1 00:00:00,040 --> 00:00:01,640 FEMALE ANNOUNCER: The following content is provided 2 00:00:01,640 --> 00:00:03,690 under a Creative Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,980 offer high quality educational resources for free. 5 00:00:09,980 --> 00:00:12,830 To make a donation or to view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:31,690 --> 00:00:32,420 HERBERT GROSS: Hi. 9 00:00:32,420 --> 00:00:35,590 In our last few lectures we were trying to establish the 10 00:00:35,590 --> 00:00:39,810 identity of integral calculus and differential calculus in 11 00:00:39,810 --> 00:00:43,730 their own right, independently of one another, and then by 12 00:00:43,730 --> 00:00:47,230 the fundamental theorems of integral calculus to show the 13 00:00:47,230 --> 00:00:50,300 amazing relationship between these two subjects. 14 00:00:50,300 --> 00:00:54,680 Now what we would like to do today is to emphasize this 15 00:00:54,680 --> 00:00:58,330 topic in terms of an application unlike what we've 16 00:00:58,330 --> 00:00:59,600 been doing before. 17 00:00:59,600 --> 00:01:04,220 In particular, what we will do today is discuss the question 18 00:01:04,220 --> 00:01:06,140 of finding volumes. 19 00:01:06,140 --> 00:01:08,550 And in doing this, several interesting things should 20 00:01:08,550 --> 00:01:12,310 happen, not the least of which is that we will rederive 21 00:01:12,310 --> 00:01:15,140 certain results that we've been taking for granted about 22 00:01:15,140 --> 00:01:17,180 solid regions-- 23 00:01:17,180 --> 00:01:19,060 volumes of regions-- for quite awhile. 24 00:01:19,060 --> 00:01:22,380 And also, it will give us an excellent chance to understand 25 00:01:22,380 --> 00:01:25,640 what we really mean by a mathematical structure. 26 00:01:25,640 --> 00:01:28,330 Well, at any rate, to emphasize the structure part, 27 00:01:28,330 --> 00:01:32,690 I've called today's lesson, '3-dimensional Area'. 28 00:01:32,690 --> 00:01:35,400 See, instead of calling it volume, I call it 29 00:01:35,400 --> 00:01:38,580 3-dimensional area, and the reason for this is I'd like to 30 00:01:38,580 --> 00:01:41,860 show you how one can study volumes in a completely 31 00:01:41,860 --> 00:01:44,800 analogous way to how we studied areas. 32 00:01:44,800 --> 00:01:47,480 Remember, we had three basic assumptions for area. 33 00:01:47,480 --> 00:01:52,960 I'm now going to assume three similar assumptions for volume 34 00:01:52,960 --> 00:01:55,930 except where I have to amend them by necessity. 35 00:01:55,930 --> 00:01:58,630 And the only place this amendment has to take place is 36 00:01:58,630 --> 00:02:03,450 whereas the rectangle was the basic building block of areas, 37 00:02:03,450 --> 00:02:06,530 the so-called cylinder will be the basic 38 00:02:06,530 --> 00:02:08,130 building block of volumes. 39 00:02:08,130 --> 00:02:10,605 Let me take a moment to digress here and explain to 40 00:02:10,605 --> 00:02:12,740 you mathematically what the 41 00:02:12,740 --> 00:02:15,390 mathematician calls a cylinder. 42 00:02:15,390 --> 00:02:20,160 We start with any closed curve, say, in a plane, and we 43 00:02:20,160 --> 00:02:23,670 then take a line perpendicular to that plane. 44 00:02:23,670 --> 00:02:28,320 And with that line we trace along the curve. 45 00:02:28,320 --> 00:02:31,660 And we then take another plane parallel to the plane that the 46 00:02:31,660 --> 00:02:35,030 curve is in and slice this thing off someplace. 47 00:02:35,030 --> 00:02:41,260 In other words, by definition, a cylinder has congruent cross 48 00:02:41,260 --> 00:02:43,250 sections all the way through. 49 00:02:43,250 --> 00:02:46,930 And what we're saying is that-- in fact, the familiar 50 00:02:46,930 --> 00:02:49,130 form of a cylinder is the one where the cross 51 00:02:49,130 --> 00:02:50,260 section is a circle. 52 00:02:50,260 --> 00:02:52,390 That's called the right circular cylinder. 53 00:02:52,390 --> 00:02:55,120 Remember, the volume of a right circular cylinder is 'pi 54 00:02:55,120 --> 00:02:58,230 r squared h', the area of the cross 55 00:02:58,230 --> 00:02:59,860 section times the height. 56 00:02:59,860 --> 00:03:01,760 Well, that's the generalization that we make. 57 00:03:01,760 --> 00:03:05,270 In other words, our first assumption is that for any 58 00:03:05,270 --> 00:03:09,080 cylinder the volume of the cylinder is the 59 00:03:09,080 --> 00:03:13,730 cross-sectional area times the height. 60 00:03:13,730 --> 00:03:16,500 Or you could call it the area of the base times the height, 61 00:03:16,500 --> 00:03:19,870 since the cross-sectional area is the same for all slices. 62 00:03:19,870 --> 00:03:21,180 OK? 63 00:03:21,180 --> 00:03:25,480 The next assumption says, if we think of volume as meaning 64 00:03:25,480 --> 00:03:28,960 the amount of space only in three dimensions, whereas area 65 00:03:28,960 --> 00:03:31,650 means the amount of space in two dimensions, our next 66 00:03:31,650 --> 00:03:34,700 assumption is that if the three dimensional region 'R' 67 00:03:34,700 --> 00:03:37,860 is contained in the three dimensional region 'S', then 68 00:03:37,860 --> 00:03:42,270 the volume of 'R' is less than or equal to the volume of 'S'. 69 00:03:42,270 --> 00:03:46,460 And finally, we assume an analogous result about the 70 00:03:46,460 --> 00:03:49,470 area of the whole equals the sum of the areas of the parts. 71 00:03:49,470 --> 00:03:54,310 We assume that if a region is made up of the union of 'n' 72 00:03:54,310 --> 00:03:56,640 regions which do not overlap-- 73 00:03:56,640 --> 00:03:59,460 notice the union notation here. 74 00:03:59,460 --> 00:04:04,010 The union of 'R sub 1' up to 'R sub n' and if the 'R's do 75 00:04:04,010 --> 00:04:08,730 not overlap, then the volume of the region 'R' is the sum 76 00:04:08,730 --> 00:04:12,860 of the volumes of the constituent parts. 77 00:04:12,860 --> 00:04:15,360 In other words, notice that except for the fact that 78 00:04:15,360 --> 00:04:20,910 cylinder replaces rectangle, the basic axioms for studying 79 00:04:20,910 --> 00:04:24,030 volume are precisely the same as the axioms 80 00:04:24,030 --> 00:04:25,270 for studying area. 81 00:04:25,270 --> 00:04:28,370 In particular, then, what this means is that structurally the 82 00:04:28,370 --> 00:04:32,510 same results that we were able to show for area should follow 83 00:04:32,510 --> 00:04:35,810 word for word, essentially, for volumes. 84 00:04:35,810 --> 00:04:38,360 And I thought what we would do is start with a rather 85 00:04:38,360 --> 00:04:39,870 familiar example. 86 00:04:39,870 --> 00:04:40,780 You may recall-- 87 00:04:40,780 --> 00:04:43,030 of course, we've used this result many times-- 88 00:04:43,030 --> 00:04:47,110 that the volume of a cylinder is given by 1/3-- 89 00:04:47,110 --> 00:04:51,810 that the volume of a cone is '1/3 pi r squared h' where 'r' 90 00:04:51,810 --> 00:04:55,080 is the radius of the base and 'h' is the height. 91 00:04:55,080 --> 00:04:57,800 And you may remember that in solid geometry as a 92 00:04:57,800 --> 00:05:00,780 traditional high school curriculum went, these results 93 00:05:00,780 --> 00:05:03,890 were given but seldom if ever proved. 94 00:05:03,890 --> 00:05:07,040 So what I thought we would do now is see how we can use 95 00:05:07,040 --> 00:05:10,950 these axioms to arrive at these results, and to get the 96 00:05:10,950 --> 00:05:13,750 spirit of what we've been trying to do, I will do this 97 00:05:13,750 --> 00:05:17,800 by integral calculus at least first and then by differential 98 00:05:17,800 --> 00:05:19,150 calculus later. 99 00:05:19,150 --> 00:05:20,960 But the idea is something like this. 100 00:05:20,960 --> 00:05:25,610 To visualize the cone, the radius of whose base is 'r' 101 00:05:25,610 --> 00:05:29,600 and whose height is 'h', we can think of the straight line 102 00:05:29,600 --> 00:05:34,420 that joins the origin to the point (r, h), this region 103 00:05:34,420 --> 00:05:38,370 here, this right triangle. 104 00:05:38,370 --> 00:05:41,410 And we can think of that as being revolved about the 105 00:05:41,410 --> 00:05:44,280 x-axis to give the cone. 106 00:05:44,280 --> 00:05:47,910 Now when we were dealing with areas, you may recall that we 107 00:05:47,910 --> 00:05:53,060 broke things down into rectangles that were too big, 108 00:05:53,060 --> 00:05:56,460 rectangles that were too small, and we computed, say, 109 00:05:56,460 --> 00:06:00,000 'U sub n' and 'L sub n', et cetera. 110 00:06:00,000 --> 00:06:02,460 We can do the same thing now. 111 00:06:02,460 --> 00:06:06,710 What we do is we again circumscribe rectangles. 112 00:06:06,710 --> 00:06:11,230 Now the idea is whatever volume is traced out by this 113 00:06:11,230 --> 00:06:14,960 piece here, whatever volume is traced out when we rotate this 114 00:06:14,960 --> 00:06:20,280 triangle about the x-axis, that volume will be less than 115 00:06:20,280 --> 00:06:24,670 the volume generated by this particular rectangle, because, 116 00:06:24,670 --> 00:06:27,430 you see, the volume that we're looking for is contained 117 00:06:27,430 --> 00:06:31,270 inside the rectangle when we revolve this particular thing. 118 00:06:31,270 --> 00:06:34,070 Now notice that this particular rectangle when 119 00:06:34,070 --> 00:06:37,290 revolved gives me a right circular cylinder, and we're 120 00:06:37,290 --> 00:06:39,750 assuming that we know how to find the volume of a cylinder. 121 00:06:39,750 --> 00:06:42,760 It's the cross sectional area times the height. 122 00:06:42,760 --> 00:06:46,160 Let's focus our attention on what I call the k-th region 123 00:06:46,160 --> 00:06:48,670 here and see what this thing looks like. 124 00:06:48,670 --> 00:06:51,480 First of all, notice that if we've divided this length, 125 00:06:51,480 --> 00:06:55,880 which is 'h' units long into 'n' equal parts, each of these 126 00:06:55,880 --> 00:06:59,430 pieces is 'h' over 'n'. 127 00:06:59,430 --> 00:07:03,460 Secondly, notice that the radius of the base of the 128 00:07:03,460 --> 00:07:05,930 cylinder that we're going to get-- well, let's see. 129 00:07:05,930 --> 00:07:08,290 It's going to be this y-coordinate. 130 00:07:08,290 --> 00:07:12,300 Given the x-coordinate, 'y' is determined by multiplying the 131 00:07:12,300 --> 00:07:15,160 x-coordinate by 'r/h'. 132 00:07:15,160 --> 00:07:18,300 The x-coordinate is 'kh' over 'n'. 133 00:07:18,300 --> 00:07:20,770 I multiply that by 'r/h'. 134 00:07:20,770 --> 00:07:23,530 That gives me 'kr' over 'n'. 135 00:07:23,530 --> 00:07:25,640 That's the height of this-- 136 00:07:25,640 --> 00:07:26,900 the radius of the base of the cylinder that 137 00:07:26,900 --> 00:07:28,310 we're going to revolve. 138 00:07:28,310 --> 00:07:30,390 Now what is the volume of this cylinder? 139 00:07:30,390 --> 00:07:35,600 The area of the base is 'pi y squared', and I'm now going to 140 00:07:35,600 --> 00:07:40,040 multiply that by the height, which is 'h/n'. 141 00:07:40,040 --> 00:07:42,830 And if I do that, I obtain what? 142 00:07:42,830 --> 00:07:50,310 The volume of this particular cylinder is 'pi r squared h' 143 00:07:50,310 --> 00:07:53,450 times 'k squared' over 'n cubed'. 144 00:07:53,450 --> 00:07:58,200 And now, if I add up all of these volumes as 'k' goes from 145 00:07:58,200 --> 00:08:04,040 1 to 'n', that will give me a bunch of stacked cylinders 146 00:08:04,040 --> 00:08:08,240 which enclose my cone. 147 00:08:08,240 --> 00:08:10,520 In other words, an answer that will be too 148 00:08:10,520 --> 00:08:12,210 large will be what? 149 00:08:12,210 --> 00:08:17,610 This sum as 'k' goes from 1 to 'n', notice that this is the 150 00:08:17,610 --> 00:08:21,510 only portion that depends on 'k', so the upper 151 00:08:21,510 --> 00:08:22,310 approximation-- 152 00:08:22,310 --> 00:08:24,390 in other words, the volume that's too large to be the 153 00:08:24,390 --> 00:08:25,410 right answer-- 154 00:08:25,410 --> 00:08:29,180 is 'pi r squared h' over 'n cubed' times the sum as 'k' 155 00:08:29,180 --> 00:08:31,270 goes from 1 to 'n', 'k squared''. 156 00:08:31,270 --> 00:08:35,000 Now you notice I always stick to problems where we have 157 00:08:35,000 --> 00:08:37,530 something fairly simple like this, because this limit 158 00:08:37,530 --> 00:08:40,340 process, as we've mentioned in the previous lectures, becomes 159 00:08:40,340 --> 00:08:43,970 very, very difficult to do in general, the beauty or one of 160 00:08:43,970 --> 00:08:45,990 the beauties of our fundamental theorem. 161 00:08:45,990 --> 00:08:50,870 But the idea is I do know that this sum is 'n' times 'n + 1' 162 00:08:50,870 --> 00:08:54,370 times '2n + 1' over 6. 163 00:08:54,370 --> 00:08:58,730 Now distributing the 'n cubed' one factor at a time, the way 164 00:08:58,730 --> 00:09:02,970 we have before, I can now write that this is '1/6 pi r 165 00:09:02,970 --> 00:09:04,260 squared h'. 166 00:09:04,260 --> 00:09:08,000 'n + 1' over 'n' is '1 + '1/n''. 167 00:09:08,000 --> 00:09:11,840 '2n + 1' over 'n' is '2 + '1/n''. 168 00:09:11,840 --> 00:09:15,000 And I find that my upper approximation is given by this 169 00:09:15,000 --> 00:09:16,250 expression. 170 00:09:16,250 --> 00:09:19,860 And if I now take the limit of 'U sub n' as 'n' goes to 171 00:09:19,860 --> 00:09:22,500 infinity, this factor approaches 1. 172 00:09:22,500 --> 00:09:24,370 This factor approaches 2. 173 00:09:24,370 --> 00:09:28,940 Therefore, our entire product approaches in the limit '1/3 174 00:09:28,940 --> 00:09:31,740 pi r squared h', which is the familiar 175 00:09:31,740 --> 00:09:34,640 result of solid geometry. 176 00:09:34,640 --> 00:09:37,000 Of course, we've taken a lot for granted over here. 177 00:09:37,000 --> 00:09:40,040 What we've really proven here is not that the volume of a 178 00:09:40,040 --> 00:09:42,860 cone is '1/3 pi r squared h'. 179 00:09:42,860 --> 00:09:46,340 What we have proven is at the limit of 'U sub n' as 'n' 180 00:09:46,340 --> 00:09:49,820 approaches infinity, it's '1/3 pi r squared h'. 181 00:09:49,820 --> 00:09:52,840 The question that comes up is how do you know that as these 182 00:09:52,840 --> 00:09:56,960 divisions get small that the volume-- 183 00:09:56,960 --> 00:09:59,980 the upper approximation gets arbitrarily close to the 184 00:09:59,980 --> 00:10:01,100 correct answer. 185 00:10:01,100 --> 00:10:04,640 And again, notice how we can reason analogously to what we 186 00:10:04,640 --> 00:10:06,040 did in the case of area. 187 00:10:06,040 --> 00:10:11,820 Namely, what we could do next, you see, is take the smallest 188 00:10:11,820 --> 00:10:14,310 cylinder that can be inscribed here. 189 00:10:14,310 --> 00:10:18,090 In other words, this would give us an approximation which 190 00:10:18,090 --> 00:10:19,960 is too small. 191 00:10:19,960 --> 00:10:27,650 The total error is no more than the solid generated by 192 00:10:27,650 --> 00:10:31,440 this hatch region revolving about the x-axis. 193 00:10:31,440 --> 00:10:39,220 Notice, however, that each of these pieces fits very nicely 194 00:10:39,220 --> 00:10:43,740 in here, so that the total error between an approximation 195 00:10:43,740 --> 00:10:46,110 which is too big and an approximation which is too 196 00:10:46,110 --> 00:10:50,430 small is this height, which is 'r'. 197 00:10:50,430 --> 00:10:52,590 OK? 198 00:10:52,590 --> 00:10:54,540 Let's see, the cross-sectional area-- this is 'r'. 199 00:10:54,540 --> 00:10:58,210 So 'pi r squared' is the area of the base. 200 00:10:58,210 --> 00:11:01,370 The height is 'h/n'. 201 00:11:01,370 --> 00:11:05,430 Notice that 'r' and 'h' are given constants, therefore, as 202 00:11:05,430 --> 00:11:09,200 'n' goes to infinity, the numerator stays constant. 203 00:11:09,200 --> 00:11:12,650 The denominator goes to infinity. 204 00:11:12,650 --> 00:11:14,670 The difference goes to 0. 205 00:11:14,670 --> 00:11:18,490 In other words, again, we can show that 'U sub n' and 'L sub 206 00:11:18,490 --> 00:11:21,160 n' have a common limit. 207 00:11:21,160 --> 00:11:23,770 In fact, we can generalize this result rather nicely. 208 00:11:27,390 --> 00:11:29,660 Take this drawing to be whatever you'd like it to be. 209 00:11:29,660 --> 00:11:35,780 I've simply tried to visualize here a solid region. 210 00:11:35,780 --> 00:11:37,390 This is a 3-dimensional region. 211 00:11:37,390 --> 00:11:40,870 It has various cross sections. 212 00:11:40,870 --> 00:11:43,980 And I know that as I look at it in the 'x' direction, the 213 00:11:43,980 --> 00:11:47,510 region begins at 'x' equals 'a' and terminates at 'x' 214 00:11:47,510 --> 00:11:48,520 equals 'b'. 215 00:11:48,520 --> 00:11:51,420 And the question is how can I find the volume of this 216 00:11:51,420 --> 00:11:57,060 particular region, assuming I know the cross-sectional area 217 00:11:57,060 --> 00:11:58,420 for any slice? 218 00:11:58,420 --> 00:11:59,770 And the idea, again, is what? 219 00:11:59,770 --> 00:12:04,600 We can slice this solid up into 'n' parts, which I call 220 00:12:04,600 --> 00:12:07,410 'delta V1' up to 'delta Vn'. 221 00:12:07,410 --> 00:12:10,480 The sum of these would be the true volume that 222 00:12:10,480 --> 00:12:12,020 we're looking for. 223 00:12:12,020 --> 00:12:17,550 Now, I focus my attention on the k-th piece here, and 224 00:12:17,550 --> 00:12:22,210 again, what I do is I inscribe and circumscribe cylinders, 225 00:12:22,210 --> 00:12:25,800 one of which is contained entirely within 'delta V sub 226 00:12:25,800 --> 00:12:29,680 k', and the other of which surrounds 'delta V sub k'. 227 00:12:29,680 --> 00:12:33,150 In other words, what I do is is I find the biggest possible 228 00:12:33,150 --> 00:12:36,840 cross-sectional area I have in this interval, and I denote 229 00:12:36,840 --> 00:12:40,840 the 'x' value at which that occurs by 'M sub k'. 230 00:12:40,840 --> 00:12:46,190 I find the value of 'x', which I call 'm sub k', at which I 231 00:12:46,190 --> 00:12:48,970 get the smallest cross-sectional area. 232 00:12:48,970 --> 00:12:52,160 And therefore, the inscribed cylinder has 233 00:12:52,160 --> 00:12:55,090 volume given by this. 234 00:12:55,090 --> 00:13:00,300 The circumscribed cylinder has volume given by this, and the 235 00:13:00,300 --> 00:13:03,690 piece that I'm looking for, the true volume, is caught 236 00:13:03,690 --> 00:13:05,220 between these two. 237 00:13:05,220 --> 00:13:09,300 Therefore, if I add these up as 'k' goes from 1 to 'n', 238 00:13:09,300 --> 00:13:13,770 I've caught 'V' between 'U sub n' and 'L sub n'. 239 00:13:13,770 --> 00:13:18,170 Assuming only that the area is a continuous function, the 240 00:13:18,170 --> 00:13:22,120 difference between the largest cross section and the smallest 241 00:13:22,120 --> 00:13:27,670 cross section approaches 0 as 'delta x sub k' approaches 0. 242 00:13:27,670 --> 00:13:30,690 In essence then, the same as we did before, what we can 243 00:13:30,690 --> 00:13:34,900 show is that as 'n' goes to infinity, both 'U sub n' and 244 00:13:34,900 --> 00:13:40,840 'L sub n' approach a common limit, and therefore, the 'V' 245 00:13:40,840 --> 00:13:44,870 being caught between these two is equal to the common limit. 246 00:13:44,870 --> 00:13:46,390 Now here's the interesting point. 247 00:13:46,390 --> 00:13:49,530 When we talked about the definite integral, no one 248 00:13:49,530 --> 00:13:53,250 asked physically what the function 'f' was or what the 249 00:13:53,250 --> 00:13:57,010 'c sub k's that we're using here were, only that they be 250 00:13:57,010 --> 00:14:00,170 in the proper interval and 'f' be a continuous function. 251 00:14:00,170 --> 00:14:04,430 Notice that we're assuming that 'A' is continuous. 252 00:14:04,430 --> 00:14:07,320 In essence then, by the definition of the definite 253 00:14:07,320 --> 00:14:09,900 integral, that volume is just what? 254 00:14:09,900 --> 00:14:12,970 The definite integral from 'a' to 'b', 'f of x' 'dx'. 255 00:14:12,970 --> 00:14:17,700 In other words, it's this sum taken in the limit as 'n' 256 00:14:17,700 --> 00:14:20,590 approaches infinity, or another way of saying that, as 257 00:14:20,590 --> 00:14:25,400 the maximum 'delta X sub k' approaches 0. 258 00:14:25,400 --> 00:14:27,800 That, by the way, is the integral calculus approach. 259 00:14:27,800 --> 00:14:30,580 If we want the differential calculus approach, remember 260 00:14:30,580 --> 00:14:32,610 what we do, we say look it. 261 00:14:32,610 --> 00:14:37,610 The change in volume is less than or equal to the maximum 262 00:14:37,610 --> 00:14:42,150 cross-sectional area times 'delta X' and greater than or 263 00:14:42,150 --> 00:14:44,890 equal to the minimum cross-sectional area 264 00:14:44,890 --> 00:14:46,420 times 'delta X'. 265 00:14:46,420 --> 00:14:49,820 Same as we did for area, you see, we divide 266 00:14:49,820 --> 00:14:51,700 through by 'delta X'. 267 00:14:51,700 --> 00:14:55,830 We have that 'delta V' divided by 'delta X' is caught between 268 00:14:55,830 --> 00:15:02,970 'A of M', 'A of m', where 'm' and 'M' are in this interval. 269 00:15:02,970 --> 00:15:08,370 And now you see as 'delta X' approaches 0, 'm' and 'M' both 270 00:15:08,370 --> 00:15:09,370 approach 'x'. 271 00:15:09,370 --> 00:15:11,860 You see the same procedure as we had before. 272 00:15:11,860 --> 00:15:14,780 You see what we're saying going back to this diagram is, 273 00:15:14,780 --> 00:15:22,040 for example, here is a 'delta X', and 'm' and 'M' are points 274 00:15:22,040 --> 00:15:27,280 in here, and as 'delta X' goes to 0, 'm' and 'M' both 275 00:15:27,280 --> 00:15:29,630 approach the end point 'x'. 276 00:15:29,630 --> 00:15:34,170 And since 'A' is assumed to be continuous, if 'M' and 'm' 277 00:15:34,170 --> 00:15:39,710 approach 'x', 'A of M', 'A of m' approach 'A of x', and we 278 00:15:39,710 --> 00:15:44,980 arrive at, by differential calculus, that 'dV dx' is 'A 279 00:15:44,980 --> 00:15:50,600 of x', and therefore, 'V' is again equal to integral ''A of 280 00:15:50,600 --> 00:15:53,500 x' 'dx'' as 'x' goes from 'a' to 'b'. 281 00:15:53,500 --> 00:15:57,610 Where now by differential calculus, this means what? 282 00:15:57,610 --> 00:16:05,890 It simply means 'G of b' minus 'G of a', where 'G' is any 283 00:16:05,890 --> 00:16:09,000 function whose derivative is 'f'. 284 00:16:09,000 --> 00:16:11,580 Now again, I have to go through this thing rather 285 00:16:11,580 --> 00:16:15,550 hurriedly because I want to get some examples done. 286 00:16:15,550 --> 00:16:18,530 But what I hope is that we went slowly enough so that you 287 00:16:18,530 --> 00:16:22,460 can again sense how we're using integral calculus, 288 00:16:22,460 --> 00:16:25,910 differential calculus, and the relationship between them. 289 00:16:25,910 --> 00:16:29,220 Let's, at any rate, illustrate some of these results more 290 00:16:29,220 --> 00:16:31,610 concretely in terms of-- 291 00:16:31,610 --> 00:16:36,160 well, first of all, let's talk about one particular type of 292 00:16:36,160 --> 00:16:39,920 solid, what's called a solid of revolution. 293 00:16:39,920 --> 00:16:42,970 That's the particular type of solid where you have a region 294 00:16:42,970 --> 00:16:47,130 in the 'xy' plane, and you take that region and rotate it 295 00:16:47,130 --> 00:16:51,580 either around the x-axis or the y-axis, thus generating a 296 00:16:51,580 --> 00:16:54,240 3-dimensional region. 297 00:16:54,240 --> 00:16:57,710 See, in other words, a plane area is rotated through 360 298 00:16:57,710 --> 00:17:01,500 degrees either with respect to the x-axis or the y-axis. 299 00:17:01,500 --> 00:17:03,950 I'll consider the x-axis here. 300 00:17:03,950 --> 00:17:06,880 Notice that this is a special case of what we've just 301 00:17:06,880 --> 00:17:10,339 studied, namely, in this particular case, if 'y' equals 302 00:17:10,339 --> 00:17:13,710 'f of x' is a continuous function, notice that every 303 00:17:13,710 --> 00:17:15,910 cross section here, every cross 304 00:17:15,910 --> 00:17:19,240 section will be a circle. 305 00:17:19,240 --> 00:17:22,670 The area of the circle is 'pi y squared'. 306 00:17:22,670 --> 00:17:24,920 That's 'pi 'f of x' squared'. 307 00:17:24,920 --> 00:17:28,180 And therefore, according to our fundamental theorem, since 308 00:17:28,180 --> 00:17:29,770 the area is continuous-- 309 00:17:29,770 --> 00:17:31,380 and why is the area continuous? 310 00:17:31,380 --> 00:17:34,280 Well, if 'f' is continuous, remember the product of 311 00:17:34,280 --> 00:17:36,190 continuous functions is continuous. 312 00:17:36,190 --> 00:17:39,170 If 'f' is continuous, 'f squared' is continuous. 313 00:17:39,170 --> 00:17:42,330 So according to our result, the volume of the region 'R' 314 00:17:42,330 --> 00:17:45,010 is just the integral from 'a' to 'b', 'pi 'f 315 00:17:45,010 --> 00:17:46,660 of x' squared' 'dx'. 316 00:17:46,660 --> 00:17:48,960 And you see, using differential calculus, all we 317 00:17:48,960 --> 00:17:52,520 need now is a function whose derivative is 'pi 'f of x' 318 00:17:52,520 --> 00:17:54,890 squared', and we call that function 'G'. 319 00:17:54,890 --> 00:17:57,770 We compute 'G of b' minus 'G of a' and that gives us the 320 00:17:57,770 --> 00:18:00,120 volume that we're looking for. 321 00:18:00,120 --> 00:18:02,830 Remember, when we talked about areas, we mentioned this was 322 00:18:02,830 --> 00:18:04,020 highly specialized. 323 00:18:04,020 --> 00:18:06,550 What if you had a region like this? 324 00:18:06,550 --> 00:18:09,540 And again, sparing the details, observe that if we 325 00:18:09,540 --> 00:18:13,290 have a region like this, we can draw in the lines where 326 00:18:13,290 --> 00:18:15,130 the curve doubles back. 327 00:18:15,130 --> 00:18:21,010 We can now visualize this as the volume-- 328 00:18:21,010 --> 00:18:22,630 the difference of two volumes. 329 00:18:22,630 --> 00:18:26,970 Namely, we can find this volume and subtract from that 330 00:18:26,970 --> 00:18:30,440 volume this volume. 331 00:18:30,440 --> 00:18:33,590 See, in other words, both of these have the right form. 332 00:18:33,590 --> 00:18:36,430 And by this difference, what's left? 333 00:18:36,430 --> 00:18:39,170 The difference of the big volume minus the small volume 334 00:18:39,170 --> 00:18:41,070 is the volume generated by 'R'. 335 00:18:41,070 --> 00:18:44,750 And as I say, these are rather simple details that we can 336 00:18:44,750 --> 00:18:48,640 check out computationally in terms of exercises, but the 337 00:18:48,640 --> 00:18:50,990 reason I wanted to mention the solid of revolution is that 338 00:18:50,990 --> 00:18:55,110 not only is this a rather common and important category, 339 00:18:55,110 --> 00:18:58,250 but it also happens to be the type of solid that we opened 340 00:18:58,250 --> 00:18:59,450 our program with. 341 00:18:59,450 --> 00:19:03,700 Remember, the cone may be viewed as what? 342 00:19:03,700 --> 00:19:08,330 The solid generated by a particular right triangle 343 00:19:08,330 --> 00:19:11,110 being revolved about the x-axis. 344 00:19:11,110 --> 00:19:13,860 In fact, I thought we could try that same problem now 345 00:19:13,860 --> 00:19:17,350 doing it by the antiderivative method. 346 00:19:17,350 --> 00:19:21,030 Namely, we take this particular region 'R' and 347 00:19:21,030 --> 00:19:22,980 notice now, if we revolve this about the x-axis-- 348 00:19:25,520 --> 00:19:28,080 let's see, the cross-sectional area will be what? 349 00:19:28,080 --> 00:19:31,470 Well, it's a circle of radius 'y'. 350 00:19:31,470 --> 00:19:34,430 For a given value of 'x', 'y' is equal to 'r' 351 00:19:34,430 --> 00:19:36,040 times 'x' over 'h'. 352 00:19:36,040 --> 00:19:38,080 See the slope of this line is 'r/h'. 353 00:19:38,080 --> 00:19:39,920 It passes through the origin here. 354 00:19:39,920 --> 00:19:43,860 So the cross-sectional area is 'y squared' times pi. 355 00:19:43,860 --> 00:19:47,950 That's 'pi 'r squared h' squared' over 'h squared' 356 00:19:47,950 --> 00:19:49,720 times 'x squared'. 357 00:19:49,720 --> 00:19:53,080 And to find that volume, I simply integrate this 358 00:19:53,080 --> 00:19:56,190 between 0 and 'h'. 359 00:19:56,190 --> 00:19:59,600 Recalling that pi, 'r', and 'h' are constants, I can take 360 00:19:59,600 --> 00:20:01,900 the constants outside of the integral sign. 361 00:20:01,900 --> 00:20:05,140 The integral of 'x squared', meaning what? 362 00:20:05,140 --> 00:20:09,070 The inverse derivative is '1/3 x cubed'. 363 00:20:09,070 --> 00:20:14,590 If I evaluate that between 0 and 'h', I get '1/3 h cubed'. 364 00:20:14,590 --> 00:20:18,820 The 'h cubed' in the numerator cancels the 'h squared' in the 365 00:20:18,820 --> 00:20:22,000 denominator, leaving a factor of 'h' in the numerator, and I 366 00:20:22,000 --> 00:20:26,000 wind up, as I saw before, that the volume of this cone is 367 00:20:26,000 --> 00:20:28,380 '1/3 pi r squared h'. 368 00:20:28,380 --> 00:20:30,840 And this is nice that I get the same answer as by the 369 00:20:30,840 --> 00:20:33,300 limit method, because according to the fundamental 370 00:20:33,300 --> 00:20:36,280 theorem, the first fundamental theorem, this is precisely 371 00:20:36,280 --> 00:20:37,560 what was supposed to happen. 372 00:20:37,560 --> 00:20:40,890 In other words, I can do these either by limits or by 373 00:20:40,890 --> 00:20:41,680 derivatives. 374 00:20:41,680 --> 00:20:44,430 I want you to see these things side by side, because in 375 00:20:44,430 --> 00:20:47,120 certain cases, as I've emphasized in the previous 376 00:20:47,120 --> 00:20:51,140 lectures, there will be times when we cannot, by 377 00:20:51,140 --> 00:20:55,000 differential calculus, find a function 'G' whose derivative 378 00:20:55,000 --> 00:20:57,350 is equal to a given function 'f of x'. 379 00:20:57,350 --> 00:20:59,940 But enough about that for the time being. 380 00:20:59,940 --> 00:21:03,620 The next question that comes up gives us a review of what 381 00:21:03,620 --> 00:21:05,140 happens with inverse functions. 382 00:21:05,140 --> 00:21:07,690 It's a rather interesting type of situation. 383 00:21:07,690 --> 00:21:10,060 It's called the method of cylindrical shells and it's 384 00:21:10,060 --> 00:21:11,930 motivated by the following. 385 00:21:11,930 --> 00:21:15,210 Let's suppose again we're given a very nice region 'R'. 386 00:21:15,210 --> 00:21:16,550 What do I mean by very nice? 387 00:21:16,550 --> 00:21:19,080 Well, to simplify the computation, even though it 388 00:21:19,080 --> 00:21:23,670 doesn't change the theory at all, I'm assuming that 'y' 389 00:21:23,670 --> 00:21:26,730 equals 'f of x' is an increasing curve. 390 00:21:26,730 --> 00:21:29,190 In other words, I'm even assuming that we have a 391 00:21:29,190 --> 00:21:32,200 one-to-one function here. 392 00:21:32,200 --> 00:21:36,220 Now the idea is here's this nice region and instead of 393 00:21:36,220 --> 00:21:38,950 revolving this about the x-axis, I would like to 394 00:21:38,950 --> 00:21:41,880 revolve it about the y-axis. 395 00:21:41,880 --> 00:21:47,000 Now you see, to use the method of revolution here, to revolve 396 00:21:47,000 --> 00:21:50,470 this about the y-axis, essentially what I do is I 397 00:21:50,470 --> 00:21:55,060 pick a washer-shaped region, you see? 398 00:21:55,060 --> 00:21:59,265 I have to compute the volume generated by the 'y' part. 399 00:21:59,265 --> 00:22:01,810 See, in other words, I do this as two separate parts. 400 00:22:01,810 --> 00:22:06,740 I find the volume of the big piece minus the volume of the 401 00:22:06,740 --> 00:22:10,330 small piece, and what's left is the volume 402 00:22:10,330 --> 00:22:11,680 generated by 'R'. 403 00:22:11,680 --> 00:22:13,970 But notice a rather difficult computational 404 00:22:13,970 --> 00:22:15,510 thing occurs here. 405 00:22:15,510 --> 00:22:20,160 Namely, notice that this length here has to be 406 00:22:20,160 --> 00:22:25,010 expressed as 'x' goes from one value to another value. 407 00:22:25,010 --> 00:22:29,510 Now, you see, if this is 'b', and this is 'a', you see, 408 00:22:29,510 --> 00:22:31,170 notice what's happening here, how our 409 00:22:31,170 --> 00:22:32,770 strips are being chosen. 410 00:22:32,770 --> 00:22:37,360 You see, for a given strip, the final 'x' value is 'b', 411 00:22:37,360 --> 00:22:39,590 but what is the initial 'x' value? 412 00:22:39,590 --> 00:22:41,840 See, down here the 'x' value is 'a', but 413 00:22:41,840 --> 00:22:43,260 what happens up here? 414 00:22:43,260 --> 00:22:45,670 In other words, how do you find what the 'x' value is for 415 00:22:45,670 --> 00:22:47,160 a given value of 'y' here? 416 00:22:47,160 --> 00:22:52,660 Well, you see what you must do is invert the relationship. 417 00:22:52,660 --> 00:22:54,040 Now even though I've picked a case 418 00:22:54,040 --> 00:22:55,610 where the inverse exists-- 419 00:22:55,610 --> 00:22:57,570 see, this is a one-to-one function-- 420 00:22:57,570 --> 00:23:00,910 we've already had ample examples in which we've shown 421 00:23:00,910 --> 00:23:04,040 that computationally it's extremely difficult if it's 422 00:23:04,040 --> 00:23:10,010 even possible to explicitly perform the inversion. 423 00:23:10,010 --> 00:23:11,860 And this is where the method of cylindrical 424 00:23:11,860 --> 00:23:13,290 shells comes from. 425 00:23:13,290 --> 00:23:16,230 Essentially what the method of cylindrical shells says is 426 00:23:16,230 --> 00:23:19,080 wouldn't it have been nice if we chose our generating 427 00:23:19,080 --> 00:23:20,920 element to be this way? 428 00:23:20,920 --> 00:23:23,750 In other words, what we say is look at this piece 429 00:23:23,750 --> 00:23:25,240 of area over here. 430 00:23:25,240 --> 00:23:30,980 One way of visualizing this solid being rotated is to 431 00:23:30,980 --> 00:23:35,390 think of this particular region being rotated about the 432 00:23:35,390 --> 00:23:38,990 x-axis, and it generates a certain volume. 433 00:23:38,990 --> 00:23:41,610 By the way, what volume will it generate? 434 00:23:41,610 --> 00:23:45,090 The volume that it will generate will be less than the 435 00:23:45,090 --> 00:23:49,310 volume that this rectangle generates but greater than the 436 00:23:49,310 --> 00:23:52,660 volume that this rectangle here generates. 437 00:23:52,660 --> 00:23:54,800 Now what is the volume generated 438 00:23:54,800 --> 00:23:56,410 by the large rectangle? 439 00:23:56,410 --> 00:23:59,160 And, by the way, notice that I mean by the volume generated 440 00:23:59,160 --> 00:24:03,570 by the rectangle think of this as being a slab of a certain 441 00:24:03,570 --> 00:24:06,530 amount of material and I rotate that slab around 442 00:24:06,530 --> 00:24:08,080 through 360 degrees. 443 00:24:08,080 --> 00:24:11,420 The volume I'm thinking of is the volume of the material in 444 00:24:11,420 --> 00:24:13,930 that slab, not the material that's enclosed. 445 00:24:13,930 --> 00:24:15,490 It would be like, if you're thinking in 446 00:24:15,490 --> 00:24:17,910 terms of a tin can. 447 00:24:17,910 --> 00:24:21,150 I'm not thinking of the volume enclosed by the tin can. 448 00:24:21,150 --> 00:24:23,850 I'm thinking of the volume of the tin itself that 449 00:24:23,850 --> 00:24:24,930 makes up the can. 450 00:24:24,930 --> 00:24:28,410 Well, you see again, to go through this thing as rapidly 451 00:24:28,410 --> 00:24:32,010 as possible but still hitting the main points, you see, 452 00:24:32,010 --> 00:24:34,920 notice that the volume that I'm looking for, what is the 453 00:24:34,920 --> 00:24:38,050 volume that's cut out by this big rectangle? 454 00:24:38,050 --> 00:24:44,240 Well, notice that the area of the base from-- if I look at 455 00:24:44,240 --> 00:24:51,170 this as being this cylinder minus this cylinder, the 456 00:24:51,170 --> 00:24:55,560 volume of the big cylinder is pi times ''x + delta X' 457 00:24:55,560 --> 00:24:58,560 squared' times the height here, which is 'f 458 00:24:58,560 --> 00:25:01,090 of 'x + delta X'. 459 00:25:01,090 --> 00:25:04,300 And the volume of the hollow part from 460 00:25:04,300 --> 00:25:06,090 here to here is what? 461 00:25:06,090 --> 00:25:07,780 It's 'pi x squared'-- 462 00:25:07,780 --> 00:25:09,420 that's the radius of the base-- 463 00:25:09,420 --> 00:25:12,850 times the height, which is still 'f of 'x + delta X''. 464 00:25:12,850 --> 00:25:19,550 In other words, 'delta V' is bounded above by this volume. 465 00:25:19,550 --> 00:25:22,150 In other words, as messy as this looks, that's only what? 466 00:25:22,150 --> 00:25:25,610 That's the volume of the region 467 00:25:25,610 --> 00:25:28,370 generated by this big rectangle. 468 00:25:28,370 --> 00:25:30,970 If we take the smallest rectangle, namely the one 469 00:25:30,970 --> 00:25:34,660 that's inscribed inside this region, we get the same 470 00:25:34,660 --> 00:25:38,230 results, except that the height is now replaced by 'f 471 00:25:38,230 --> 00:25:42,300 of x' rather than by 'f of 'x + delta X''. 472 00:25:42,300 --> 00:25:45,670 In other words, we catch 'delta V' between two 473 00:25:45,670 --> 00:25:48,820 expressions involving 'x'. 474 00:25:48,820 --> 00:25:51,210 By the way, notice how the bracketed expression 475 00:25:51,210 --> 00:25:55,370 simplifies the 'pi x squared' here cancels with the 'pi x 476 00:25:55,370 --> 00:25:59,700 squared' here leaving inside the parentheses just '2x delta 477 00:25:59,700 --> 00:26:02,130 X' plus 'delta X squared'. 478 00:26:02,130 --> 00:26:04,900 In other words, simplifying this thing, I can now show 479 00:26:04,900 --> 00:26:09,910 that 'delta V' is caught between these two expressions 480 00:26:09,910 --> 00:26:13,520 now, this expression here, which is too big, and this 481 00:26:13,520 --> 00:26:16,490 expression here, which is too small. 482 00:26:16,490 --> 00:26:19,470 Now I divide by 'delta X'. 483 00:26:19,470 --> 00:26:23,470 The usual procedure to find 'dV dx', it's 'delta V' 484 00:26:23,470 --> 00:26:24,960 divided by 'delta X'. 485 00:26:24,960 --> 00:26:28,140 Then I will take the limit as 'delta X' approaches 0. 486 00:26:28,140 --> 00:26:30,240 You see, so I divide through by 'delta X'. 487 00:26:30,240 --> 00:26:33,040 We're assuming, of course, that 'delta X' is not 0. 488 00:26:33,040 --> 00:26:36,660 That's what the limit means as 'delta X' approaches 0. 489 00:26:36,660 --> 00:26:40,330 You see, it's not zero, but it gets arbitrarily close to 0. 490 00:26:40,330 --> 00:26:44,680 Notice then that my 'delta V' divided by 'delta X' is caught 491 00:26:44,680 --> 00:26:51,190 between pi times '2x + delta X' times 'f of 'x + delta X'', 492 00:26:51,190 --> 00:26:56,480 and pi times '2x + delta X' times 'f of x'. 493 00:26:56,480 --> 00:26:59,350 And I now let 'delta X' approach 0. 494 00:26:59,350 --> 00:27:00,580 And here's the key point. 495 00:27:00,580 --> 00:27:04,470 As 'delta X' approaches 0, notice that the left hand side 496 00:27:04,470 --> 00:27:09,540 becomes '2 pi x' times 'f of x'. 497 00:27:09,540 --> 00:27:13,140 Notice also what happens to the right hand side. 498 00:27:13,140 --> 00:27:17,200 This factor, as 'delta X' approaches 0, becomes '2x'. 499 00:27:17,200 --> 00:27:22,220 And because 'f' is continuous, as 'delta X' approaches 0, 'f 500 00:27:22,220 --> 00:27:26,370 of 'x + delta X'' approaches 'f of x'. 501 00:27:26,370 --> 00:27:29,510 In other words, then, in the limit, as 'delta X' approaches 502 00:27:29,510 --> 00:27:36,560 0, I have that 'dV dx' on the one hand can't be any greater 503 00:27:36,560 --> 00:27:38,780 than '2 pi x' times 'f of x'. 504 00:27:38,780 --> 00:27:41,640 On the other hand, it can't be any less than '2 pi 505 00:27:41,640 --> 00:27:43,410 x' times 'f of x'. 506 00:27:43,410 --> 00:27:48,770 Consequently, it must equal '2 pi x' times 'f of x'. 507 00:27:48,770 --> 00:27:52,600 Therefore, if this is 'dV dx', then 'V' itself is the 508 00:27:52,600 --> 00:27:57,830 integral of this thing evaluated between 'a' and 'b', 509 00:27:57,830 --> 00:27:59,620 because that's where we're adding these things up from. 510 00:27:59,620 --> 00:28:04,550 In other words, if we're using differential calculus, this is 511 00:28:04,550 --> 00:28:10,230 'G of b' minus 'G of a' where 'G prime' equals 'f'. 512 00:28:10,230 --> 00:28:13,090 If we're using integral calculus, we've found the 'U 513 00:28:13,090 --> 00:28:17,510 sub n' and 'L sub n' and we've caught 'V' between 'U sub n' 514 00:28:17,510 --> 00:28:19,220 and 'L sub n'. 515 00:28:19,220 --> 00:28:23,180 But, in any event, what we've shown rigorously now is that 516 00:28:23,180 --> 00:28:25,500 by the cylindrical shell method-- and we'll illustrate 517 00:28:25,500 --> 00:28:28,900 these with examples to finish off today's lesson-- 518 00:28:28,900 --> 00:28:32,500 that the volume is given by integral from 'a' 519 00:28:32,500 --> 00:28:34,830 to 'b' '2 pi x'-- 520 00:28:34,830 --> 00:28:37,120 and let me just replace 'f of x' by 'y' to 521 00:28:37,120 --> 00:28:38,900 make my diagram simpler-- 522 00:28:38,900 --> 00:28:40,470 times 'dx'. 523 00:28:40,470 --> 00:28:42,940 And if you want to think of this in what I call the 524 00:28:42,940 --> 00:28:45,690 traditional engineering point of view where you think of a 525 00:28:45,690 --> 00:28:49,600 thin rectangle generating a volume, what we're saying is 526 00:28:49,600 --> 00:28:53,760 if you think of a little thin piece like this being revolved 527 00:28:53,760 --> 00:28:57,760 to generate, say, some material in a tin can, notice 528 00:28:57,760 --> 00:29:01,310 that the amount of material in here will be what? 529 00:29:01,310 --> 00:29:06,030 Well, when you unroll this thing-- 530 00:29:06,030 --> 00:29:07,350 see, this thing sort of like this. 531 00:29:07,350 --> 00:29:12,600 When you unroll this thing, the radius is 'x', so the 532 00:29:12,600 --> 00:29:16,300 circumference when you unroll it will be '2 pi x'. 533 00:29:16,300 --> 00:29:17,870 The height is 'y'. 534 00:29:17,870 --> 00:29:21,940 So the cross-sectional area is '2 pi x y' and the 535 00:29:21,940 --> 00:29:23,860 thickness is 'dx'. 536 00:29:23,860 --> 00:29:28,550 So if I multiply that by dx, that gives me the volume 537 00:29:28,550 --> 00:29:29,990 generated by this piece. 538 00:29:29,990 --> 00:29:32,270 And then in the proud tradition of the sigma 539 00:29:32,270 --> 00:29:35,780 notation, which I'll come back to in the next lecture to show 540 00:29:35,780 --> 00:29:37,700 how dangerous this really is, but the 541 00:29:37,700 --> 00:29:39,120 shortcut method is what? 542 00:29:39,120 --> 00:29:42,480 Add up all of these contributions as 'x' goes from 543 00:29:42,480 --> 00:29:45,370 'a' to 'b'. 544 00:29:45,370 --> 00:29:47,950 At any rate, that's called the method of cylindrical shells. 545 00:29:47,950 --> 00:29:51,230 Essentially, one uses cylindrical shells when we 546 00:29:51,230 --> 00:29:54,000 think of a generating element being parallel 547 00:29:54,000 --> 00:29:55,700 to the axis of rotation. 548 00:29:55,700 --> 00:29:58,790 We use revolution, when it's perpendicular 549 00:29:58,790 --> 00:30:00,220 to the axis of rotation. 550 00:30:00,220 --> 00:30:04,530 Which of the two is easier depends on the particular 551 00:30:04,530 --> 00:30:07,830 computational technique necessitated by the 552 00:30:07,830 --> 00:30:10,930 relationship between the variables in the problem. 553 00:30:10,930 --> 00:30:14,050 Well, at any rate, let's do a couple of examples. 554 00:30:14,050 --> 00:30:17,340 The first example I'd like to do is to take that same region 555 00:30:17,340 --> 00:30:21,680 'R', namely, the right triangle whose 556 00:30:21,680 --> 00:30:24,000 legs are 'r' and 'h'. 557 00:30:24,000 --> 00:30:28,150 We've already solved this problem of finding the volume 558 00:30:28,150 --> 00:30:30,620 when we rotate this about the x-axis. 559 00:30:30,620 --> 00:30:33,830 What I'd now like to do is see what volume is generated by 560 00:30:33,830 --> 00:30:36,990 this as I rotate it about the y-axis. 561 00:30:36,990 --> 00:30:40,320 And again, I find that I can do this problem in several 562 00:30:40,320 --> 00:30:43,580 ways, but I thought it was an easy enough problem to do by 563 00:30:43,580 --> 00:30:46,660 cylindrical shells, because, as we so often do, I thought 564 00:30:46,660 --> 00:30:49,240 the first problem that we do by cylindrical shells should 565 00:30:49,240 --> 00:30:51,650 be one that we can check by another method. 566 00:30:51,650 --> 00:30:55,260 But at any rate, using cylindrical shells, let's see 567 00:30:55,260 --> 00:30:57,330 what happens over here. 568 00:30:57,330 --> 00:30:58,530 The volume is what? 569 00:30:58,530 --> 00:31:05,020 It's the integral from 0 to 'h', '2 pi x' times this 570 00:31:05,020 --> 00:31:09,160 height, which is 'rx' over 'h' integrated 571 00:31:09,160 --> 00:31:10,790 with respect to 'x'. 572 00:31:10,790 --> 00:31:13,900 That's just '2 pi r' over 'h'-- we can take that 573 00:31:13,900 --> 00:31:15,880 outside, because that's a constant factor-- 574 00:31:15,880 --> 00:31:17,980 integral 'x squared dx'. 575 00:31:17,980 --> 00:31:20,930 The integral of 'x squared' is '1/3 x cubed'. 576 00:31:20,930 --> 00:31:23,540 We evaluate that between 0 and 'h'. 577 00:31:23,540 --> 00:31:25,690 That gives us '1/3 h cubed'. 578 00:31:25,690 --> 00:31:28,760 We cancel the 'h' in the denominator with one of the 579 00:31:28,760 --> 00:31:31,740 'h's in the numerator, and we find that the volume that's 580 00:31:31,740 --> 00:31:37,220 generated is '2/3 pi r h squared', not 'r squared h', 581 00:31:37,220 --> 00:31:40,735 'r h squared', not 1/3, 2/3. 582 00:31:40,735 --> 00:31:43,910 Remember, by the way, what this thing looks like. 583 00:31:43,910 --> 00:31:45,290 I think you can visualize this. 584 00:31:45,290 --> 00:31:49,700 This is a cylinder with a cone cut out of it. 585 00:31:49,700 --> 00:31:52,190 See, in other words, if this thing had been solid, we'd 586 00:31:52,190 --> 00:31:55,340 have called it a right circular cylinder, and then 587 00:31:55,340 --> 00:31:59,320 what's missing is the cone shaped region over here. 588 00:31:59,320 --> 00:32:01,530 In fact, that's how we can check this. 589 00:32:01,530 --> 00:32:05,090 See, what would the volume be that's 590 00:32:05,090 --> 00:32:07,300 generated by this rectangle? 591 00:32:07,300 --> 00:32:10,990 This would be a cylinder the radius of whose base is 'h', 592 00:32:10,990 --> 00:32:14,540 whose height is 'r', and the volume of that cylinder is 'pi 593 00:32:14,540 --> 00:32:16,230 h squared r'. 594 00:32:16,230 --> 00:32:19,260 The cone that's missing, the cone that was cut out of this 595 00:32:19,260 --> 00:32:22,920 thing, has the radius of a space equal to 'h' and its 596 00:32:22,920 --> 00:32:26,510 height equal to 'r', so its volume is '1/3 597 00:32:26,510 --> 00:32:29,090 pi h squared r'. 598 00:32:29,090 --> 00:32:31,420 And, therefore, the volume that's left when we subtract 599 00:32:31,420 --> 00:32:35,210 this off is '2/3 pi h squared r', which 600 00:32:35,210 --> 00:32:37,640 does check with this. 601 00:32:37,640 --> 00:32:41,630 By the way, just as an aside, notice that the region 'R' 602 00:32:41,630 --> 00:32:45,020 generates a different volume if we rotate it about the 603 00:32:45,020 --> 00:32:49,620 y-axis than it did if we rotate it about the x-axis. 604 00:32:49,620 --> 00:32:53,980 Numerically, what we're saying is that we just found that the 605 00:32:53,980 --> 00:32:57,230 volume when rotated around the y-axis is '2/3 606 00:32:57,230 --> 00:32:58,940 pi h squared r'. 607 00:32:58,940 --> 00:33:02,790 We know that when we revolve that about the x-axis, it's 608 00:33:02,790 --> 00:33:05,470 '1/3 pi r squared h', and these two 609 00:33:05,470 --> 00:33:07,580 expressions are not identical. 610 00:33:07,580 --> 00:33:12,320 In fact, if we divide both sides by 'pi r h' over 3, we 611 00:33:12,320 --> 00:33:16,830 find that equality holds only if we have that highly 612 00:33:16,830 --> 00:33:19,540 specialized case that '2h' equals 'r', which is not 613 00:33:19,540 --> 00:33:20,710 really important. 614 00:33:20,710 --> 00:33:22,550 I just threw that in as an aside. 615 00:33:22,550 --> 00:33:25,830 But I do want you to notice that the same area, of course, 616 00:33:25,830 --> 00:33:29,050 generates different volumes depending on what you rotate 617 00:33:29,050 --> 00:33:30,650 it with respect to. 618 00:33:30,650 --> 00:33:33,270 Well, at any rate, at least this was a problem that we 619 00:33:33,270 --> 00:33:34,600 could check by another method. 620 00:33:34,600 --> 00:33:37,350 Let me just use cylindrical shells for a problem which is 621 00:33:37,350 --> 00:33:40,130 slightly tougher but one that can still be checked by 622 00:33:40,130 --> 00:33:41,420 another method. 623 00:33:41,420 --> 00:33:43,330 Let's take the following region. 624 00:33:43,330 --> 00:33:47,620 Let's take the curve 'y' equals '2x - x squared' 625 00:33:47,620 --> 00:33:51,340 between 'x' equals 0 and 'x' equals 2. 626 00:33:51,340 --> 00:33:55,250 Leaving the details as a rather trivial exercise, it is 627 00:33:55,250 --> 00:33:59,710 not difficult to see that this is the parabola that peaks at 628 00:33:59,710 --> 00:34:06,270 1, 1 and crosses the x-axis at 'x' equals 0 and 'x' equals 2. 629 00:34:06,270 --> 00:34:10,310 If I now want to compute this volume as I rotate the region 630 00:34:10,310 --> 00:34:13,120 'R' about the y-axis-- 631 00:34:13,120 --> 00:34:15,469 see, I'm going to rotate this about the y-axis. 632 00:34:15,469 --> 00:34:18,320 I want to find out what volume is generated by this region 633 00:34:18,320 --> 00:34:19,699 'R' in this case. 634 00:34:19,699 --> 00:34:25,530 Remember I can use either cylindrical shells or I can 635 00:34:25,530 --> 00:34:29,780 use revolution here. 636 00:34:29,780 --> 00:34:32,270 Notice the problem I'm going to be in. 637 00:34:32,270 --> 00:34:35,159 Notice that this particular function is single valued but 638 00:34:35,159 --> 00:34:36,239 not one-to-one. 639 00:34:36,239 --> 00:34:38,830 When I try to find these two 'x' values I'm going to run 640 00:34:38,830 --> 00:34:40,060 into multi-values. 641 00:34:40,060 --> 00:34:41,489 I'm going to have to invert. 642 00:34:41,489 --> 00:34:44,770 All sorts of computational skills are going to 643 00:34:44,770 --> 00:34:46,000 come into play here. 644 00:34:46,000 --> 00:34:49,060 On the other hand, if I take my generating element parallel 645 00:34:49,060 --> 00:34:52,510 to the y-axis, I have a very simple expression for this, 646 00:34:52,510 --> 00:34:55,960 and now that indicates what? 647 00:34:55,960 --> 00:34:59,320 By the method of cylindrical shells, this will be the 648 00:34:59,320 --> 00:35:02,210 integral from 0 to 2. 649 00:35:02,210 --> 00:35:05,930 The generating arm is 'x', so that cuts out '2 pi x'. 650 00:35:05,930 --> 00:35:09,470 The height is 'y', which is '2x - x squared'. 651 00:35:09,470 --> 00:35:10,850 The thickness is 'dx'. 652 00:35:10,850 --> 00:35:13,240 In other words, mechanically I must evaluate 653 00:35:13,240 --> 00:35:15,230 this particular integral. 654 00:35:15,230 --> 00:35:16,000 OK? 655 00:35:16,000 --> 00:35:19,580 At any rate, it's factoring out the 2 pi and just 656 00:35:19,580 --> 00:35:22,090 observing that the integral of '2 x 657 00:35:22,090 --> 00:35:24,040 squared' is '2/3 x cubed'. 658 00:35:24,040 --> 00:35:25,340 The integral of 'x' to the fourth is 659 00:35:25,340 --> 00:35:27,070 '1/4 'x to the 4th''. 660 00:35:27,070 --> 00:35:32,190 Evaluating that between 0 and 2, I get 2 pi 661 00:35:32,190 --> 00:35:35,490 times 16/3 minus 4. 662 00:35:35,490 --> 00:35:37,300 The lower limit is 0 here. 663 00:35:37,300 --> 00:35:42,330 This just comes out to be 16 minus 12, 4/3 times 2 pi. 664 00:35:42,330 --> 00:35:44,530 That's 8 pi over 3. 665 00:35:44,530 --> 00:35:48,150 By the way, before I go any further with this, let's make 666 00:35:48,150 --> 00:35:50,350 the interesting observation. 667 00:35:50,350 --> 00:35:53,480 See, 8 pi over 3 is the volume generated by this whole thing 668 00:35:53,480 --> 00:35:57,140 being revolved about the y-axis. 669 00:35:57,140 --> 00:36:00,340 If I'd drawn in this line, which is a line of symmetry, 670 00:36:00,340 --> 00:36:02,980 notice that these two areas are congruent. 671 00:36:02,980 --> 00:36:04,780 These two regions are congruent. 672 00:36:04,780 --> 00:36:08,710 However, it's also interesting to observe that the volume 673 00:36:08,710 --> 00:36:12,700 generated by this piece as you revolve it about the y-axis is 674 00:36:12,700 --> 00:36:15,390 not twice the volume generated by this piece. 675 00:36:18,640 --> 00:36:21,190 See, notice that the integral from 0 to 1-- in other words, 676 00:36:21,190 --> 00:36:23,940 if we just took this region here, integrated this from 0 677 00:36:23,940 --> 00:36:32,880 to 1, we would get 5/6, 2/3 minus 1/4 times 2 pi, 5/6. 678 00:36:32,880 --> 00:36:36,920 And if we double that, we would get 5/3. 679 00:36:36,920 --> 00:36:39,450 In other words, this area is 5/6. 680 00:36:39,450 --> 00:36:44,470 Double it would be 5 pi over 6. 681 00:36:44,470 --> 00:36:48,880 Double it would be 5 pi over 3, and 5 pi over 3 is not the 682 00:36:48,880 --> 00:36:51,330 same as 8 pi over 3. 683 00:36:51,330 --> 00:36:54,060 The thing to keep in mind here is notice how the distance 684 00:36:54,060 --> 00:36:55,070 comes in again. 685 00:36:55,070 --> 00:37:01,530 You see, for example, these two lines here are symmetric 686 00:37:01,530 --> 00:37:05,110 with respect to the line 'x' equals 1. 687 00:37:05,110 --> 00:37:09,110 But notice that this generates a much larger volume than this 688 00:37:09,110 --> 00:37:11,720 because its generating arm is longer. 689 00:37:11,720 --> 00:37:13,200 It's further away. 690 00:37:13,200 --> 00:37:15,330 But at any rate, that's just an aside. 691 00:37:15,330 --> 00:37:17,790 Notice how by the method of cylindrical shells, we 692 00:37:17,790 --> 00:37:20,620 determine the volume is 8 pi over 3. 693 00:37:20,620 --> 00:37:24,620 Suppose we'd wanted to do this by the solid method-- the 694 00:37:24,620 --> 00:37:26,470 solid revolution method. 695 00:37:26,470 --> 00:37:28,850 Notice that we would first have to invert this 696 00:37:28,850 --> 00:37:29,880 relationship. 697 00:37:29,880 --> 00:37:33,260 We would first have to solve for 'x' in terms of 'y'. 698 00:37:33,260 --> 00:37:37,320 Notice that 'y' equals '2x - x squared' is the same as saying 699 00:37:37,320 --> 00:37:41,140 that ''x squared' - 2x + y' is 0. 700 00:37:41,140 --> 00:37:44,680 Using the quadratic formula, we can solve for 'x', and we 701 00:37:44,680 --> 00:37:49,430 now find that 'x' is 1 plus the square root of '1 - y' or 702 00:37:49,430 --> 00:37:52,800 1 minus the square root of '1 - y'. 703 00:37:52,800 --> 00:37:57,050 What that means, by the way, geometrically, is simply this. 704 00:37:57,050 --> 00:38:01,720 For a given value of 'y', there are two values of 'x' 705 00:38:01,720 --> 00:38:06,340 located symmetrically with respect to 'x' equals 1. 706 00:38:06,340 --> 00:38:08,200 See, they're on symmetrical portions. 707 00:38:08,200 --> 00:38:09,920 Well, this doesn't make that much difference. 708 00:38:09,920 --> 00:38:11,300 The thing now is what? 709 00:38:11,300 --> 00:38:13,680 What is my cross-sectional area? 710 00:38:13,680 --> 00:38:19,130 My cross-sectional area is pi times this length squared 711 00:38:19,130 --> 00:38:22,680 minus pi times this length squared. 712 00:38:22,680 --> 00:38:26,280 That's pi times 1 plus the square root of ''1 - y' 713 00:38:26,280 --> 00:38:30,280 squared' minus pi times the square root of 1 minus the 714 00:38:30,280 --> 00:38:32,670 square root of ''1 - y' squared'. 715 00:38:32,670 --> 00:38:35,790 When I square this and subtract, all but the middle 716 00:38:35,790 --> 00:38:37,180 term drops out. 717 00:38:37,180 --> 00:38:40,640 In other words, I have twice the square root of '1 - y' 718 00:38:40,640 --> 00:38:44,210 here minus twice the square root of '1 - y' here. 719 00:38:44,210 --> 00:38:48,610 When I subtract, I get 4 times the square root of '1 - y'. 720 00:38:48,610 --> 00:38:50,560 I multiply that by pi. 721 00:38:50,560 --> 00:38:53,180 That's my cross-sectional area. 722 00:38:53,180 --> 00:38:56,260 And now to find the volume, I just integrate that as 'y' 723 00:38:56,260 --> 00:38:58,800 goes from 0 to 1. 724 00:38:58,800 --> 00:39:02,100 You see, and if I carry out this integration, noticing 725 00:39:02,100 --> 00:39:06,620 that the integral of ''1 - y' to the 1/2' is minus 2/3. 726 00:39:06,620 --> 00:39:08,800 Remember, the derivative of '1 - y' with respect to 727 00:39:08,800 --> 00:39:11,610 'y' is minus 1. 728 00:39:11,610 --> 00:39:13,390 ''1 - y' to the 3/2'. 729 00:39:13,390 --> 00:39:15,510 Evaluate that between 0 and 1. 730 00:39:15,510 --> 00:39:17,220 The upper limit gives me 0. 731 00:39:17,220 --> 00:39:19,370 The lower limit is minus 2/3. 732 00:39:19,370 --> 00:39:20,550 I subtract the lower limit. 733 00:39:20,550 --> 00:39:21,800 It gives me 2/3. 734 00:39:21,800 --> 00:39:25,860 4 pi times 2/3 is 8 pi over 3, the same 735 00:39:25,860 --> 00:39:27,940 answer as I got before. 736 00:39:27,940 --> 00:39:30,760 Notice, by the way, that this was messy, but we 737 00:39:30,760 --> 00:39:31,770 could handle it. 738 00:39:31,770 --> 00:39:33,580 If this had been much tougher-- 739 00:39:33,580 --> 00:39:37,680 say a 6 over here or something like that instead of a 2-- 740 00:39:37,680 --> 00:39:41,010 how would we have solved for 'x' in terms of 'y'? 741 00:39:41,010 --> 00:39:43,370 You see, in other words, this would have been a case where 742 00:39:43,370 --> 00:39:46,250 the shell method would have been necessitated because of 743 00:39:46,250 --> 00:39:48,780 the impossibility of the algebra. 744 00:39:48,780 --> 00:39:51,660 But at any rate, we have plenty of opportunity to 745 00:39:51,660 --> 00:39:55,550 illustrate that in terms of exercises and supplementary 746 00:39:55,550 --> 00:39:58,100 notes and reading material and what have you. 747 00:39:58,100 --> 00:40:00,330 That is actually the easiest part. 748 00:40:00,330 --> 00:40:02,840 The hard part is to understand the significance of what's 749 00:40:02,840 --> 00:40:07,170 going on, so I thought that to summarize today's lecture, 750 00:40:07,170 --> 00:40:10,620 let's keep in mind that whether you call it area or 751 00:40:10,620 --> 00:40:14,530 whether you call it volume or whether you call it distance 752 00:40:14,530 --> 00:40:18,870 traveled in velocity, the fact remains that if 'f' is a 753 00:40:18,870 --> 00:40:21,800 function continuous on the closed interval from 'a' to 754 00:40:21,800 --> 00:40:27,930 'b', and we partition that interval into 'n' parts, and 755 00:40:27,930 --> 00:40:32,270 we form the sum as 'k' goes from 1 to 'n', 'f of 'c sub 756 00:40:32,270 --> 00:40:37,150 k'' times 'delta x sub k', where 'c sub k' is in the k-th 757 00:40:37,150 --> 00:40:40,750 interval, and 'delta x sub k' is just 'x sub k' 758 00:40:40,750 --> 00:40:43,130 minus 'x sub 'k - 1''. 759 00:40:43,130 --> 00:40:48,200 If we take that limit as the largest 'delta x' approaches 0 760 00:40:48,200 --> 00:40:52,960 and call that 'Q', that limit 'Q' exists. 761 00:40:52,960 --> 00:40:55,940 Symbolically, it's written by the definite integral from 'a' 762 00:40:55,940 --> 00:41:01,660 to 'b', ''f of x' dx', and, more to the point, if you 763 00:41:01,660 --> 00:41:05,200 happen to know differential calculus, you can compute 'Q' 764 00:41:05,200 --> 00:41:09,430 just by computing 'G of b' minus 'G of a', where 'G 765 00:41:09,430 --> 00:41:13,530 prime' is any function whose derivative is 'f'. 766 00:41:13,530 --> 00:41:14,290 OK? 767 00:41:14,290 --> 00:41:16,800 Now again, this is why I'm calling it a summary. 768 00:41:16,800 --> 00:41:20,570 If you separate this out from all of the computational stuff 769 00:41:20,570 --> 00:41:23,010 that we did in today's lecture, this is the part 770 00:41:23,010 --> 00:41:24,680 that's left. 771 00:41:24,680 --> 00:41:25,500 OK? 772 00:41:25,500 --> 00:41:29,080 And what I want to do next time is to show you that 773 00:41:29,080 --> 00:41:32,170 things are not quite this straightforward all the time, 774 00:41:32,170 --> 00:41:34,860 that certain nice things have been happening here that allow 775 00:41:34,860 --> 00:41:36,820 us, essentially, to get away with murder. 776 00:41:36,820 --> 00:41:40,570 And what I mean by that will become clearer next time, but 777 00:41:40,570 --> 00:41:42,220 until next time then goodbye. 778 00:41:45,130 --> 00:41:47,510 MALE ANNOUNCER: Funding for the publication of this video 779 00:41:47,510 --> 00:41:52,380 was provided by the Gabriella and Paul Rosenbaum Foundation. 780 00:41:52,380 --> 00:41:56,560 Help OCW continue to provide free and open access to MIT 781 00:41:56,560 --> 00:42:00,760 courses by making a donation at ocw.mit.edu/donate.