1 00:00:06,690 --> 00:00:08,606 CHRISTINE BREINER: Welcome back to recitation. 2 00:00:08,606 --> 00:00:12,210 In this video I'd like us to work on the following problem. 3 00:00:12,210 --> 00:00:14,220 We're going to let capital D denote 4 00:00:14,220 --> 00:00:18,470 the portion of the solid sphere of radius 1 that's centered at 5 00:00:18,470 --> 00:00:23,110 (0, 0, 1), which also lies about the plane z equal 1. 6 00:00:23,110 --> 00:00:26,090 And then I'd like us to first supply the limits for D 7 00:00:26,090 --> 00:00:27,610 in spherical coordinates. 8 00:00:27,610 --> 00:00:30,370 In other words, I want you to determine the values 9 00:00:30,370 --> 00:00:34,560 for rho, theta, and phi that will give us all of D. 10 00:00:34,560 --> 00:00:36,490 And then, I would like us to just set up 11 00:00:36,490 --> 00:00:39,300 the integral for the average distance of a point in D 12 00:00:39,300 --> 00:00:40,540 from the origin. 13 00:00:40,540 --> 00:00:42,360 So there are two parts to this problem. 14 00:00:42,360 --> 00:00:46,100 The first is to determine what values of rho, theta, and phi 15 00:00:46,100 --> 00:00:51,200 describe this solid region D. And then second 16 00:00:51,200 --> 00:00:54,940 is just set up the integral for the average distance of a point 17 00:00:54,940 --> 00:00:57,590 in that region from the origin. 18 00:00:57,590 --> 00:00:59,670 So why don't you pause the video, work on those, 19 00:00:59,670 --> 00:01:01,490 and then when you're ready to see my solutions, 20 00:01:01,490 --> 00:01:02,823 you can bring the video back up. 21 00:01:12,000 --> 00:01:13,230 OK, welcome back. 22 00:01:13,230 --> 00:01:16,870 Well, I would say from looking at this problem, actually part 23 00:01:16,870 --> 00:01:20,330 a is potentially a little bit more hazardous for some of us 24 00:01:20,330 --> 00:01:21,380 than part b. 25 00:01:21,380 --> 00:01:25,310 Once we know the bounds that describe D, 26 00:01:25,310 --> 00:01:27,950 it's not too hard to set up this integral. 27 00:01:27,950 --> 00:01:29,590 So the hard part of this problem is 28 00:01:29,590 --> 00:01:34,690 understanding how to write this solid region D in spherical 29 00:01:34,690 --> 00:01:35,460 coordinates. 30 00:01:35,460 --> 00:01:37,460 But it's actually really not that hard either, 31 00:01:37,460 --> 00:01:40,429 so I'm going to try and take us through it in a reasonable way. 32 00:01:40,429 --> 00:01:41,970 So the first thing: I'm going to draw 33 00:01:41,970 --> 00:01:44,441 a very rough picture of the region 34 00:01:44,441 --> 00:01:45,940 so we understand what it looks like. 35 00:01:45,940 --> 00:01:47,490 So in order to do part a. 36 00:01:47,490 --> 00:01:49,460 And if you did not do this, I would highly 37 00:01:49,460 --> 00:01:51,780 recommend that next time you encounter such a problem, 38 00:01:51,780 --> 00:01:54,100 you begin by drawing yourself a picture. 39 00:01:54,100 --> 00:01:55,740 Even if it's not a great picture, 40 00:01:55,740 --> 00:01:59,010 it will give you some intuition about what's happening. 41 00:01:59,010 --> 00:02:03,728 So let me first draw the axes. 42 00:02:03,728 --> 00:02:05,630 And ultimately what I have-- say, 43 00:02:05,630 --> 00:02:08,010 here is the point (0, 0, 1). 44 00:02:08,010 --> 00:02:16,390 I have a sphere that's looking-- in the zy-plane, 45 00:02:16,390 --> 00:02:18,970 it looks something like this. 46 00:02:18,970 --> 00:02:23,120 And so it has this depth here. 47 00:02:23,120 --> 00:02:25,660 And that's the solid sphere I want to be considering. 48 00:02:25,660 --> 00:02:27,870 And then I'm going to be removing the bottom half. 49 00:02:27,870 --> 00:02:31,320 I'm only going to be looking at the part that is above the z 50 00:02:31,320 --> 00:02:32,540 equals 1 plane. 51 00:02:32,540 --> 00:02:38,260 So actually it's going to be all of the circular slices that 52 00:02:38,260 --> 00:02:42,500 are in the top half, the upper hemisphere of this sphere. 53 00:02:42,500 --> 00:02:44,810 And so it's this solid region there. 54 00:02:44,810 --> 00:02:46,240 That's D. 55 00:02:46,240 --> 00:02:48,840 And what I want to do is to determine 56 00:02:48,840 --> 00:02:51,480 rho and theta and phi. 57 00:02:51,480 --> 00:02:53,800 Actually, the theta is the easy one, right? 58 00:02:53,800 --> 00:02:58,837 Because theta, at each value here, I go all the way around 59 00:02:58,837 --> 00:02:59,795 in the theta direction. 60 00:03:02,860 --> 00:03:07,360 At any given height or radius, I want to go all the way around 61 00:03:07,360 --> 00:03:09,120 in the theta direction from 0 to 2*pi. 62 00:03:09,120 --> 00:03:12,670 So my theta bounds are the easy ones. 63 00:03:12,670 --> 00:03:16,200 I'm covering 0 to 2*pi in theta. 64 00:03:16,200 --> 00:03:23,760 Because if I cut off the back half of the sphere, 65 00:03:23,760 --> 00:03:26,240 I want to only have a restricted value of theta. 66 00:03:26,240 --> 00:03:29,330 But because I'm covering all the way around 67 00:03:29,330 --> 00:03:31,890 and my restriction is only in the bottom half, 68 00:03:31,890 --> 00:03:33,540 my theta values haven't changed. 69 00:03:33,540 --> 00:03:38,870 So theta is easy: 0 and 2*pi. 70 00:03:38,870 --> 00:03:43,030 Now the harder ones are going to be rho and phi. 71 00:03:43,030 --> 00:03:46,730 But in fact, actually, phi is not that hard either. 72 00:03:46,730 --> 00:03:50,230 I notice phi is the angle that I make 73 00:03:50,230 --> 00:03:52,795 from the z-axis to any given point. 74 00:03:52,795 --> 00:03:55,170 So I notice that I certainly am including the point where 75 00:03:55,170 --> 00:03:58,680 phi is 0, and then I'm going all the way down to a point 76 00:03:58,680 --> 00:04:02,380 out here, which is a 45 degree angle with the z-axis. 77 00:04:02,380 --> 00:04:03,930 So phi is also easy. 78 00:04:03,930 --> 00:04:06,850 It's actually just between 0 and pi over 4. 79 00:04:13,780 --> 00:04:16,180 And the rho will be the slightly harder part. 80 00:04:16,180 --> 00:04:18,210 So that's really the only really tricky part 81 00:04:18,210 --> 00:04:21,070 in this problem is determining the rho value. 82 00:04:21,070 --> 00:04:24,540 Now, the rho value, to get the outer boundary, 83 00:04:24,540 --> 00:04:27,430 we'll look at that part first. 84 00:04:27,430 --> 00:04:30,010 Well, the boundary of this sphere 85 00:04:30,010 --> 00:04:34,041 here has a certain equation in x, y, and z that we know, 86 00:04:34,041 --> 00:04:34,540 right? 87 00:04:34,540 --> 00:04:38,710 It's x squared plus y squared plus the quantity z minus 1 88 00:04:38,710 --> 00:04:41,360 squared equals 1. 89 00:04:41,360 --> 00:04:44,040 I mean, that's just the equation for a sphere of radius 90 00:04:44,040 --> 00:04:45,711 1 centered at (0, 0, 1). 91 00:04:45,711 --> 00:04:47,460 So I'm going to write that here, and we're 92 00:04:47,460 --> 00:04:49,360 going to show how we can manipulate that. 93 00:04:55,430 --> 00:04:56,310 Right? 94 00:04:56,310 --> 00:04:58,730 x squared plus y squared is r squared. 95 00:04:58,730 --> 00:05:02,290 r squared is rho squared sine squared phi. 96 00:05:02,290 --> 00:05:07,180 So I can replace this by rho squared sine squared phi. 97 00:05:07,180 --> 00:05:08,750 If you didn't know that immediately, 98 00:05:08,750 --> 00:05:10,770 you could make the substitution for x and y 99 00:05:10,770 --> 00:05:13,990 in spherical coordinates, and it simplifies to this. 100 00:05:13,990 --> 00:05:17,500 So either way, if you didn't know r squared was this, 101 00:05:17,500 --> 00:05:20,050 you can get it from just doing the substitution. 102 00:05:20,050 --> 00:05:24,020 And then z is going to be rho cosine phi. 103 00:05:24,020 --> 00:05:28,430 So here, I'm going to have a rho cosine phi minus 1 104 00:05:28,430 --> 00:05:30,650 quantity squared equals 1. 105 00:05:30,650 --> 00:05:33,330 This, in the spherical coordinates, 106 00:05:33,330 --> 00:05:38,490 is describing the boundary of this entire sphere, right? 107 00:05:38,490 --> 00:05:40,780 And so I can actually simplify this. 108 00:05:40,780 --> 00:05:41,980 It's not too hard. 109 00:05:41,980 --> 00:05:45,690 If I square this, I get a little cancellation. 110 00:05:45,690 --> 00:05:48,540 And then because I want my rho to be-- I'm 111 00:05:48,540 --> 00:05:51,070 assuming in this region, rho is greater than 0-- 112 00:05:51,070 --> 00:05:52,450 I can do a little simplification. 113 00:05:52,450 --> 00:05:58,112 I come up with the fact that rho is equal to 2 cosine phi. 114 00:05:58,112 --> 00:06:00,070 And let's describe exactly where that is. 115 00:06:00,070 --> 00:06:04,690 That's the entire boundary of this entire sphere 116 00:06:04,690 --> 00:06:08,470 is described by rho is equal to 2 cosine phi. 117 00:06:08,470 --> 00:06:10,810 And so I want to think about what my bounds are for rho. 118 00:06:10,810 --> 00:06:14,620 Actually, I'm going to grab a piece of colored chalk. 119 00:06:14,620 --> 00:06:17,410 If I start at the origin, I think about what is rho? 120 00:06:17,410 --> 00:06:20,450 So say this is a point on the boundary of the sphere. 121 00:06:20,450 --> 00:06:23,140 I am going to start my rho value-- whatever 122 00:06:23,140 --> 00:06:25,930 it is when it hits the plane z equals 1-- 123 00:06:25,930 --> 00:06:27,490 and I'm going to stop it when it hits 124 00:06:27,490 --> 00:06:29,380 the boundary of this sphere. 125 00:06:29,380 --> 00:06:33,430 So my outer boundary for rho is going to be this value. 126 00:06:33,430 --> 00:06:36,830 It's going to be determined by phi, right? 127 00:06:36,830 --> 00:06:40,582 And now I have to determine my inner boundary, right? 128 00:06:40,582 --> 00:06:42,540 And my inner boundary is actually quite simple. 129 00:06:42,540 --> 00:06:44,630 It's a very simple geometric thing. 130 00:06:44,630 --> 00:06:47,010 And so my inner boundary deals with the fact 131 00:06:47,010 --> 00:06:50,590 that if this is my plane z equals 1, 132 00:06:50,590 --> 00:06:54,300 and I look at this triangle I make right here. 133 00:06:54,300 --> 00:06:56,980 This angle down here, the bottom angle is phi, 134 00:06:56,980 --> 00:07:00,360 and this is a right angle, and the rho value I'm interested in 135 00:07:00,360 --> 00:07:02,430 is this hypotenuse, right? 136 00:07:02,430 --> 00:07:06,090 I need to figure out what the length of this is right here. 137 00:07:06,090 --> 00:07:09,610 And you can see it right away from just the fact 138 00:07:09,610 --> 00:07:18,360 that phi is this angle here, you get rho is secant phi. 139 00:07:18,360 --> 00:07:22,600 So the bottom boundary comes from just simple geometry. 140 00:07:22,600 --> 00:07:25,290 You get this length is 1 here. 141 00:07:25,290 --> 00:07:28,190 So you get rho is equal to secant phi, right? 142 00:07:28,190 --> 00:07:30,239 This length here is 1, this is the rho 143 00:07:30,239 --> 00:07:31,905 I'm interested in-- the blue part here-- 144 00:07:31,905 --> 00:07:34,830 and so rho is equal to secant phi is the lower bound. 145 00:07:34,830 --> 00:07:39,080 And it's equal to 2 cosine phi at the upper bound, OK? 146 00:07:39,080 --> 00:07:40,670 And the thing I want to be careful of 147 00:07:40,670 --> 00:07:43,820 is I'm not supposed to include-- it won't matter 148 00:07:43,820 --> 00:07:47,680 for the integral-- but I'm not supposed to include the plane. 149 00:07:47,680 --> 00:07:49,440 Let me write this, and make sure. 150 00:07:49,440 --> 00:07:53,780 Rho is going to be greater than secant phi 151 00:07:53,780 --> 00:08:00,540 and it's going to be less than or equal to 2 cosine phi, 152 00:08:00,540 --> 00:08:02,990 right? 153 00:08:02,990 --> 00:08:03,490 Right? 154 00:08:03,490 --> 00:08:06,031 So let me double-check and make sure I didn't make a geometry 155 00:08:06,031 --> 00:08:07,650 mistake here, just to be sure. 156 00:08:07,650 --> 00:08:10,710 This picture tells me that cosine phi 157 00:08:10,710 --> 00:08:12,740 is equal to 1 over rho. 158 00:08:12,740 --> 00:08:13,250 That's good. 159 00:08:13,250 --> 00:08:15,160 So rho is equal to 1 over cosine phi. 160 00:08:15,160 --> 00:08:17,000 So I get secant phi there. 161 00:08:17,000 --> 00:08:21,940 So my rho values start at the secant phi length 162 00:08:21,940 --> 00:08:23,960 and they go to the 2 cosine phi length. 163 00:08:23,960 --> 00:08:25,980 I know maybe I'm beating a dead horse here, 164 00:08:25,980 --> 00:08:27,771 but I want to make sure we understand where 165 00:08:27,771 --> 00:08:29,070 the rho values are coming from. 166 00:08:29,070 --> 00:08:30,986 So actually, I have all the bounds I need now. 167 00:08:30,986 --> 00:08:34,100 I have the theta bounds, and I have the phi bounds, 168 00:08:34,100 --> 00:08:35,590 and the rho bounds. 169 00:08:35,590 --> 00:08:38,970 Now you notice that theta and phi 170 00:08:38,970 --> 00:08:40,470 don't depend on the other variables, 171 00:08:40,470 --> 00:08:42,680 but rho depends on phi. 172 00:08:42,680 --> 00:08:45,790 So we're going to have to integrate that first. 173 00:08:45,790 --> 00:08:48,867 So now we can deal with part b. 174 00:08:48,867 --> 00:08:50,950 Part b-- let's come back over and remind ourselves 175 00:08:50,950 --> 00:08:53,350 what it said-- said set up the integral 176 00:08:53,350 --> 00:08:57,600 for the average distance of a point in D from the origin. 177 00:08:57,600 --> 00:09:00,080 So I'm taking the average value of a function. 178 00:09:00,080 --> 00:09:02,190 What is that function I'm averaging? 179 00:09:02,190 --> 00:09:04,474 How do I find the distance from the origin? 180 00:09:04,474 --> 00:09:05,890 Well, the distance from the origin 181 00:09:05,890 --> 00:09:08,098 is a great function to have in spherical coordinates, 182 00:09:08,098 --> 00:09:09,540 because it's just rho. 183 00:09:09,540 --> 00:09:12,180 So the function I'm supposed to average over 184 00:09:12,180 --> 00:09:13,880 is the function rho. 185 00:09:13,880 --> 00:09:15,880 In spherical coordinates, that's the function. 186 00:09:15,880 --> 00:09:18,010 So let me write down what we're going to have here. 187 00:09:18,010 --> 00:09:27,050 So in part b, the average distance 188 00:09:27,050 --> 00:09:34,330 is going to equal 1 divided by the volume of D 189 00:09:34,330 --> 00:09:44,000 times the triple integral over D of the function rho dV, OK? 190 00:09:44,000 --> 00:09:47,520 So now I have to write dV in the spherical coordinates, 191 00:09:47,520 --> 00:09:50,802 and I have to write D in the spherical coordinates bounds. 192 00:09:50,802 --> 00:09:53,010 And then I know I have to figure out the volume of D. 193 00:09:53,010 --> 00:09:54,640 So we're going to figure out each of these things, 194 00:09:54,640 --> 00:09:56,012 and then we'll be done. 195 00:09:56,012 --> 00:09:56,600 All right. 196 00:09:56,600 --> 00:09:58,650 So first, what is the volume of D? 197 00:09:58,650 --> 00:10:01,080 Well, the volume of D, let's think about what it is. 198 00:10:01,080 --> 00:10:06,180 It's a sphere of radius 1. 199 00:10:06,180 --> 00:10:13,400 And so the volume of a sphere of radius 1 is 4/3 pi r cubed. 200 00:10:13,400 --> 00:10:14,740 And I want half of that. 201 00:10:14,740 --> 00:10:16,260 So I want 2/3. 202 00:10:16,260 --> 00:10:19,700 Since my radius is 1, I just have to do 2/3 pi. 203 00:10:19,700 --> 00:10:24,980 So the first part is 1 divided by 2/3 pi. 204 00:10:24,980 --> 00:10:28,790 That's the volume of a half-sphere of radius 1. 205 00:10:28,790 --> 00:10:29,855 And now let's integrate. 206 00:10:33,220 --> 00:10:36,502 I'll leave a little space to write my bounds. 207 00:10:36,502 --> 00:10:37,960 I'm going to write the bounds last, 208 00:10:37,960 --> 00:10:40,080 after I have everything in order over here. 209 00:10:40,080 --> 00:10:45,490 dV is rho squared sine phi d rho d theta d phi. 210 00:10:45,490 --> 00:10:51,680 So I'm going to end up with a rho cubed sine phi d rho d 211 00:10:51,680 --> 00:10:55,750 theta d phi, right? 212 00:10:55,750 --> 00:11:01,030 The dV gave me an extra rho squared and a sine phi. 213 00:11:01,030 --> 00:11:02,830 That whole part is dV. 214 00:11:02,830 --> 00:11:05,660 And then I keep one rho from the fact 215 00:11:05,660 --> 00:11:08,100 that the distance function is rho. 216 00:11:08,100 --> 00:11:10,181 And so I get a rho cubed there. 217 00:11:10,181 --> 00:11:11,430 So hopefully that makes sense. 218 00:11:11,430 --> 00:11:19,025 Now for d rho, I know the bounds are secant phi to 2 cosine phi. 219 00:11:19,025 --> 00:11:22,820 For d theta, my bounds are 0 to 2 pi. 220 00:11:22,820 --> 00:11:25,930 And for d phi, my bounds were 0 to pi over 4. 221 00:11:28,660 --> 00:11:32,180 I didn't make you evaluate it, I'm just making you set it up. 222 00:11:32,180 --> 00:11:35,510 That actually is the solution we wanted for part b. 223 00:11:35,510 --> 00:11:38,340 I wanted to average the distance from any point in D 224 00:11:38,340 --> 00:11:39,520 to the origin. 225 00:11:39,520 --> 00:11:42,600 So I just took the average value of the function rho 226 00:11:42,600 --> 00:11:46,460 over that region D. And so that's how you finish that up. 227 00:11:46,460 --> 00:11:49,190 And so in this problem, basically we 228 00:11:49,190 --> 00:11:51,030 want you to get really familiar with how 229 00:11:51,030 --> 00:11:54,800 to do some things in these spherical coordinates, which 230 00:11:54,800 --> 00:11:56,860 are sometimes a little hard to do. 231 00:11:56,860 --> 00:11:59,450 But if you noticed, what we were doing in trying to figure out 232 00:11:59,450 --> 00:12:02,150 the bounds-- in particular, trying to figure out rho-- 233 00:12:02,150 --> 00:12:04,210 we took what we knew in the x-, y-, 234 00:12:04,210 --> 00:12:06,940 z-coordinates about certain relationships, 235 00:12:06,940 --> 00:12:11,640 and then we replaced the x-, y-, z-values by the values in terms 236 00:12:11,640 --> 00:12:16,730 of rho and theta and phi, and you can simplify to figure out 237 00:12:16,730 --> 00:12:19,880 the relationships you have between rho and theta and phi 238 00:12:19,880 --> 00:12:21,410 for the boundary value. 239 00:12:21,410 --> 00:12:24,390 So that was one of the techniques we were using there. 240 00:12:24,390 --> 00:12:28,430 And hopefully, the geometric understanding of why these 241 00:12:28,430 --> 00:12:32,130 angles go from 0 to 2*pi and 0 to pi over 4 is clear. 242 00:12:32,130 --> 00:12:35,080 And actually, the fact that this is rho 243 00:12:35,080 --> 00:12:36,650 equals 2 cosine phi should remind you 244 00:12:36,650 --> 00:12:39,230 of the two-dimensional case where you had some problem 245 00:12:39,230 --> 00:12:41,510 like r equals 2 cosine theta. 246 00:12:41,510 --> 00:12:46,140 And that drew a circle off-center from the origin. 247 00:12:46,140 --> 00:12:50,199 It's the analogous thing happening here. 248 00:12:50,199 --> 00:12:52,490 Maybe I should stop there before I say too many things. 249 00:12:52,490 --> 00:12:54,070 But again, the object of this was just 250 00:12:54,070 --> 00:12:55,880 to get really comfortable with spherical coordinates, 251 00:12:55,880 --> 00:12:57,100 and I hope it's helped you. 252 00:12:57,100 --> 00:12:58,700 I'll stop there.