OK, in the x, y plane. 00:07:58.450 --> 00:08:02.070 I'm not going to be totally exactly, but this curve is 00:08:02.070 --> 00:08:03.830 going to look something like-- 00:08:06.565 --> 00:08:07.870 there's 2-- 00:08:07.870 --> 00:08:09.145 it's going to look something like-- 00:08:13.700 --> 00:08:16.620 oops, I over-shot, we'll make that negative 1-- 00:08:16.620 --> 00:08:18.470 something like this. 00:08:18.470 --> 00:08:19.710 And what's the point? 00:08:19.710 --> 00:08:24.050 The point is that I start at radius 2, and by the time I 00:08:24.050 --> 00:08:28.240 get to theta equals pi I've gone down. 00:08:28.240 --> 00:08:29.740 And so my radius is 1. 00:08:29.740 --> 00:08:33.850 This has radius 1 and angle pi. 00:08:33.850 --> 00:08:36.200 So that represents this part of the curve. 00:08:36.200 --> 00:08:38.400 That's this part of the curve. 00:08:38.400 --> 00:08:41.900 Now, what's nice about this drawing is that we know this 00:08:41.900 --> 00:08:44.070 part of the curve and this part of the curve should look 00:08:44.070 --> 00:08:45.310 exactly the same. 00:08:45.310 --> 00:08:48.620 So once I've drawn half of this, I'm going to know 00:08:48.620 --> 00:08:49.110 everything. 00:08:49.110 --> 00:08:51.500 Once I've gone from 0 to 2 pi, I can 00:08:51.500 --> 00:08:52.930 just reflect it, basically. 00:08:52.930 --> 00:08:56.050 We'll see what I mean by reflect in this case, but the 00:08:56.050 --> 00:08:58.690 same radii are happening again and some sort 00:08:58.690 --> 00:09:00.670 of symmetric fashion. 00:09:00.670 --> 00:09:00.920 OK. 00:09:00.920 --> 00:09:02.240 So we've got 0 to pi. 00:09:02.240 --> 00:09:04.650 Now what happens between pi and 2 pi? 00:09:04.650 --> 00:09:08.290 Notice pi to 2 pi in the theta direction on the x, y plane is 00:09:08.290 --> 00:09:09.445 I start in this direction-- 00:09:09.445 --> 00:09:12.150 I don't know if you can see that, let me come 00:09:12.150 --> 00:09:13.250 over to this side-- 00:09:13.250 --> 00:09:16.630 I start in this direction for pi, and I'm 00:09:16.630 --> 00:09:17.620 going to rotate down. 00:09:17.620 --> 00:09:21.490 This is 3 pi over 2, and this is 2 pi. 00:09:21.490 --> 00:09:22.470 Those are my angles. 00:09:22.470 --> 00:09:24.310 So what are my radii? 00:09:24.310 --> 00:09:30.740 Well, I start at radius 1 and I'm going to radius 0. 00:09:30.740 --> 00:09:35.470 And so what happens is I'm coming through this negative 1 00:09:35.470 --> 00:09:40.140 and I'm coming around, and then by the time I get-- 00:09:40.140 --> 00:09:42.310 it's going to be something like this-- by the time I get 00:09:42.310 --> 00:09:45.260 to 2 pi, my radius is 0. 00:09:45.260 --> 00:09:48.430 Now just to make sure this curve makes sense to you, you 00:09:48.430 --> 00:09:53.440 could pick a place, an angle, maybe like right here. 00:09:53.440 --> 00:09:55.430 I don't know if that's easy to see, but maybe that angle 00:09:55.430 --> 00:09:56.210 right there. 00:09:56.210 --> 00:09:59.480 That angle is between 3 pi over 2 and 2 pi. 00:09:59.480 --> 00:10:01.280 Is there positive radius there? 00:10:01.280 --> 00:10:02.900 Yeah, there is positive radius there. 00:10:02.900 --> 00:10:06.540 So in fact, this curve does come into this fourth quadrant 00:10:06.540 --> 00:10:09.090 here and then curve back in. 00:10:09.090 --> 00:10:09.410 OK? 00:10:09.410 --> 00:10:11.550 It does curve back in. 00:10:11.550 --> 00:10:11.850 All right. 00:10:11.850 --> 00:10:16.020 So then what happens between 2 pi and 3 pi? 00:10:16.020 --> 00:10:16.150 OK. 00:10:16.150 --> 00:10:18.310 Hopefully this picture is clear so far. 00:10:18.310 --> 00:10:20.240 I'm going to come back to the other side. 00:10:20.240 --> 00:10:20.460 OK. 00:10:20.460 --> 00:10:22.390 What happens between 2 pi and 3 pi? 00:10:22.390 --> 00:10:24.020 2 pi, we're here. 00:10:24.020 --> 00:10:26.720 And 3 pi, we're back over here again. 00:10:26.720 --> 00:10:30.490 And notice that the radius is going to be doing the same 00:10:30.490 --> 00:10:33.200 kind of growth that it did between 2 pi and 3 pi as it 00:10:33.200 --> 00:10:36.632 did decay from pi to 2 pi. 00:10:36.632 --> 00:10:37.500 OK? 00:10:37.500 --> 00:10:40.680 Because the radii now, there's a symmetry with how the radii 00:10:40.680 --> 00:10:46.340 behave. So from 2 pi to 3 pi, I start off with radius 0 and 00:10:46.340 --> 00:10:48.120 I have a small radius. 00:10:48.120 --> 00:10:53.870 And then once I get to 3 pi over here, I'm going to have 00:10:53.870 --> 00:10:55.070 radius 1 again. 00:10:55.070 --> 00:10:56.940 I'm going to be at radius 1, which is going to correspond 00:10:56.940 --> 00:10:57.840 to this point. 00:10:57.840 --> 00:11:00.580 So I'm going to have exactly the same picture, which is 00:11:00.580 --> 00:11:03.265 dangerous because I probably would do it wrong. 00:11:03.265 --> 00:11:06.840 But I'll try and draw it this way and then talk about it. 00:11:06.840 --> 00:11:09.480 Hopefully that looks about the same. 00:11:09.480 --> 00:11:13.570 So this curve coming through here was from pi to 2 pi. 00:11:13.570 --> 00:11:17.750 This curve coming through here was from to 2 pi to 3 pi. 00:11:17.750 --> 00:11:20.240 And then to finish it off, 3 pi to 4 pi is going to look 00:11:20.240 --> 00:11:22.310 like this curve here. 00:11:22.310 --> 00:11:24.260 So I come through-- 00:11:24.260 --> 00:11:27.440 ooh, this is where it starts to get really dangerous, but 00:11:27.440 --> 00:11:29.980 let's say that's pretty close-- 00:11:29.980 --> 00:11:33.160 so there's my 3 pi to 4 pi. 00:11:33.160 --> 00:11:36.410 It's again, the same growth the way it was decaying 00:11:36.410 --> 00:11:38.260 between 0 and pi. 00:11:38.260 --> 00:11:43.940 So this is your picture in the x, y plane of the curve, r 00:11:43.940 --> 00:11:47.150 equals 1 plus cosine theta over 2. 00:11:47.150 --> 00:11:50.900 Now, we haven't calculated any area problems, yet. 00:11:50.900 --> 00:11:54.210 So what I'd like us to do is I'm going to shade two 00:11:54.210 --> 00:12:00.430 regions, and I want us to just write down the integral form 00:12:00.430 --> 00:12:04.070 to find the area for each of these regions. 00:12:04.070 --> 00:12:09.600 So the first region of interest is the pink region. 00:12:09.600 --> 00:12:12.750 I'm going to ask us to find the area of the pink region, 00:12:12.750 --> 00:12:16.420 and then I'm going to ask us to find the area of everything 00:12:16.420 --> 00:12:17.670 else, the blue region. 00:12:20.450 --> 00:12:24.560 So let's think about how to do that. 00:12:24.560 --> 00:12:26.980 And I think I'm going to have to come over to the other side 00:12:26.980 --> 00:12:28.890 to write down the integrals, but I'll be 00:12:28.890 --> 00:12:30.440 coming back and forth. 00:12:30.440 --> 00:12:35.700 So just to remind you, what you saw in lecture was that dA 00:12:35.700 --> 00:12:40.550 is equal to r squared over 2 d theta. 00:12:40.550 --> 00:12:42.740 That's what dA is. 00:12:42.740 --> 00:12:46.450 And so this is going to be an integral in theta, and we know 00:12:46.450 --> 00:12:48.650 what r is as a function of theta. 00:12:48.650 --> 00:12:52.890 And so if I want to find-- actually, I should even use my 00:12:52.890 --> 00:12:54.330 colors appropriately. 00:12:54.330 --> 00:12:59.890 I should say, the pink area is going to 00:12:59.890 --> 00:13:01.860 be equal to an integral. 00:13:01.860 --> 00:13:03.780 And I'm going to write the r squared. 00:13:03.780 --> 00:13:05.010 I know what r squared is. 00:13:05.010 --> 00:13:10.320 It's 1 plus cosine theta over 2 squared, and 00:13:10.320 --> 00:13:12.560 then an over 2 d theta. 00:13:12.560 --> 00:13:14.810 And then what is important about this? 00:13:14.810 --> 00:13:15.910 It's our bounds. 00:13:15.910 --> 00:13:16.090 Right? 00:13:16.090 --> 00:13:17.800 Our bounds are what's important. 00:13:17.800 --> 00:13:21.760 And so let's go back and let's look at our picture. 00:13:21.760 --> 00:13:25.680 What are the bounds on theta for the pink region? 00:13:25.680 --> 00:13:28.540 So where does that pink region start and stop? 00:13:28.540 --> 00:13:32.530 And maybe we even need to look at this top graph, also. 00:13:32.530 --> 00:13:36.750 So if we think about it, we went from 0 to pi to get this 00:13:36.750 --> 00:13:38.120 outer curve. 00:13:38.120 --> 00:13:39.440 So how do we get the inner curve? 00:13:39.440 --> 00:13:44.070 The inner curve started at theta equals pi, went to here, 00:13:44.070 --> 00:13:45.390 went to here-- 00:13:45.390 --> 00:13:47.900 that was theta equals 3 pi. 00:13:47.900 --> 00:13:52.160 So it went all the way from pi to 3 pi. 00:13:52.160 --> 00:13:55.410 Now, if you're paying good attention, you can say, well, 00:13:55.410 --> 00:13:59.550 Christine, we know that this region is totally symmetric. 00:13:59.550 --> 00:14:02.860 So why don't I just take the area from pi to 2 pi and 00:14:02.860 --> 00:14:04.470 multiply it by 2? 00:14:04.470 --> 00:14:05.340 And you can. 00:14:05.340 --> 00:14:06.910 You can do it either way. 00:14:06.910 --> 00:14:09.900 So you can either take the integral from all the way from 00:14:09.900 --> 00:14:13.880 pi to 3 pi, which corresponds to starting at this angle, 00:14:13.880 --> 00:14:16.880 going all the way around, and coming back to there, which 00:14:16.880 --> 00:14:18.240 takes you all the way around this curve. 00:14:18.240 --> 00:14:21.780 Or you can go from pi to 2 pi and multiply that by 2. 00:14:21.780 --> 00:14:25.470 So let me come back and let me write that down. 00:14:25.470 --> 00:14:31.970 It's either pi to 3 pi, this, or you just write it as 00:14:31.970 --> 00:14:34.310 integral from pi to 2 pi. 00:14:34.310 --> 00:14:38.430 And if I multiply this by 2, the 2 drops out. 00:14:38.430 --> 00:14:40.477 The 2 in the denominator drops out. 00:14:44.680 --> 00:14:46.720 I'm not going to solve this problem for you, but I do want 00:14:46.720 --> 00:14:48.850 to point out the kinds of terms you would have. You 00:14:48.850 --> 00:14:51.060 would have a constant term when you square this. 00:14:51.060 --> 00:14:54.370 You would have a term that was 2 cosine theta over 2, which 00:14:54.370 --> 00:14:56.980 is easy to integrate by a u-substitution. 00:14:56.980 --> 00:14:59.990 And then you would have a cosine squared theta over 2, 00:14:59.990 --> 00:15:03.160 which you'd want to use the double angle formula or the 00:15:03.160 --> 00:15:06.310 half angle formula that you've seen used to manipulate these 00:15:06.310 --> 00:15:08.040 integrals that involve just a cosine 00:15:08.040 --> 00:15:09.680 squared or a sine squared. 00:15:09.680 --> 00:15:11.600 So that would be your strategy. 00:15:11.600 --> 00:15:13.175 OK, now let's look at the blue area. 00:15:13.175 --> 00:15:14.425 OK. 00:15:19.090 --> 00:15:22.110 So to find the blue area, again, I know all that matters 00:15:22.110 --> 00:15:24.520 is really the bounds. 00:15:24.520 --> 00:15:26.515 We're going to see we have to do a little extra work. 00:15:29.600 --> 00:15:31.000 But this is our first setup. 00:15:31.000 --> 00:15:33.090 And now let's go look at the bounds. 00:15:33.090 --> 00:15:33.360 OK. 00:15:33.360 --> 00:15:35.880 So we go back to the curve. 00:15:35.880 --> 00:15:37.020 All right. 00:15:37.020 --> 00:15:38.000 What is the blue area? 00:15:38.000 --> 00:15:41.730 Well, the blue area, that's a little harder. 00:15:41.730 --> 00:15:43.680 So let's see what happens. 00:15:43.680 --> 00:15:49.490 If I were to take theta from 0 to pi, what would happen? 00:15:49.490 --> 00:15:52.370 I would not only pick up the blue area, but I'd pick up 00:15:52.370 --> 00:15:54.100 this pink stuff inside. 00:15:54.100 --> 00:15:55.640 But I don't want the pink stuff. 00:15:55.640 --> 00:15:57.290 I just want the blue stuff. 00:15:57.290 --> 00:16:00.020 So what am I going to have to do to find the blue area just, 00:16:00.020 --> 00:16:01.980 say, from 0 up to pi? 00:16:01.980 --> 00:16:05.080 I'm going to have to find the area from 0 to pi, and then 00:16:05.080 --> 00:16:06.520 I'm going to have to subtract off the 00:16:06.520 --> 00:16:07.722 area of this component. 00:16:07.722 --> 00:16:09.210 OK? 00:16:09.210 --> 00:16:10.660 But we know, actually, how to find the 00:16:10.660 --> 00:16:12.490 area of this component. 00:16:12.490 --> 00:16:12.790 OK. 00:16:12.790 --> 00:16:14.120 So hopefully this makes sense. 00:16:14.120 --> 00:16:16.350 Because let's think about, when I'm finding area, I'm 00:16:16.350 --> 00:16:18.420 going from the origin and I'm coming out and I have the 00:16:18.420 --> 00:16:20.020 radius out there. 00:16:20.020 --> 00:16:24.930 So when I integrate this dA from 0 to pi for theta, I'm 00:16:24.930 --> 00:16:28.800 picking up pieces taht come out, little sectors that come 00:16:28.800 --> 00:16:31.880 out like this between pi over 2 and pi. 00:16:31.880 --> 00:16:34.960 So I'm getting more area than I want if I just let theta go 00:16:34.960 --> 00:16:36.740 between 0 and pi. 00:16:36.740 --> 00:16:39.670 So I have to calculate all of it, and then I have to take 00:16:39.670 --> 00:16:41.970 away the extra stuff. 00:16:41.970 --> 00:16:42.250 OK. 00:16:42.250 --> 00:16:45.410 So the blue area is actually the bigger area subtracting 00:16:45.410 --> 00:16:47.180 the smaller area. 00:16:47.180 --> 00:16:48.890 And so how am I going to write this? 00:16:48.890 --> 00:16:51.380 If we come back over, I'm just going to take 2 times this 00:16:51.380 --> 00:16:52.440 whole thing. 00:16:52.440 --> 00:16:54.220 So I'm going to take 2 times this. 00:16:54.220 --> 00:16:57.580 And I know I have to integrate it from 0 to pi. 00:16:57.580 --> 00:17:00.850 And I'm taking 2 times because it's symmetric, remember. 00:17:00.850 --> 00:17:04.310 And then I'm going to subtract track off this one that's 2 00:17:04.310 --> 00:17:06.360 times the thing from pi to 2 pi. 00:17:06.360 --> 00:17:08.610 Now, you might say, well, Christine, the pink stuff I'm 00:17:08.610 --> 00:17:13.780 interested in between 0 and pi is actually theta between not 00:17:13.780 --> 00:17:15.830 pi and 2 pi, but 2 pi and 3 pi. 00:17:15.830 --> 00:17:17.880 But again, there's all this symmetry in the problem. 00:17:17.880 --> 00:17:20.270 So it doesn't really matter. 00:17:20.270 --> 00:17:23.360 But if you're a stickler, I guess I'll even write it this 00:17:23.360 --> 00:17:24.790 way just to make sure. 00:17:24.790 --> 00:17:27.200 So I'll write it as 2 pi to 3 pi so that 00:17:27.200 --> 00:17:28.129 everyone's very happy. 00:17:28.129 --> 00:17:29.379 OK? 00:17:34.920 --> 00:17:38.240 So again, what do we do? 00:17:38.240 --> 00:17:39.460 These 2's divide out. 00:17:39.460 --> 00:17:43.370 So from 0 to pi of r squared d theta, that's going to give me 00:17:43.370 --> 00:17:46.890 the area of the blue plus the pink. 00:17:46.890 --> 00:17:51.010 And then 2 pi to 3 pi of 1 plus cosine over theta over 2 00:17:51.010 --> 00:17:54.510 squared d theta is going to give me the area of the pink. 00:17:54.510 --> 00:17:58.600 So the blue plus the pink is over here, and then the pink 00:17:58.600 --> 00:17:59.320 is over here. 00:17:59.320 --> 00:18:02.280 So when I subtract that off I just get the blue. 00:18:02.280 --> 00:18:03.030 OK. 00:18:03.030 --> 00:18:06.220 Let me, again, just go back one more time and point out 00:18:06.220 --> 00:18:08.190 what we did at the very beginning to remind us what 00:18:08.190 --> 00:18:10.540 was happening. 00:18:10.540 --> 00:18:11.540 And then I will finish. 00:18:11.540 --> 00:18:13.700 So let's come back over here. 00:18:13.700 --> 00:18:18.190 So the idea was to graph this curve that was in r is a 00:18:18.190 --> 00:18:19.620 function of theta. 00:18:19.620 --> 00:18:21.620 And I was supposed to understand what that looked 00:18:21.620 --> 00:18:24.140 like in the x, y plane. 00:18:24.140 --> 00:18:30.680 And so my trick was to take the relationship between r and 00:18:30.680 --> 00:18:33.050 theta and graph that explicitly 00:18:33.050 --> 00:18:35.160 in an r theta plane-- 00:18:35.160 --> 00:18:37.370 so I let theta vary in the horizontal direction and r 00:18:37.370 --> 00:18:39.090 vary in the vertical direction-- 00:18:39.090 --> 00:18:41.160 and I can do that very easily. 00:18:41.160 --> 00:18:44.830 And then translate that into the x, y plane. 00:18:44.830 --> 00:18:49.380 So my curve, again, went the big part, the little part 00:18:49.380 --> 00:18:52.390 here, little part here, big part here. 00:18:52.390 --> 00:18:53.760 That was the order. 00:18:53.760 --> 00:18:58.045 So if you need arrows on it, this was the order. 00:19:00.820 --> 00:19:04.890 And then once I had that, the problem was about areas. 00:19:04.890 --> 00:19:08.340 And there was a lot of symmetry in this problem, but 00:19:08.340 --> 00:19:13.560 the main point I wanted to show was just knowing where 00:19:13.560 --> 00:19:16.570 your theta starts and stops is not enough to determine an 00:19:16.570 --> 00:19:20.520 area of a region if that region is excluding some part 00:19:20.520 --> 00:19:22.780 that would be potentially counted twice. 00:19:22.780 --> 00:19:25.720 So that was the reason I wanted you to calculate not 00:19:25.720 --> 00:19:30.190 just the pink area, but also see that the blue is not from 00:19:30.190 --> 00:19:33.510 theta from 0 to pi, but you have to subtract off this 00:19:33.510 --> 00:19:36.230 extra stuff that you counted. 00:19:36.230 --> 00:19:39.270 And I guess that is where I will stop.