1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,950 Commons license. 3 00:00:03,950 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:09,465 continue to offer high quality educational resources for free. 5 00:00:09,465 --> 00:00:12,550 To make a donation, or to view additional materials 6 00:00:12,550 --> 00:00:15,850 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,850 --> 00:00:21,750 at ocw.mit.edu 8 00:00:21,750 --> 00:00:27,110 PROFESSOR: Today, I'm going to continue the idea of setting up 9 00:00:27,110 --> 00:00:29,790 integrals. 10 00:00:29,790 --> 00:00:39,090 And what we'll deal with is volumes by slices. 11 00:00:39,090 --> 00:00:47,240 By slicing. 12 00:00:47,240 --> 00:00:52,010 And it's lucky that this is after lunch. 13 00:00:52,010 --> 00:00:53,960 Maybe it's after breakfast for some of you, 14 00:00:53,960 --> 00:00:59,050 because there's the typical way of introducing this subject is 15 00:00:59,050 --> 00:01:00,380 with a food analogy. 16 00:01:00,380 --> 00:01:02,790 There's a lot of ways of slicing up food. 17 00:01:02,790 --> 00:01:05,420 And we'll give a few more examples than just this one. 18 00:01:05,420 --> 00:01:09,560 But, suppose you have, well, suppose 19 00:01:09,560 --> 00:01:13,870 you have a loaf of bread here. 20 00:01:13,870 --> 00:01:21,077 So here's our loaf of bread, and I 21 00:01:21,077 --> 00:01:23,160 hope that looks a little bit like a loaf of bread. 22 00:01:23,160 --> 00:01:25,670 It's supposed to be sitting on the kitchen 23 00:01:25,670 --> 00:01:28,920 counter ready to be eaten. 24 00:01:28,920 --> 00:01:32,260 And in order to figure out how much bread there is there, 25 00:01:32,260 --> 00:01:35,290 one way of doing it is to cut it into slices. 26 00:01:35,290 --> 00:01:37,980 Now, you probably know that bread is often 27 00:01:37,980 --> 00:01:40,020 sliced like this. 28 00:01:40,020 --> 00:01:42,080 There are even machines to do it. 29 00:01:42,080 --> 00:01:48,450 And with this setup here, I'll draw the slice 30 00:01:48,450 --> 00:01:52,830 with a little bit of a more colorful decoration. 31 00:01:52,830 --> 00:01:57,980 So here's our red slice of bread. 32 00:01:57,980 --> 00:02:00,150 It's coming around like this. 33 00:02:00,150 --> 00:02:03,280 And it comes back down behind. 34 00:02:03,280 --> 00:02:06,960 So here's our bread slice. 35 00:02:06,960 --> 00:02:21,930 And what I'd like to figure out is its volume. 36 00:02:21,930 --> 00:02:24,850 So first of all, there's the thickness of the bread. 37 00:02:24,850 --> 00:02:26,950 Which is this dimension, the thickness 38 00:02:26,950 --> 00:02:35,490 is this dimension dx here. 39 00:02:35,490 --> 00:02:37,730 And the only other dimension that I'm going to give, 40 00:02:37,730 --> 00:02:40,900 because this is a very qualitative analysis for now, 41 00:02:40,900 --> 00:02:47,610 is what I'll call the area. 42 00:02:47,610 --> 00:02:52,660 And that's the area on the face of the slice. 43 00:02:52,660 --> 00:02:57,150 And so the area of one slice, which 44 00:02:57,150 --> 00:03:00,240 I'll denote by delta V, that's a chunk of volume, 45 00:03:00,240 --> 00:03:05,430 is approximately the area times the change in x. 46 00:03:05,430 --> 00:03:13,630 And in the limit, that's going to be something like this. 47 00:03:13,630 --> 00:03:16,029 And maybe the areas of the slices vary. 48 00:03:16,029 --> 00:03:18,320 There might be a little hole in the middle of the bread 49 00:03:18,320 --> 00:03:18,819 somewhere. 50 00:03:18,819 --> 00:03:21,010 Maybe it gets a little small on one side. 51 00:03:21,010 --> 00:03:24,170 So it might change as x changes. 52 00:03:24,170 --> 00:03:31,760 And the whole volume you get by adding up. 53 00:03:31,760 --> 00:03:37,580 So if you like, this is one slice. 54 00:03:37,580 --> 00:03:40,830 And this is the sum. 55 00:03:40,830 --> 00:03:44,810 And you should think of it in a sort of intuitive way 56 00:03:44,810 --> 00:03:47,070 as being analogous to the Riemann sum, where you would 57 00:03:47,070 --> 00:03:49,390 take each slice individually. 58 00:03:49,390 --> 00:03:54,300 And that would look like this. 59 00:03:54,300 --> 00:04:01,720 Alright, so that's just a superficial and intuitive way 60 00:04:01,720 --> 00:04:02,830 of looking at it. 61 00:04:02,830 --> 00:04:04,810 Now, we're only going to talk about one 62 00:04:04,810 --> 00:04:06,325 kind of systematic slice. 63 00:04:06,325 --> 00:04:07,960 It's already on your problem set, 64 00:04:07,960 --> 00:04:11,810 you had an example of a slice of some region. 65 00:04:11,810 --> 00:04:14,990 But we're only going to talk systematically about something 66 00:04:14,990 --> 00:04:32,330 called solids of revolution. 67 00:04:32,330 --> 00:04:34,430 The idea here is this. 68 00:04:34,430 --> 00:04:37,930 Suppose you have some shape, some graph, 69 00:04:37,930 --> 00:04:40,700 which maybe looks like this. 70 00:04:40,700 --> 00:04:44,220 And now I'm going to revolve it. 71 00:04:44,220 --> 00:04:46,880 This is the x-axis and this is the y-axis. 72 00:04:46,880 --> 00:04:58,470 In this case, I'm going to revolve it around the x-axis. 73 00:04:58,470 --> 00:05:04,760 If you do that, then the shape that you get 74 00:05:04,760 --> 00:05:08,310 is maybe like this. 75 00:05:08,310 --> 00:05:09,930 If I can draw it a little bit. 76 00:05:09,930 --> 00:05:12,600 It's maybe a football. 77 00:05:12,600 --> 00:05:17,015 So that's the shape that you get if you take this piece of disk 78 00:05:17,015 --> 00:05:18,540 and you revolve it around. 79 00:05:18,540 --> 00:05:20,500 If you had made this copy underneath, 80 00:05:20,500 --> 00:05:22,360 it still would have been the same region. 81 00:05:22,360 --> 00:05:27,950 So we only pay attention to what's above the axis here. 82 00:05:27,950 --> 00:05:30,600 So that's the basic idea. 83 00:05:30,600 --> 00:05:33,530 Now, I'm going to apply the method of slices 84 00:05:33,530 --> 00:05:35,940 to figure out the volume of such regions 85 00:05:35,940 --> 00:05:37,510 and give you a general formula. 86 00:05:37,510 --> 00:05:40,240 And then apply it in a specific case. 87 00:05:40,240 --> 00:05:44,254 I want to take one little slice of this football, 88 00:05:44,254 --> 00:05:45,920 maybe a football wouldn't work too well, 89 00:05:45,920 --> 00:05:50,380 maybe we should go back to a loaf of bread. 90 00:05:50,380 --> 00:05:55,180 Anyway, the key point is that you never really 91 00:05:55,180 --> 00:05:58,270 have to draw a 3-D picture. 92 00:05:58,270 --> 00:05:59,820 And 3-D pictures are awful. 93 00:05:59,820 --> 00:06:01,600 They're very hard to deal with. 94 00:06:01,600 --> 00:06:03,460 And it's hard to visualize with them. 95 00:06:03,460 --> 00:06:06,010 And one of the reasons why we're dealing with solids 96 00:06:06,010 --> 00:06:10,020 of revolution is that we don't have as many visualization 97 00:06:10,020 --> 00:06:10,730 problems. 98 00:06:10,730 --> 00:06:13,170 So we're only going to deal with this. 99 00:06:13,170 --> 00:06:15,240 And then you have to imagine from 100 00:06:15,240 --> 00:06:17,902 the two-dimensional cross-section what 101 00:06:17,902 --> 00:06:19,610 the three-dimensional picture looks like. 102 00:06:19,610 --> 00:06:21,372 So we'll do a little bit of an exercise 103 00:06:21,372 --> 00:06:22,830 with the three-dimensional picture. 104 00:06:22,830 --> 00:06:24,990 But ultimately, you should be used to, 105 00:06:24,990 --> 00:06:27,770 getting used to, drawing 2-D diagrams always. 106 00:06:27,770 --> 00:06:29,710 To depict the three-dimensional situation. 107 00:06:29,710 --> 00:06:34,310 Since it's much harder to draw. 108 00:06:34,310 --> 00:06:38,540 The first step is to consider what this slice is over here. 109 00:06:38,540 --> 00:06:45,400 And again it's going to have width dx. 110 00:06:45,400 --> 00:06:49,630 And we're going to consider what it looks like over 111 00:06:49,630 --> 00:06:51,560 on the 3-D picture. 112 00:06:51,560 --> 00:06:54,980 So it starts out being more or less like this. 113 00:06:54,980 --> 00:06:57,210 But then we're going to sweep it around. 114 00:06:57,210 --> 00:06:59,280 We're revolving around the x-axis, 115 00:06:59,280 --> 00:07:01,370 so it's spinning around this way. 116 00:07:01,370 --> 00:07:03,210 And if you take this and think of it 117 00:07:03,210 --> 00:07:06,130 as being on a hinge, which is down on the x-axis, 118 00:07:06,130 --> 00:07:09,170 it's going to swing down and swoop around and come back. 119 00:07:09,170 --> 00:07:10,750 Swing around. 120 00:07:10,750 --> 00:07:14,140 And that traces out something over here. 121 00:07:14,140 --> 00:07:15,900 Which I'm going to draw this way. 122 00:07:15,900 --> 00:07:22,960 It traces out a disk. 123 00:07:22,960 --> 00:07:29,050 So it's hard for me to draw, and I'm not going to try too hard. 124 00:07:29,050 --> 00:07:32,240 Maybe I drew it more like a wheel, looking like a wheel. 125 00:07:32,240 --> 00:07:35,850 But anyway, it's this little flat disk here. 126 00:07:35,850 --> 00:07:38,110 And so the method that I'm describing 127 00:07:38,110 --> 00:07:44,050 for figuring out the volume is called the method of disks. 128 00:07:44,050 --> 00:07:56,910 This is going to be our first method. 129 00:07:56,910 --> 00:07:58,630 Now I'm going to apply the reasoning 130 00:07:58,630 --> 00:08:02,130 that I have up on the previous blackboard here. 131 00:08:02,130 --> 00:08:06,180 Namely, I need to get the volume of this chunk. 132 00:08:06,180 --> 00:08:08,850 And the way I'm going to get the volume of this chunk 133 00:08:08,850 --> 00:08:13,225 is by figuring out its thickness and its area, 134 00:08:13,225 --> 00:08:14,880 its cross-sectional area. 135 00:08:14,880 --> 00:08:19,490 And that's not too difficult to do. 136 00:08:19,490 --> 00:08:25,810 If this height, so this height is what we usually call y. 137 00:08:25,810 --> 00:08:28,140 And y is usually a function of x, it's varying. 138 00:08:28,140 --> 00:08:31,030 And this particular distance is y. 139 00:08:31,030 --> 00:08:35,500 Then the area of the face is easy to calculate. 140 00:08:35,500 --> 00:08:42,100 Because it's a circle, or a disk if you like, with this radius. 141 00:08:42,100 --> 00:08:48,540 So its area is pi y^2. 142 00:08:48,540 --> 00:08:53,900 So that's, if you like, one of the dimensions. 143 00:08:53,900 --> 00:08:59,190 And then the thickness is dx. 144 00:08:59,190 --> 00:09:03,360 So the incremental volume is this. 145 00:09:03,360 --> 00:09:09,140 So this is the method of disks. 146 00:09:09,140 --> 00:09:11,480 And this is the integrand. 147 00:09:11,480 --> 00:09:14,410 Now, there's one peculiar thing about this formula. 148 00:09:14,410 --> 00:09:17,065 And there are more peculiar things about this formula. 149 00:09:17,065 --> 00:09:18,690 But there's one peculiar thing that you 150 00:09:18,690 --> 00:09:20,130 should notice immediately. 151 00:09:20,130 --> 00:09:22,840 Which is that I'm integrating with respect to x. 152 00:09:22,840 --> 00:09:26,260 And I haven't yet told you what y is. 153 00:09:26,260 --> 00:09:32,280 Well, that will depend on what function y = f(x) I use. 154 00:09:32,280 --> 00:09:34,222 So we have to plug that in eventually. 155 00:09:34,222 --> 00:09:36,180 If we're actually going to calculate something, 156 00:09:36,180 --> 00:09:37,700 we're going to have to figure that out. 157 00:09:37,700 --> 00:09:39,070 There's another very important point 158 00:09:39,070 --> 00:09:41,340 which is that in order to get a definite integral, something 159 00:09:41,340 --> 00:09:42,560 I haven't mentioned, we're going to have 160 00:09:42,560 --> 00:09:44,340 to figure out where we're starting 161 00:09:44,340 --> 00:09:46,050 and we're ending the picture. 162 00:09:46,050 --> 00:09:52,250 Which is something we dealt with last time in 2-D pictures. 163 00:09:52,250 --> 00:09:55,400 So let's deal with an example. 164 00:09:55,400 --> 00:10:05,220 And we'll switch over from a football to a soccer ball. 165 00:10:05,220 --> 00:10:10,800 I'm going to take a circle, and we'll say it has radius a. 166 00:10:10,800 --> 00:10:13,750 So this is 0 and this is a. 167 00:10:13,750 --> 00:10:17,260 And I'm putting it in this particular spot for a reason. 168 00:10:17,260 --> 00:10:19,820 You can do this in lots of different ways, 169 00:10:19,820 --> 00:10:22,990 but I'm picking this one to make a certain exercise 170 00:10:22,990 --> 00:10:24,970 on your homework easier for you. 171 00:10:24,970 --> 00:10:31,330 Because I'm doing half of it for you right now. 172 00:10:31,330 --> 00:10:32,340 Appreciate it, yeah. 173 00:10:32,340 --> 00:10:36,180 I'm sure especially today it's appreciated. 174 00:10:36,180 --> 00:10:40,330 So again, the formula has to do with keeping 175 00:10:40,330 --> 00:10:43,450 track of these slices here. 176 00:10:43,450 --> 00:10:45,830 And we're sweeping things around. 177 00:10:45,830 --> 00:10:48,940 So the full region that we're talking about 178 00:10:48,940 --> 00:10:57,300 is the volume of the ball of radius a. 179 00:10:57,300 --> 00:10:58,900 That's what our goal is, to figure out 180 00:10:58,900 --> 00:11:04,870 what the volume of the ball of radius a is. 181 00:11:04,870 --> 00:11:09,670 Alright, again as I say, the thing sweeps around. 182 00:11:09,670 --> 00:11:15,800 Coming out of the blackboard, spinning around on this x-axis. 183 00:11:15,800 --> 00:11:18,480 So the setup is the following. 184 00:11:18,480 --> 00:11:21,130 It's always the same. 185 00:11:21,130 --> 00:11:23,210 Here's our formula. 186 00:11:23,210 --> 00:11:26,240 And we need to figure out what's going on with that formula. 187 00:11:26,240 --> 00:11:32,830 And so we need to solve for y as a function of x. 188 00:11:32,830 --> 00:11:36,180 And in order to do that, what we're going to do 189 00:11:36,180 --> 00:11:41,890 is just write down the equation for the circle. 190 00:11:41,890 --> 00:11:45,800 This is the circle. 191 00:11:45,800 --> 00:11:49,180 It's centered at (a, 0), so that's its formula. 192 00:11:49,180 --> 00:11:51,002 And now there's one nice thing, which 193 00:11:51,002 --> 00:11:53,460 is that we really didn't need it to find the formula for y, 194 00:11:53,460 --> 00:11:57,140 we only needed to find the formula for y squared. 195 00:11:57,140 --> 00:12:00,240 So let's just solve for y^2 and we won't have to solve 196 00:12:00,240 --> 00:12:01,820 a quadratic or anything. 197 00:12:01,820 --> 00:12:06,260 Take a square root, that's nice. 198 00:12:06,260 --> 00:12:12,580 This is a^2 - (x^2 - 2ax + a^2). 199 00:12:12,580 --> 00:12:15,170 The a^2's cancel. 200 00:12:15,170 --> 00:12:18,720 These two terms cancel. 201 00:12:18,720 --> 00:12:30,330 And so the formula here is 2ax - x^2. 202 00:12:30,330 --> 00:12:34,670 Alright now, that is what's known as the integrand. 203 00:12:34,670 --> 00:12:38,890 Well, except for this factor of pi here. 204 00:12:38,890 --> 00:12:42,900 And so the answer for the volume is going to be the integral 205 00:12:42,900 --> 00:12:51,050 of pi times this integrand, (2ax - x^2) dx. 206 00:12:51,050 --> 00:12:54,450 And that's just the same thing as this. 207 00:12:54,450 --> 00:12:57,536 But now there's also the issue of the limits. 208 00:12:57,536 --> 00:12:59,160 Which is a completely separate problem, 209 00:12:59,160 --> 00:13:03,720 which we also have to solve. 210 00:13:03,720 --> 00:13:08,650 The range of x is, from this leftmost point 211 00:13:08,650 --> 00:13:10,130 to the rightmost point. 212 00:13:10,130 --> 00:13:13,970 So x varies, starts at 0, and it goes all the way up 213 00:13:13,970 --> 00:13:16,310 to what, what's the top value here. 214 00:13:16,310 --> 00:13:18,340 2a. 215 00:13:18,340 --> 00:13:23,290 And so now I have a completely specified integral. 216 00:13:23,290 --> 00:13:25,960 Again, and this was the theme last time, 217 00:13:25,960 --> 00:13:27,540 the whole goal is to get ourselves 218 00:13:27,540 --> 00:13:30,390 to a complete formula for something 219 00:13:30,390 --> 00:13:32,021 with an integrand and limits. 220 00:13:32,021 --> 00:13:33,520 And then we'll be able to calculate. 221 00:13:33,520 --> 00:13:40,480 Now, we have clear sailing to the end of the problem. 222 00:13:40,480 --> 00:13:42,110 So let's just finish it off. 223 00:13:42,110 --> 00:13:45,910 We have the volume is, if I take the antiderivative of that, 224 00:13:45,910 --> 00:13:53,850 that's pi ax^2, whose derivative is 2ax, minus x^3 / 3. 225 00:13:53,850 --> 00:13:57,250 That's the thing whose derivative is -x^2. 226 00:13:57,250 --> 00:14:02,290 Evaluated at 0 and 2a. 227 00:14:02,290 --> 00:14:09,120 And that is equal to pi times, let's see. 228 00:14:09,120 --> 00:14:09,930 2^2 = 4. 229 00:14:09,930 --> 00:14:19,130 So this is 4a^3 - 8a^3 / 3, right? 230 00:14:19,130 --> 00:14:28,000 And so all told, that is, let's see, (12/3 - 8/3) pi a^3. 231 00:14:28,000 --> 00:14:36,670 Which is maybe a familiar formula, 4/3 pi a^3. 232 00:14:36,670 --> 00:14:41,540 So it worked, we got it right. 233 00:14:41,540 --> 00:14:48,060 Let me just point out a couple of other things 234 00:14:48,060 --> 00:14:51,020 about this formula. 235 00:14:51,020 --> 00:14:57,740 The first one is that from this point of view, 236 00:14:57,740 --> 00:15:01,910 we've actually accomplished more then just finding 237 00:15:01,910 --> 00:15:04,490 the volume of the ball. 238 00:15:04,490 --> 00:15:07,270 We've also found the volume of a bunch 239 00:15:07,270 --> 00:15:11,520 of intermediate regions, which I can draw schematically 240 00:15:11,520 --> 00:15:12,690 this way. 241 00:15:12,690 --> 00:15:18,780 If I chop this thing, and this portion is x here, 242 00:15:18,780 --> 00:15:23,160 then the antiderivative here, this region here, 243 00:15:23,160 --> 00:15:26,970 which maybe I'll fill in with this region 244 00:15:26,970 --> 00:15:31,970 here, which I'm going to call V(x), 245 00:15:31,970 --> 00:15:35,500 is the volume of the portion of the sphere. 246 00:15:35,500 --> 00:15:53,160 Volume of portion of width x of the ball. 247 00:15:53,160 --> 00:16:02,010 And, well, the formula for it is that it's the volume equals pi 248 00:16:02,010 --> 00:16:05,530 (ax ^2 - x^3 / 3). 249 00:16:05,530 --> 00:16:06,654 That's it. 250 00:16:06,654 --> 00:16:08,320 So we've got something which is actually 251 00:16:08,320 --> 00:16:09,600 a lot more information. 252 00:16:09,600 --> 00:16:16,140 For instance, if you plug in x = a, 253 00:16:16,140 --> 00:16:18,500 not surprisingly, and this is a good idea 254 00:16:18,500 --> 00:16:20,940 to do because it checks that we've actually 255 00:16:20,940 --> 00:16:23,020 got a correct formula here. 256 00:16:23,020 --> 00:16:28,480 So if you like you can call this a double-check. 257 00:16:28,480 --> 00:16:36,220 If you check V(a), this should be the volume of a half-ball. 258 00:16:36,220 --> 00:16:38,250 That's halfway. 259 00:16:38,250 --> 00:16:40,660 If I go over here and I only go up to a, 260 00:16:40,660 --> 00:16:42,640 that's exactly half of the ball. 261 00:16:42,640 --> 00:16:45,400 That had better be half, so let's just see. 262 00:16:45,400 --> 00:16:53,830 V(a) in this case is pi, and then I have (a^3 - a^3 / 3). 263 00:16:53,830 --> 00:17:00,330 And that turns out to be pi times a total of 2/3 a^3, 264 00:17:00,330 --> 00:17:07,540 which is indeed half. 265 00:17:07,540 --> 00:17:12,490 Now, on your problem set, what you're going to want to look at 266 00:17:12,490 --> 00:17:17,870 is this full formula here. 267 00:17:17,870 --> 00:17:21,420 Of this chunk. 268 00:17:21,420 --> 00:17:27,140 And what it's going to be good for is a real-life problem. 269 00:17:27,140 --> 00:17:30,510 That is, a problem that really came up 270 00:17:30,510 --> 00:17:35,930 over the summer, and this fall, at a couple of universities 271 00:17:35,930 --> 00:17:38,050 near here, where people were trying 272 00:17:38,050 --> 00:17:42,610 to figure out a phenomenon which is well-known. 273 00:17:42,610 --> 00:17:56,680 Namely, if you have a bunch of particles in a fluid, 274 00:17:56,680 --> 00:18:03,310 and maybe the size of these things is 1 micron. 275 00:18:03,310 --> 00:18:06,260 That is, the radius is 1 micron. 276 00:18:06,260 --> 00:18:09,200 And then you have a bunch of other little particles, 277 00:18:09,200 --> 00:18:11,330 which are a lot smaller. 278 00:18:11,330 --> 00:18:15,750 Maybe 10 nanometers. 279 00:18:15,750 --> 00:18:23,470 Then what happens is that the particles, the big particles, 280 00:18:23,470 --> 00:18:25,030 like to hug each other. 281 00:18:25,030 --> 00:18:28,120 They like to clump together, they're very nice. 282 00:18:28,120 --> 00:18:29,990 Friendly characters. 283 00:18:29,990 --> 00:18:34,250 So what's the explanation for this? 284 00:18:34,250 --> 00:18:38,340 The explanation is that actually they 285 00:18:38,340 --> 00:18:40,842 are not quite as friendly as they might seem. 286 00:18:40,842 --> 00:18:42,800 What's really happening is that the little guys 287 00:18:42,800 --> 00:18:45,240 are shoving them around. 288 00:18:45,240 --> 00:18:47,380 And pushing them together. 289 00:18:47,380 --> 00:18:49,700 And they have sharp elbows, the little ones 290 00:18:49,700 --> 00:18:50,830 and they're pushing them. 291 00:18:50,830 --> 00:18:55,480 Don't like them to be around and they're pushing them together. 292 00:18:55,480 --> 00:18:58,910 But there's actually another possibility. 293 00:18:58,910 --> 00:19:01,550 Which is that they also will stick 294 00:19:01,550 --> 00:19:06,180 to the sides of the container. 295 00:19:06,180 --> 00:19:09,250 So there are two things that actually happen here. 296 00:19:09,250 --> 00:19:12,790 And if you want to get a quantitative handle 297 00:19:12,790 --> 00:19:16,340 on how much of this happens, it has 298 00:19:16,340 --> 00:19:19,500 to do with how much space these things take up. 299 00:19:19,500 --> 00:19:22,390 And so the issue is some kind of overlap 300 00:19:22,390 --> 00:19:26,000 between a band around one sphere and a band 301 00:19:26,000 --> 00:19:27,400 around the other sphere. 302 00:19:27,400 --> 00:19:32,780 And this overlap region is what you have to calculate. 303 00:19:32,780 --> 00:19:36,600 You have to calculate what's in here. 304 00:19:36,600 --> 00:19:41,380 And you can do that using this formula here. 305 00:19:41,380 --> 00:19:46,300 It's not even difficult. So this is, if you cut it in half, 306 00:19:46,300 --> 00:19:48,290 turns out to be two of these guys. 307 00:19:48,290 --> 00:19:51,630 And then you're on your way to figuring out this problem. 308 00:19:51,630 --> 00:19:53,770 And the question is, which do they prefer. 309 00:19:53,770 --> 00:19:55,300 Do they prefer to touch each other, 310 00:19:55,300 --> 00:19:57,250 or do they prefer to touch the wall. 311 00:19:57,250 --> 00:19:59,050 Do they all cluster to the wall. 312 00:19:59,050 --> 00:20:02,030 So you can actually see this in solutions, what they do. 313 00:20:02,030 --> 00:20:04,020 And the question is, to what extent 314 00:20:04,020 --> 00:20:07,580 do they prefer one configuration to the other. 315 00:20:07,580 --> 00:20:11,600 So that's a real, live problem, which really comes up. 316 00:20:11,600 --> 00:20:13,500 Came up just this year. 317 00:20:13,500 --> 00:20:15,420 And frequently comes up. 318 00:20:15,420 --> 00:20:23,640 Which is solved by our first calculation. 319 00:20:23,640 --> 00:20:27,350 So now I have, I called this Method 1. 320 00:20:27,350 --> 00:20:30,970 For solids of revolution, which is called the method of disks. 321 00:20:30,970 --> 00:20:38,670 And now I need to tell you about the other standard method. 322 00:20:38,670 --> 00:21:07,850 Which is called the method of shells. 323 00:21:07,850 --> 00:21:12,350 So this is our second method. 324 00:21:12,350 --> 00:21:18,260 I'm going to illustrate this one with a holiday-themed example 325 00:21:18,260 --> 00:21:19,740 here. 326 00:21:19,740 --> 00:21:23,310 This is supposed to be a witches' cauldron. 327 00:21:23,310 --> 00:21:28,280 Whoops, witch, witches, well. 328 00:21:28,280 --> 00:21:33,970 Maybe more than one witch will have this cauldron here. 329 00:21:33,970 --> 00:21:35,560 So here's this shape. 330 00:21:35,560 --> 00:21:44,500 And we're going to figure out how much liquid is in here. 331 00:21:44,500 --> 00:21:47,760 I'm going to plot this. 332 00:21:47,760 --> 00:21:54,210 Maybe I'll put this down just a bit lower here. 333 00:21:54,210 --> 00:21:58,750 And I'm going to make it a parabola. 334 00:21:58,750 --> 00:22:01,510 This is y = x^2. 335 00:22:01,510 --> 00:22:08,940 And I'm going to make the top height be y = a. 336 00:22:08,940 --> 00:22:10,260 So here's my situation. 337 00:22:10,260 --> 00:22:14,750 And I want to figure out how much liquid is in here. 338 00:22:14,750 --> 00:22:18,050 Now, the reason why I presented the problem in this form, 339 00:22:18,050 --> 00:22:20,780 of course, is really to get you used to these things. 340 00:22:20,780 --> 00:22:23,490 And the first new thing that I want you to get used to 341 00:22:23,490 --> 00:22:28,160 is the idea that now we can also revolve around the y-axis, not 342 00:22:28,160 --> 00:22:30,580 just the x-axis. 343 00:22:30,580 --> 00:22:40,550 So this one is going to be revolved around the y-axis. 344 00:22:40,550 --> 00:22:41,550 And that's what happens. 345 00:22:41,550 --> 00:22:43,790 If you spin the parabola around, you 346 00:22:43,790 --> 00:22:54,030 get this kind of shape, this kind of solid shape here. 347 00:22:54,030 --> 00:22:59,280 Now, I'm going to use the same kind 348 00:22:59,280 --> 00:23:01,530 of slicing that I did before. 349 00:23:01,530 --> 00:23:04,370 But it's going to look totally different. 350 00:23:04,370 --> 00:23:09,180 Namely, I'll draw it in red again. 351 00:23:09,180 --> 00:23:12,820 I'm going to take a little slice over here. 352 00:23:12,820 --> 00:23:16,580 And now I want to imagine what happens if it gets 353 00:23:16,580 --> 00:23:23,060 revolved around the y-axis. 354 00:23:23,060 --> 00:23:24,699 This time it's not a disk. 355 00:23:24,699 --> 00:23:26,740 Actually, if I revolve this around the other way, 356 00:23:26,740 --> 00:23:28,140 it would have had a hole in it. 357 00:23:28,140 --> 00:23:30,410 Which is also possible to do. 358 00:23:30,410 --> 00:23:34,050 That's practically the same as the method of disks. 359 00:23:34,050 --> 00:23:38,387 You'll maybe discuss that in recitation. 360 00:23:38,387 --> 00:23:40,470 Anyway, we're going to revolve it around this way. 361 00:23:40,470 --> 00:23:43,760 So again, I need to sweep it around, 362 00:23:43,760 --> 00:23:47,080 swing it around like this. 363 00:23:47,080 --> 00:23:54,590 And I'll draw the shape. 364 00:23:54,590 --> 00:23:56,180 It's going to sweep around in a circle 365 00:23:56,180 --> 00:24:00,300 and maybe it'll have a little bit of thickness to it. 366 00:24:00,300 --> 00:24:06,620 And this is the thing that people call a shell. 367 00:24:06,620 --> 00:24:12,360 This is the so-called shell of the method. 368 00:24:12,360 --> 00:24:14,900 I would maybe call it a cylinder, 369 00:24:14,900 --> 00:24:16,690 and another way of thinking about it 370 00:24:16,690 --> 00:24:22,350 is that you can maybe wrap up a piece of paper. 371 00:24:22,350 --> 00:24:27,310 Like this. 372 00:24:27,310 --> 00:24:29,440 There it is, there's a cylinder you can see. 373 00:24:29,440 --> 00:24:31,440 Very thin, right? 374 00:24:31,440 --> 00:24:34,990 Very thin. 375 00:24:34,990 --> 00:24:37,590 Now, the reason why I used the piece of paper 376 00:24:37,590 --> 00:24:39,870 as an example of this is that I'm 377 00:24:39,870 --> 00:24:43,220 going to have to figure out the volume of this thing. 378 00:24:43,220 --> 00:24:56,110 Its thickness, as usual, is equal to dx. 379 00:24:56,110 --> 00:25:01,710 Its height is what? 380 00:25:01,710 --> 00:25:05,570 Well, actually I have to use the diagram to see that. 381 00:25:05,570 --> 00:25:15,340 The top value is a, and the bottom value is what we call y. 382 00:25:15,340 --> 00:25:22,210 So the height is equal to, I'm sorry, a - y. 383 00:25:22,210 --> 00:25:25,127 Now, again this is an incredibly risky thing here. 384 00:25:25,127 --> 00:25:26,960 And I've done this before I pointed this out 385 00:25:26,960 --> 00:25:29,340 on the very first day of lecture. 386 00:25:29,340 --> 00:25:34,040 The letter y represents a lot of different things. 387 00:25:34,040 --> 00:25:39,460 And in disguise, when I call this y, I mean y = x^2. 388 00:25:39,460 --> 00:25:41,200 In other words, the interesting curve. 389 00:25:41,200 --> 00:25:45,080 I don't mean y = a, which is the other part. 390 00:25:45,080 --> 00:25:55,080 In general, you might think of it as being equal to y_(top) - 391 00:25:55,080 --> 00:25:57,780 y_(bottom). 392 00:25:57,780 --> 00:26:00,140 So there are, of course, two y's involved. 393 00:26:00,140 --> 00:26:01,390 In disguise. 394 00:26:01,390 --> 00:26:04,450 But we have a shorthand, and the sort 395 00:26:04,450 --> 00:26:06,290 of uninteresting one we call by its-- 396 00:26:06,290 --> 00:26:09,340 we just evaluate immediately, and the interesting one 397 00:26:09,340 --> 00:26:14,330 we leave as the symbol y. 398 00:26:14,330 --> 00:26:18,030 Now, the last bit that I have to do, I claim, 399 00:26:18,030 --> 00:26:22,330 in order to figure this out, is the circumference. 400 00:26:22,330 --> 00:26:26,130 And the reason for that is that if I think of this thing 401 00:26:26,130 --> 00:26:30,260 as like this tube, or piece of paper 402 00:26:30,260 --> 00:26:33,500 here, in order to figure out how much stuff there is here, 403 00:26:33,500 --> 00:26:39,160 all I have to do is unfold it. 404 00:26:39,160 --> 00:26:42,629 Its size, the whole quantity of paper 405 00:26:42,629 --> 00:26:44,670 here is the same whether it's rolled up like this 406 00:26:44,670 --> 00:26:46,090 or whether it's stretched out. 407 00:26:46,090 --> 00:26:56,110 So if I unwrap it, it looks like what? 408 00:26:56,110 --> 00:26:58,790 Well, it looks like kind of a slab? 409 00:26:58,790 --> 00:27:06,550 Right, it looks like just a slab like this. 410 00:27:06,550 --> 00:27:08,860 And again, the thickness is dx. 411 00:27:08,860 --> 00:27:10,650 The height is a - y. 412 00:27:10,650 --> 00:27:13,990 And now we can see that the length here 413 00:27:13,990 --> 00:27:16,100 is all the way around. 414 00:27:16,100 --> 00:27:19,710 It's the circumference. 415 00:27:19,710 --> 00:27:24,100 So this is going to be the circumference when I unwrap it. 416 00:27:24,100 --> 00:27:26,520 And in order to figure out the circumference, 417 00:27:26,520 --> 00:27:29,510 I need to figure out the radius. 418 00:27:29,510 --> 00:27:34,800 So the radius is, on this diagram, is right over there. 419 00:27:34,800 --> 00:27:38,160 This is the radius. 420 00:27:38,160 --> 00:27:42,430 And that distance is x. 421 00:27:42,430 --> 00:27:45,850 So this length here is x. 422 00:27:45,850 --> 00:27:56,440 And so this circumference is 2 pi x. 423 00:27:56,440 --> 00:28:00,520 And this height is still ay. a - y, sorry. 424 00:28:00,520 --> 00:28:13,610 And the thickness is still dx. 425 00:28:13,610 --> 00:28:17,550 So in total, we're just going to multiply these numbers together 426 00:28:17,550 --> 00:28:27,110 to get the total volume. 427 00:28:27,110 --> 00:28:31,260 We have, in other words, dV is equal to the product 428 00:28:31,260 --> 00:28:37,040 of the (2 pi x) dimension dimension, the (8 - y) 429 00:28:37,040 --> 00:28:42,150 dimension, and the dx dimension. 430 00:28:42,150 --> 00:28:43,790 Incidentally, dimensional analysis 431 00:28:43,790 --> 00:28:46,700 is very useful and important in these problems. 432 00:28:46,700 --> 00:28:49,167 You can see that there are three lengths being 433 00:28:49,167 --> 00:28:50,000 multiplied together. 434 00:28:50,000 --> 00:28:52,140 So we'll get a volume in end. 435 00:28:52,140 --> 00:28:52,840 Something cubic. 436 00:28:52,840 --> 00:28:54,839 And we will be coming back to that, because it's 437 00:28:54,839 --> 00:28:57,690 a quite subtle issue sometimes. 438 00:28:57,690 --> 00:29:01,720 So here's the formula, and let's simplify it a little bit. 439 00:29:01,720 --> 00:29:08,690 We have 2 pi x times, remember, first I have to substitute, 440 00:29:08,690 --> 00:29:11,450 otherwise I'm never going to be able to integrate. 441 00:29:11,450 --> 00:29:22,460 And then I rewrite that as 2 pi (ax - x^2, whoops, x^3) dx. 442 00:29:22,460 --> 00:29:25,470 Better not get that wrong. 443 00:29:25,470 --> 00:29:29,280 And now the last little bit here, that I had better 444 00:29:29,280 --> 00:29:31,570 be careful about in order to figure out 445 00:29:31,570 --> 00:29:37,900 what the total volume is, is the limits. 446 00:29:37,900 --> 00:29:42,020 So the volume is going to be the integral of this quantity 2 pi 447 00:29:42,020 --> 00:29:48,600 (ax - x^3) dx. 448 00:29:48,600 --> 00:29:58,660 And now I have to pay attention to what the limits are. 449 00:29:58,660 --> 00:30:02,130 Now, here you have to be careful. 450 00:30:02,130 --> 00:30:06,770 x is possibly the-- You have to always go back 451 00:30:06,770 --> 00:30:08,200 to the 2-D diagram. 452 00:30:08,200 --> 00:30:10,810 I went to it immediately, but that's the whole point. 453 00:30:10,810 --> 00:30:14,990 Is that everything gets read off from this diagram here. 454 00:30:14,990 --> 00:30:17,380 When you take this guy and you sweep it around, 455 00:30:17,380 --> 00:30:20,330 you take care of everything that's to the left. 456 00:30:20,330 --> 00:30:22,830 So we only have to count what's to the right. 457 00:30:22,830 --> 00:30:24,580 We don't have to count anything over here. 458 00:30:24,580 --> 00:30:27,740 Because it's taken care of when we sweep around. 459 00:30:27,740 --> 00:30:31,490 So the starting place is going to be x = 0. 460 00:30:31,490 --> 00:30:34,810 That's where we start. 461 00:30:34,810 --> 00:30:41,070 And where we end is the farthest, rightmost spot for x. 462 00:30:41,070 --> 00:30:43,680 Which is down here. 463 00:30:43,680 --> 00:30:45,930 You've got to watch out about where that is. 464 00:30:45,930 --> 00:30:48,300 In the y variable, it's up at y = a. 465 00:30:48,300 --> 00:30:53,080 But in the x variable, we can see that it's what? 466 00:30:53,080 --> 00:30:56,230 It's the square root of a. 467 00:30:56,230 --> 00:30:58,700 So these limits, this is where you're 468 00:30:58,700 --> 00:31:01,767 going to focus all your attention on the integrand 469 00:31:01,767 --> 00:31:02,850 and getting it just right. 470 00:31:02,850 --> 00:31:04,790 And then you're going to lose your steam 471 00:31:04,790 --> 00:31:06,830 and not pay attention to the limits. 472 00:31:06,830 --> 00:31:09,619 They're equally important. 473 00:31:09,619 --> 00:31:12,035 You've no hope of getting the right answer without getting 474 00:31:12,035 --> 00:31:14,060 the limits right. 475 00:31:14,060 --> 00:31:17,020 So this is the integral from 0 to square root of a. 476 00:31:17,020 --> 00:31:26,463 Again, that's just because y = a and y = x^2 implies x = square 477 00:31:26,463 --> 00:31:27,950 root a. 478 00:31:27,950 --> 00:31:30,300 That's that upper limit there. 479 00:31:30,300 --> 00:31:33,900 And now, we're ready to carry out this, 480 00:31:33,900 --> 00:31:35,490 to evaluate this integral. 481 00:31:35,490 --> 00:31:39,650 So we get 2-- sorry, we get 2 pi ax, that's pi ax^2. 482 00:31:42,680 --> 00:31:47,060 Maybe I'll leave the 2's in there. 483 00:31:47,060 --> 00:31:52,210 2 pi ax^2 is the antiderivative of this ax^2 / 2, 484 00:31:52,210 --> 00:31:59,810 and then here x^4 / 4, evaluated at 0 and square root of a. 485 00:31:59,810 --> 00:32:05,560 And finally, let's see, what is that. 486 00:32:05,560 --> 00:32:14,410 That's 2 pi (a^2 / 2 - a^2 / 4). 487 00:32:14,410 --> 00:32:19,900 Which is a total of 1/4, right, 2 pi a^2 / 4. 488 00:32:19,900 --> 00:32:23,990 Which is pi/2 a^2. 489 00:32:33,210 --> 00:32:33,920 Yes, question. 490 00:32:33,920 --> 00:32:40,322 STUDENT: [INAUDIBLE] 491 00:32:40,322 --> 00:32:41,030 PROFESSOR: Right. 492 00:32:41,030 --> 00:32:43,570 So the question is, why did I integrate 493 00:32:43,570 --> 00:32:46,890 only from the middle to this end, 494 00:32:46,890 --> 00:32:48,560 instead of all the way from over here, 495 00:32:48,560 --> 00:32:52,150 minus square root of a, all the way to the plus end. 496 00:32:52,150 --> 00:32:55,220 And the reason is that you have to look at what's 497 00:32:55,220 --> 00:32:57,470 happening with the rotation. 498 00:32:57,470 --> 00:33:00,820 This red guy, when I swept it around, 499 00:33:00,820 --> 00:33:04,350 I counted the stuff to the right and the left. 500 00:33:04,350 --> 00:33:09,680 So in other words, if I just rotate the right half of this, 501 00:33:09,680 --> 00:33:12,210 I'm covering the left half. 502 00:33:12,210 --> 00:33:15,970 So if I counted the stuff from minus square root of a, 503 00:33:15,970 --> 00:33:17,190 I would be doing it twice. 504 00:33:17,190 --> 00:33:19,910 I would be doubling what I need. 505 00:33:19,910 --> 00:33:21,040 So it's too much. 506 00:33:21,040 --> 00:33:23,450 Another way of saying it is if I wanted 507 00:33:23,450 --> 00:33:25,810 to take the whole region, if I rotate it 508 00:33:25,810 --> 00:33:29,380 around only 180 degrees, only pi, that 509 00:33:29,380 --> 00:33:32,610 would fill up the whole region, if I did both halves. 510 00:33:32,610 --> 00:33:35,000 And then instead of a circumference, instead of a 2 511 00:33:35,000 --> 00:33:36,850 pi x, I could use a pi x. 512 00:33:36,850 --> 00:33:39,180 But then I would have double what I had. 513 00:33:39,180 --> 00:33:41,980 So there are two ways of looking at it. 514 00:33:41,980 --> 00:33:45,140 The same goes, actually, for the football case. 515 00:33:45,140 --> 00:33:47,930 When I have that football, I didn't count the bottom part. 516 00:33:47,930 --> 00:33:50,940 Because when I swung it around the x-axis, 517 00:33:50,940 --> 00:33:57,870 the top part sufficed and I could ignore the bottom half. 518 00:33:57,870 --> 00:33:59,140 Yeah, another question. 519 00:33:59,140 --> 00:34:04,010 STUDENT: [INAUDIBLE] 520 00:34:04,010 --> 00:34:05,260 PROFESSOR: Ooh, good question. 521 00:34:05,260 --> 00:34:08,290 The question is, when do you know, how do you 522 00:34:08,290 --> 00:34:12,910 know when to take the rectangle to be vertical or horizontal. 523 00:34:12,910 --> 00:34:15,850 So far we've only done vertical rectangles. 524 00:34:15,850 --> 00:34:18,350 And I'm going to do a horizontal example in a second. 525 00:34:18,350 --> 00:34:23,560 And the answer to the question of when you do it is this. 526 00:34:23,560 --> 00:34:26,300 You can always set it up both ways. 527 00:34:26,300 --> 00:34:29,560 One way may be a difficult calculation 528 00:34:29,560 --> 00:34:33,130 and one way may be an easier calculation. 529 00:34:33,130 --> 00:34:36,079 Yesterday, we did it - or, sorry, the last time. 530 00:34:36,079 --> 00:34:37,370 Yeah, I guess it was yesterday. 531 00:34:37,370 --> 00:34:39,890 We did it with-- and the horizontal and the vertical 532 00:34:39,890 --> 00:34:41,954 were quite different in character. 533 00:34:41,954 --> 00:34:43,870 One of them was really a mess, and one of them 534 00:34:43,870 --> 00:34:45,130 was a little easier. 535 00:34:45,130 --> 00:34:47,620 So very often, one will be easier than the other. 536 00:34:47,620 --> 00:34:50,790 Every once in a while, one of them is impossible 537 00:34:50,790 --> 00:34:52,640 and the other one is possible. 538 00:34:52,640 --> 00:34:55,210 In other words, the difference in difficulty can be extreme. 539 00:34:55,210 --> 00:34:58,670 So you don't know that in advance. 540 00:34:58,670 --> 00:34:59,870 Yeah, another question. 541 00:34:59,870 --> 00:35:00,990 STUDENT: [INAUDIBLE] 542 00:35:00,990 --> 00:35:02,790 PROFESSOR: The question is, did we 543 00:35:02,790 --> 00:35:06,580 just find the volume when you rotate 544 00:35:06,580 --> 00:35:11,510 this green region around. 545 00:35:11,510 --> 00:35:17,290 Or, did we find the volume when we rotate this whole region. 546 00:35:17,290 --> 00:35:19,711 In other words, just the right half or the right and the 547 00:35:19,711 --> 00:35:20,210 left half. 548 00:35:20,210 --> 00:35:21,870 The answer is both. 549 00:35:21,870 --> 00:35:25,280 The region that you get is the same. 550 00:35:25,280 --> 00:35:27,114 You always get this cauldron, whether you 551 00:35:27,114 --> 00:35:29,030 take this right half when you rotate it around 552 00:35:29,030 --> 00:35:32,974 or you take both and you rotate it around. 553 00:35:32,974 --> 00:35:35,140 So the answer to both of those questions is the same 554 00:35:35,140 --> 00:35:35,730 and it's this. 555 00:35:35,730 --> 00:35:37,170 Yes. 556 00:35:37,170 --> 00:35:43,207 STUDENT: [INAUDIBLE] 557 00:35:43,207 --> 00:35:45,290 PROFESSOR: So that if you rotated them both around 558 00:35:45,290 --> 00:35:47,580 and you only wanted to cover things once, 559 00:35:47,580 --> 00:35:49,350 you would rotate halfway around. 560 00:35:49,350 --> 00:35:52,070 Only by 180 degrees. 561 00:35:52,070 --> 00:35:53,040 That's true. 562 00:35:53,040 --> 00:35:55,379 But you can rotate around as many times as you want. 563 00:35:55,379 --> 00:35:56,920 You're still covering the same thing. 564 00:35:56,920 --> 00:36:03,870 Over and over and over and over again. 565 00:36:03,870 --> 00:36:05,570 So let's go on. 566 00:36:05,570 --> 00:36:08,920 I have a very subtle point that I need to discuss with you 567 00:36:08,920 --> 00:36:23,640 in order to go on to the next application. 568 00:36:23,640 --> 00:36:25,820 So here's my, and I do want to get 569 00:36:25,820 --> 00:36:29,190 to the point of horizontal cross-sections as well. 570 00:36:29,190 --> 00:36:32,450 So let's continue here. 571 00:36:32,450 --> 00:36:37,860 So the first thing that I want to point out to you now 572 00:36:37,860 --> 00:36:50,230 is, I want you to beware of units. 573 00:36:50,230 --> 00:36:54,570 There's something a little fishy in this problem. 574 00:36:54,570 --> 00:37:01,660 And it can be summarized in the character of the answer, which 575 00:37:01,660 --> 00:37:05,320 is just a little bit not clear. 576 00:37:05,320 --> 00:37:09,860 Namely, it looks like it's a^2, right? 577 00:37:09,860 --> 00:37:12,710 And we know that a is in units and we 578 00:37:12,710 --> 00:37:14,950 should've gotten cubic units. 579 00:37:14,950 --> 00:37:18,520 So there's something a little bit tricky about this question. 580 00:37:18,520 --> 00:37:22,190 And so I want to illustrate the paradox right now. 581 00:37:22,190 --> 00:37:26,050 So, suppose that a = 100 centimeters. 582 00:37:26,050 --> 00:37:29,080 And suppose the units are centimeters. 583 00:37:29,080 --> 00:37:33,610 Then the formula for the volume is pi/2 100^2. 584 00:37:36,640 --> 00:37:40,336 and the units we must take are centimeters cubed. 585 00:37:40,336 --> 00:37:41,710 Despite the fact that you kind of 586 00:37:41,710 --> 00:37:43,290 want to square the centimeters. 587 00:37:43,290 --> 00:37:45,740 But that's not what this problem says. 588 00:37:45,740 --> 00:37:50,200 OK, so this is the situation that we've got. 589 00:37:50,200 --> 00:37:54,750 Now, if you work out what this is, 590 00:37:54,750 --> 00:37:58,710 to figure out what the volume of this cauldron is, 591 00:37:58,710 --> 00:38:04,010 what you find is that it's pi / 2 times, well, 592 00:38:04,010 --> 00:38:10,160 10^4 is 10 * 1,000 centimeters cubed. 593 00:38:10,160 --> 00:38:12,410 And those are otherwise known as liters. 594 00:38:12,410 --> 00:38:15,560 So this is approximately 10 pi / 2 liters, 595 00:38:15,560 --> 00:38:20,710 which is about 16 liters. 596 00:38:20,710 --> 00:38:23,230 And so that's how much was in the cauldron 597 00:38:23,230 --> 00:38:27,190 under this choice of units. 598 00:38:27,190 --> 00:38:30,010 Now, I'm going to make another choice of units now. 599 00:38:30,010 --> 00:38:32,420 And we're going to make a comparison. 600 00:38:32,420 --> 00:38:38,510 Suppose that the units are 1 meter. 601 00:38:38,510 --> 00:38:40,370 Looks like it should be the same, 602 00:38:40,370 --> 00:38:47,260 but if I calculate the volume, it's going to be pi / 2, 603 00:38:47,260 --> 00:38:51,610 and 1^2 times meters cubed. 604 00:38:51,610 --> 00:38:55,600 That's what the formula tells us to do. 605 00:38:55,600 --> 00:39:04,860 And if you calculate that out, that's pi / 2 (100 cm)^3. 606 00:39:04,860 --> 00:39:09,660 And with this unit notation we really do 607 00:39:09,660 --> 00:39:12,240 want to cube the centimeters and cube the 100. 608 00:39:12,240 --> 00:39:14,880 So we get pi / 2. 609 00:39:14,880 --> 00:39:18,540 And here we get 10 ^ 6 cm^3. 610 00:39:18,540 --> 00:39:26,990 And that comes out to pi / 2 * 1000 liters. 611 00:39:26,990 --> 00:39:35,560 Or, in other words, about 1600 liters. 612 00:39:35,560 --> 00:39:40,420 So I'd like to ask you first to contemplate this. 613 00:39:40,420 --> 00:39:42,490 And this is a paradox. 614 00:39:42,490 --> 00:39:44,360 And this is a serious paradox. 615 00:39:44,360 --> 00:39:45,910 If you really want to apply problems, 616 00:39:45,910 --> 00:39:49,650 you actually have to understand what your answers mean. 617 00:39:49,650 --> 00:39:53,411 So what do you think is going on here? 618 00:39:53,411 --> 00:39:53,910 Yeah. 619 00:39:53,910 --> 00:40:06,916 STUDENT: [INAUDIBLE] 620 00:40:06,916 --> 00:40:07,540 PROFESSOR: Yes. 621 00:40:07,540 --> 00:40:19,252 STUDENT: [INAUDIBLE] 622 00:40:19,252 --> 00:40:19,960 PROFESSOR: Right. 623 00:40:19,960 --> 00:40:22,980 So the question is, how could either of these make sense. 624 00:40:22,980 --> 00:40:26,700 How am I dealing with the units in either case. 625 00:40:26,700 --> 00:40:30,700 So now I'm going to explain to you the answer. 626 00:40:30,700 --> 00:40:32,590 Because this is really quite puzzling. 627 00:40:32,590 --> 00:40:36,470 But it has a resolution. 628 00:40:36,470 --> 00:40:49,220 The answer to this question is that both answers are correct. 629 00:40:49,220 --> 00:40:54,050 This is correct reasoning in both cases. 630 00:40:54,050 --> 00:41:00,080 What's the matter is that you have to interpret the equation 631 00:41:00,080 --> 00:41:13,800 y = x^2 in two different ways. 632 00:41:13,800 --> 00:41:17,060 Two ways. 633 00:41:17,060 --> 00:41:19,650 So let me explain what they are. 634 00:41:19,650 --> 00:41:31,950 For instance, you can take y = x^2 in centimeters. 635 00:41:31,950 --> 00:41:35,610 So y = x ^2 in centimeters. 636 00:41:35,610 --> 00:41:44,850 In which case, the picture looks like the following. 637 00:41:44,850 --> 00:41:49,100 a = 100 centimeters. 638 00:41:49,100 --> 00:41:55,990 And this distance here, which is the x value, this is 10. 639 00:41:55,990 --> 00:41:57,810 This is 10 centimeters. 640 00:41:57,810 --> 00:41:59,790 And that's what the relationship means. 641 00:41:59,790 --> 00:42:02,930 So the top of the cauldron, if you like, 642 00:42:02,930 --> 00:42:06,879 this distance here is 20 centimeters. 643 00:42:06,879 --> 00:42:08,670 This is actually very badly drawn to scale. 644 00:42:08,670 --> 00:42:10,890 It's actually very, very, deep, this thing. 645 00:42:10,890 --> 00:42:13,900 It's a rather skinny, deep one. 646 00:42:13,900 --> 00:42:19,290 So this is very much not to scale, this picture. 647 00:42:19,290 --> 00:42:27,920 The other picture, the other picture is interpreting y = x^2 648 00:42:27,920 --> 00:42:34,350 in meters. 649 00:42:34,350 --> 00:42:37,140 And that's more like what I had in mind, actually. 650 00:42:37,140 --> 00:42:40,340 I had in mind this big vat here. 651 00:42:40,340 --> 00:42:46,530 And this distance here is 1 meter, 652 00:42:46,530 --> 00:42:49,440 and then the square root of 1 is 1. 653 00:42:49,440 --> 00:42:52,770 So this distance here is also 1 meter. 654 00:42:52,770 --> 00:43:00,210 And the top is 2 meters. 655 00:43:00,210 --> 00:43:02,410 Now, it's not that crazy. 656 00:43:02,410 --> 00:43:04,410 And in fact it's easy to check that, it's pretty 657 00:43:04,410 --> 00:43:06,040 reasonable in terms of scale. 658 00:43:06,040 --> 00:43:09,600 That this thing has 16 liters in it. 659 00:43:09,600 --> 00:43:17,390 And this guy has 1600 liters in it. 660 00:43:17,390 --> 00:43:20,370 So you actually have to know what your symbols mean 661 00:43:20,370 --> 00:43:23,120 when you're dealing with these kinds of applied problems. 662 00:43:23,120 --> 00:43:27,170 And if you're ever really going to do some real consequences, 663 00:43:27,170 --> 00:43:30,260 you have to know what the units mean. 664 00:43:30,260 --> 00:43:32,870 And the problem with the equation y = x^2 is that 665 00:43:32,870 --> 00:43:37,600 it's the one that violated scaling rules. 666 00:43:37,600 --> 00:43:38,100 Yeah. 667 00:43:38,100 --> 00:43:47,264 STUDENT: [INAUDIBLE] 668 00:43:47,264 --> 00:43:47,930 PROFESSOR: Yeah. 669 00:43:47,930 --> 00:43:52,927 STUDENT: [INAUDIBLE] 670 00:43:52,927 --> 00:43:53,510 PROFESSOR: No. 671 00:43:53,510 --> 00:43:59,300 STUDENT: [INAUDIBLE] 672 00:43:59,300 --> 00:44:09,020 PROFESSOR: OK, so the question is whether the formula V = pi / 673 00:44:09,020 --> 00:44:11,520 2 a^2. 674 00:44:11,520 --> 00:44:14,430 This is the correct answer to the problem. 675 00:44:14,430 --> 00:44:17,920 But it is not consistent in units. 676 00:44:17,920 --> 00:44:21,340 If you plug in a equals some number of centimeters, 677 00:44:21,340 --> 00:44:24,060 some number of millimeters, some number of inches and so on, 678 00:44:24,060 --> 00:44:26,380 every single time you'll get a different answer. 679 00:44:26,380 --> 00:44:30,160 And they're all inconsistent. 680 00:44:30,160 --> 00:44:33,800 In other words, this formula violates scaling. 681 00:44:33,800 --> 00:44:36,960 STUDENT: [INAUDIBLE] 682 00:44:36,960 --> 00:44:44,240 PROFESSOR: If you study each step correctly, 683 00:44:44,240 --> 00:44:47,342 you will discover that these are the consistent and correct 684 00:44:47,342 --> 00:44:49,300 statements, what I'm writing on the blackboard. 685 00:44:49,300 --> 00:44:53,060 And this makes sense in a unit-less sense. 686 00:44:53,060 --> 00:44:56,850 But then if you actually stick units on them, one of them, 687 00:44:56,850 --> 00:44:58,200 they both are correct. 688 00:44:58,200 --> 00:45:00,590 And one of them describes this situation and one of them 689 00:45:00,590 --> 00:45:03,850 describes this situation. 690 00:45:03,850 --> 00:45:07,100 And it's a mistake to think of this as being 1 meter 691 00:45:07,100 --> 00:45:09,910 and cubing the meters. 692 00:45:09,910 --> 00:45:20,420 That will be an error that will cause you problems. 693 00:45:20,420 --> 00:45:23,370 This 1 is just unit-less, and then the meters cubed 694 00:45:23,370 --> 00:45:30,320 got converted. 695 00:45:30,320 --> 00:45:33,250 So I encourage you to study this on your own. 696 00:45:33,250 --> 00:45:35,770 So now I'm going to introduce the next problem. 697 00:45:35,770 --> 00:45:38,580 We'll have to solve it next time. 698 00:45:38,580 --> 00:45:41,160 But the reason why I spent all the time on units 699 00:45:41,160 --> 00:45:43,300 is that otherwise it would be impossible 700 00:45:43,300 --> 00:45:46,934 for you to believe me when I do this next calculation. 701 00:45:46,934 --> 00:45:48,350 Because we're trying to get a real 702 00:45:48,350 --> 00:45:50,360 answer out of a real question. 703 00:45:50,360 --> 00:45:53,210 And I'm going to make conversions between centimeters 704 00:45:53,210 --> 00:45:55,080 and meters back and forth. 705 00:45:55,080 --> 00:45:57,080 And we have to get it consistent in order 706 00:45:57,080 --> 00:45:58,200 to have the right answer. 707 00:45:58,200 --> 00:46:04,420 So there was a reason for illustrating this pitfall. 708 00:46:04,420 --> 00:46:08,400 So this second, the next thing that I'd like to do, 709 00:46:08,400 --> 00:46:19,560 is I'd like to boil the water in the witches' cauldron. 710 00:46:19,560 --> 00:46:23,480 This is definitely seasonally appropriate, 711 00:46:23,480 --> 00:46:25,190 since we're approaching Halloween. 712 00:46:25,190 --> 00:46:29,500 And we'll work it out fully next time. 713 00:46:29,500 --> 00:46:35,260 Now, I'm going to introduce another feature 714 00:46:35,260 --> 00:46:36,301 into the problem. 715 00:46:36,301 --> 00:46:38,550 And this is the one that I want you to understand now. 716 00:46:38,550 --> 00:46:41,527 We'll set it all up tomorrow, but right now I 717 00:46:41,527 --> 00:46:43,860 need you to understand what the new main idea that we're 718 00:46:43,860 --> 00:46:51,400 going to get. 719 00:46:51,400 --> 00:46:53,590 There is the new physical feature 720 00:46:53,590 --> 00:46:56,170 that I'm going to add to this problem is that if when you're 721 00:46:56,170 --> 00:47:01,150 boiling, when the witches are boiling this water, 722 00:47:01,150 --> 00:47:03,070 the temperature of the water is not 723 00:47:03,070 --> 00:47:06,617 the same at each level in the kettle. 724 00:47:06,617 --> 00:47:08,950 At the bottom of the kettle, where you're heating it up, 725 00:47:08,950 --> 00:47:11,150 it's at its highest temperature. 726 00:47:11,150 --> 00:47:15,291 So at the bottom it's going to be, say, 100 degrees. 727 00:47:15,291 --> 00:47:17,040 That is, it's going to be totally boiling. 728 00:47:17,040 --> 00:47:18,830 100 degrees Celsius. 729 00:47:18,830 --> 00:47:30,690 And at the top, it's going to be, say, 70 degrees. 730 00:47:30,690 --> 00:47:32,150 Right, it's very cold outside. 731 00:47:32,150 --> 00:47:37,420 In fact, it's 0 degrees outside. 732 00:47:37,420 --> 00:47:40,920 Which is the temperature at which all witches operate, 733 00:47:40,920 --> 00:47:41,900 I think. 734 00:47:41,900 --> 00:47:46,160 Anyway, so they're boiling their stuff. 735 00:47:46,160 --> 00:47:48,230 And the question that we're going to ask 736 00:47:48,230 --> 00:47:55,540 is how much heat, how much heat, do they need to do it. 737 00:47:55,540 --> 00:48:00,170 Now, the thing starts out at 0 degrees Celsius. 738 00:48:00,170 --> 00:48:02,820 And we're going to heat it up to this temperature configuration 739 00:48:02,820 --> 00:48:03,490 here. 740 00:48:03,490 --> 00:48:07,750 But it's rising from 100 down here to 70 up here. 741 00:48:07,750 --> 00:48:10,920 So the temperature is varying in height. 742 00:48:10,920 --> 00:48:12,660 And for simplicity I'm going to make 743 00:48:12,660 --> 00:48:19,700 the formula for the temperature be 70 at the top and 100 744 00:48:19,700 --> 00:48:20,440 at the bottom. 745 00:48:20,440 --> 00:48:24,070 So it's going to be 100 - 30y. 746 00:48:24,070 --> 00:48:28,240 We'll let the level be 1, here. 747 00:48:28,240 --> 00:48:29,500 Sorry, 30. 748 00:48:29,500 --> 00:48:33,160 I said 3-- I wrote 3, but I meant 30. 749 00:48:33,160 --> 00:48:39,120 So this is the situation that we have. 750 00:48:39,120 --> 00:48:43,280 Now, the point about this problem 751 00:48:43,280 --> 00:48:46,170 is we're going to figure out the total temperature, 752 00:48:46,170 --> 00:48:48,920 the total amount of heat that you need to add in order 753 00:48:48,920 --> 00:48:50,010 to heat this thing up. 754 00:48:50,010 --> 00:48:53,240 That's going to be temperature times volume. 755 00:48:53,240 --> 00:48:56,350 But some places will count more than others. 756 00:48:56,350 --> 00:48:59,720 These will be hotter, but there's less water down here. 757 00:48:59,720 --> 00:49:02,290 This is wider up here, so there's more water up here. 758 00:49:02,290 --> 00:49:05,610 So there are various things that are varying in this problem. 759 00:49:05,610 --> 00:49:09,450 Now, the only way to set up so that it works 760 00:49:09,450 --> 00:49:13,870 is to chop things up horizontally 761 00:49:13,870 --> 00:49:15,780 instead of vertically. 762 00:49:15,780 --> 00:49:18,800 Because it's on the horizontal levels 763 00:49:18,800 --> 00:49:21,090 that the temperature is constant. 764 00:49:21,090 --> 00:49:23,010 So we'll have an easy calculation 765 00:49:23,010 --> 00:49:28,320 for how much it takes to heat up a layer, a horizontal layer. 766 00:49:28,320 --> 00:49:32,850 When we rotate this guy around the y axis, 767 00:49:32,850 --> 00:49:36,660 that is which kind of shape. 768 00:49:36,660 --> 00:49:37,332 It's a disk. 769 00:49:37,332 --> 00:49:39,540 So actually this one's going to be an easier problem. 770 00:49:39,540 --> 00:49:41,820 It's going to be a disk problem, not a shell problem. 771 00:49:41,820 --> 00:49:44,111 But we're going to have to work things out with respect 772 00:49:44,111 --> 00:49:47,617 to the dy variation. 773 00:49:47,617 --> 00:49:49,950 In other words, the integral will be with respect to dy. 774 00:49:49,950 --> 00:49:51,158 So we will do that next time. 775 00:49:51,158 --> 00:49:56,410 We'll figure out how much heat it takes to boil the kettle.