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 --> 00:00:09,930 continue to offer high quality educational resources for free. 5 00:00:09,930 --> 00:00:12,550 To make a donation or to view additional materials 6 00:00:12,550 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,150 --> 00:00:21,580 at ocw.mit.edu. 8 00:00:21,580 --> 00:00:23,630 PROFESSOR: OK. 9 00:00:23,630 --> 00:00:28,710 Now, today we get to move on from integral formulas 10 00:00:28,710 --> 00:00:34,620 and methods of integration back to some geometry. 11 00:00:34,620 --> 00:00:36,870 And this is more or less going to lead 12 00:00:36,870 --> 00:00:39,270 into the kinds of tools you'll be using 13 00:00:39,270 --> 00:00:43,120 in multivariable calculus. 14 00:00:43,120 --> 00:00:45,420 The first thing that we're going to do today 15 00:00:45,420 --> 00:00:58,220 is discuss arc length. 16 00:00:58,220 --> 00:01:02,940 Like all of the cumulative sums that we've worked on, 17 00:01:02,940 --> 00:01:06,710 this one has a storyline and a picture associated to it, 18 00:01:06,710 --> 00:01:09,360 which involves dividing things up. 19 00:01:09,360 --> 00:01:11,690 If you have a roadway, if you like, 20 00:01:11,690 --> 00:01:18,290 and you have mileage markers along the road, like this, 21 00:01:18,290 --> 00:01:22,490 all the way up to, say, s_n here, 22 00:01:22,490 --> 00:01:28,740 then the length along the road is described by this parameter, 23 00:01:28,740 --> 00:01:30,560 s. 24 00:01:30,560 --> 00:01:32,930 Which is arc length. 25 00:01:32,930 --> 00:01:37,680 And if we look at a graph of this sort of thing, 26 00:01:37,680 --> 00:01:42,160 if this is the last point b, and this is the first point a, 27 00:01:42,160 --> 00:01:47,120 then you can think in terms of having points above x_1, x_2, 28 00:01:47,120 --> 00:01:50,460 x_3, etc. 29 00:01:50,460 --> 00:01:53,940 The same as we did with Riemann sums. 30 00:01:53,940 --> 00:01:56,050 And then the way that we're going 31 00:01:56,050 --> 00:02:07,540 to approximate this is by taking the straight lines between each 32 00:02:07,540 --> 00:02:09,470 of these points. 33 00:02:09,470 --> 00:02:11,360 As things get smaller and smaller, 34 00:02:11,360 --> 00:02:15,210 the straight line is going to be fairly close to the curve. 35 00:02:15,210 --> 00:02:16,980 And that's the main idea. 36 00:02:16,980 --> 00:02:20,050 So let me just depict one little chunk of this. 37 00:02:20,050 --> 00:02:20,900 Which is like this. 38 00:02:20,900 --> 00:02:24,530 One straight line, and here's the curved surface there. 39 00:02:24,530 --> 00:02:26,830 And the distance along the curved surface is what 40 00:02:26,830 --> 00:02:31,690 I'm calling delta s, the change in the length between-- 41 00:02:31,690 --> 00:02:34,920 so this would be s_2 - s_1 if I depicted that one. 42 00:02:34,920 --> 00:02:42,490 So this would be delta s is, say s. s_i - s_(i-1), 43 00:02:42,490 --> 00:02:45,240 some increment there. 44 00:02:45,240 --> 00:02:49,670 And then I can figure out what the length 45 00:02:49,670 --> 00:02:51,750 of the orange segment is. 46 00:02:51,750 --> 00:02:54,630 Because the horizontal distance is delta x. 47 00:02:54,630 --> 00:02:57,290 And the vertical distance is delta y. 48 00:02:57,290 --> 00:03:02,950 And so the formula is that the hypotenuse is delta (delta x)^2 49 00:03:02,950 --> 00:03:06,470 + (delta y)^2. 50 00:03:06,470 --> 00:03:09,420 Square root. 51 00:03:09,420 --> 00:03:11,870 And delta s is approximately that. 52 00:03:11,870 --> 00:03:15,871 So what we're saying is that (delta s)^2 is approximately 53 00:03:15,871 --> 00:03:16,370 this. 54 00:03:16,370 --> 00:03:21,870 So this is the hypotenuse. 55 00:03:21,870 --> 00:03:23,500 Squared. 56 00:03:23,500 --> 00:03:29,520 And it's very close to the length of the curve. 57 00:03:29,520 --> 00:03:34,940 And the whole idea of calculus is that in the infinitesimal, 58 00:03:34,940 --> 00:03:44,150 this is exactly correct. 59 00:03:44,150 --> 00:03:48,010 So that's what's going to happen in the limit. 60 00:03:48,010 --> 00:03:52,000 And that is the basis for calculating arc length. 61 00:03:52,000 --> 00:03:56,060 I'm going to rewrite that formula on the next board. 62 00:03:56,060 --> 00:03:59,460 But I'm going to write it in the more customary fashion. 63 00:03:59,460 --> 00:04:02,200 We've done this before, a certain amount. 64 00:04:02,200 --> 00:04:03,970 But I just want to emphasize it here 65 00:04:03,970 --> 00:04:09,390 because this handwriting is a little bit peculiar. 66 00:04:09,390 --> 00:04:11,710 This ds is really all one thing. 67 00:04:11,710 --> 00:04:15,690 What I really mean is to put the parenthesis around it. 68 00:04:15,690 --> 00:04:16,660 It's one thing. 69 00:04:16,660 --> 00:04:19,070 It's not d times s, it's ds. 70 00:04:19,070 --> 00:04:20,020 It's one thing. 71 00:04:20,020 --> 00:04:20,940 And we square it. 72 00:04:20,940 --> 00:04:22,750 But for whatever reason people have 73 00:04:22,750 --> 00:04:26,112 gotten into the habit of omitting the parentheses. 74 00:04:26,112 --> 00:04:28,070 So you're just going to have to live with that. 75 00:04:28,070 --> 00:04:31,050 And realize that this is not d of s^2 or anything like that. 76 00:04:31,050 --> 00:04:32,755 And similarly, this is a single number, 77 00:04:32,755 --> 00:04:34,130 and this is a single number. 78 00:04:34,130 --> 00:04:36,060 Infinitesimal. 79 00:04:36,060 --> 00:04:39,400 So that's just the way that this idea here 80 00:04:39,400 --> 00:04:42,194 gets written in our notation. 81 00:04:42,194 --> 00:04:43,860 And this is the first time we're dealing 82 00:04:43,860 --> 00:04:46,070 with squares of infinitesimals. 83 00:04:46,070 --> 00:04:47,600 So it's just a little different. 84 00:04:47,600 --> 00:04:49,641 But immediately the first thing we're going to do 85 00:04:49,641 --> 00:04:51,750 is take the square root. 86 00:04:51,750 --> 00:04:56,660 If I take the square root, that's the square root of dx^2 87 00:04:56,660 --> 00:04:58,640 + dy^2. 88 00:04:58,640 --> 00:05:02,400 And this is the form in which I always remember this formula. 89 00:05:02,400 --> 00:05:07,030 Let's put it in some brightly decorated form. 90 00:05:07,030 --> 00:05:12,380 But there are about four, five, six other forms 91 00:05:12,380 --> 00:05:16,110 that you'll derive from this, which all mean the same thing. 92 00:05:16,110 --> 00:05:18,479 So this is, as I say, the way I remember it. 93 00:05:18,479 --> 00:05:20,270 But there are other ways of thinking of it. 94 00:05:20,270 --> 00:05:23,070 And let's just write a couple of them down. 95 00:05:23,070 --> 00:05:27,380 The first one is that you can factor out the dx. 96 00:05:27,380 --> 00:05:29,480 So that looks like this. 97 00:05:29,480 --> 00:05:34,680 1 + (dy / dx)^2. 98 00:05:34,680 --> 00:05:37,630 And then I factored out the dx. 99 00:05:37,630 --> 00:05:39,820 So this is a variant. 100 00:05:39,820 --> 00:05:43,190 And this is the one which actually we'll 101 00:05:43,190 --> 00:05:47,490 be using in practice right now on our examples. 102 00:05:47,490 --> 00:05:55,600 So the conclusion is that the arc length, 103 00:05:55,600 --> 00:06:05,320 which if you like is this total s_n - s_0, if you like, 104 00:06:05,320 --> 00:06:09,390 is going to be equal to the integral from a to b 105 00:06:09,390 --> 00:06:16,430 of the square root of 1 + (dy/dx)^2, dx. 106 00:06:20,490 --> 00:06:27,500 In practice, it's also very often written informally 107 00:06:27,500 --> 00:06:29,090 as this. 108 00:06:29,090 --> 00:06:30,750 The integral ds. 109 00:06:30,750 --> 00:06:32,830 So the change in this little variable 110 00:06:32,830 --> 00:06:40,009 s, and this is what you'll see notationally in many textbooks. 111 00:06:40,009 --> 00:06:42,550 So that's one way of writing it, and of course the second way 112 00:06:42,550 --> 00:06:45,950 of writing it which is practically the same thing is 113 00:06:45,950 --> 00:06:50,040 square root of 1 + f'(x)^2, dx. 114 00:06:50,040 --> 00:06:52,330 Mixing in a little bit of Newton's notation. 115 00:06:52,330 --> 00:06:57,250 And this is with y = f(x). 116 00:06:57,250 --> 00:07:03,310 So this is the formula for arc length. 117 00:07:03,310 --> 00:07:05,770 And as I say, I remember it this way. 118 00:07:05,770 --> 00:07:09,060 But you're going to have to derive various variants of it. 119 00:07:09,060 --> 00:07:11,010 And you'll have to use some arithmetic 120 00:07:11,010 --> 00:07:12,650 to get to various formulas. 121 00:07:12,650 --> 00:07:15,190 And there will be more later. 122 00:07:15,190 --> 00:07:16,030 Yeah, question. 123 00:07:16,030 --> 00:07:20,820 STUDENT: [INAUDIBLE] 124 00:07:20,820 --> 00:07:27,350 PROFESSOR: OK, the question is, is f'(x)^2 equal to f''(x). 125 00:07:27,350 --> 00:07:31,020 And the answer is no. 126 00:07:31,020 --> 00:07:33,090 And let's just see what it is. 127 00:07:33,090 --> 00:07:39,490 So, for example, if f(x) = x^2, which is an example which will 128 00:07:39,490 --> 00:07:47,910 come up in a few minutes, then f'(x) = 2x and f'(x)^2 = 129 00:07:47,910 --> 00:07:52,380 (2x)^2, which is 4x^2. 130 00:07:52,380 --> 00:07:56,490 Whereas f''(x) is equal to, if I differentiate this another 131 00:07:56,490 --> 00:07:58,750 time, it's equal to 2. 132 00:07:58,750 --> 00:08:03,440 So they don't mean the same thing. 133 00:08:03,440 --> 00:08:04,700 The same thing over here. 134 00:08:04,700 --> 00:08:07,200 You can see this dy / dx, this is the rate of change 135 00:08:07,200 --> 00:08:08,190 of y with respect to x. 136 00:08:08,190 --> 00:08:09,632 The quantity squared. 137 00:08:09,632 --> 00:08:11,340 So in other words, this thing is supposed 138 00:08:11,340 --> 00:08:13,300 to mean the same as that. 139 00:08:13,300 --> 00:08:13,800 Yeah. 140 00:08:13,800 --> 00:08:19,580 Another question. 141 00:08:19,580 --> 00:08:25,660 STUDENT: [INAUDIBLE] 142 00:08:25,660 --> 00:08:27,970 PROFESSOR: So the question is, you 143 00:08:27,970 --> 00:08:30,430 got a little nervous because I left out these limits. 144 00:08:30,430 --> 00:08:32,120 And indeed, I did that on purpose 145 00:08:32,120 --> 00:08:34,530 because I didn't want to specify what was going on. 146 00:08:34,530 --> 00:08:36,590 Really, if you wrote it in terms of ds, 147 00:08:36,590 --> 00:08:38,710 you'd have to write it as starting at s_0 148 00:08:38,710 --> 00:08:42,180 and ending at s_n to be consistent with the variable s. 149 00:08:42,180 --> 00:08:45,730 But of course, if you write it in terms of another variable, 150 00:08:45,730 --> 00:08:46,940 you put that variable in. 151 00:08:46,940 --> 00:08:49,190 So this is what we do when we change variables, right? 152 00:08:49,190 --> 00:08:51,570 We have many different choices for these limits. 153 00:08:51,570 --> 00:08:54,530 And this is the clue as to which variable we use. 154 00:08:54,530 --> 00:08:59,060 STUDENT: [INAUDIBLE] 155 00:08:59,060 --> 00:09:01,720 PROFESSOR: Correct. s_0 and s_n are not 156 00:09:01,720 --> 00:09:03,310 the same thing as a and b. 157 00:09:03,310 --> 00:09:05,290 In fact, this is x_n. 158 00:09:05,290 --> 00:09:07,740 And this x_0, over here. 159 00:09:07,740 --> 00:09:08,990 That's what a and b are. 160 00:09:08,990 --> 00:09:13,242 But s_0 and s_n are mileage markers on the road. 161 00:09:13,242 --> 00:09:15,450 They're not the same thing as keeping track of what's 162 00:09:15,450 --> 00:09:16,950 happening on the x axis. 163 00:09:16,950 --> 00:09:18,710 So when we measure arc length, remember 164 00:09:18,710 --> 00:09:27,980 it's mileage along the curved path. 165 00:09:27,980 --> 00:09:32,410 So now, I need to give you some examples. 166 00:09:32,410 --> 00:09:40,820 And my first example is going to be really basic. 167 00:09:40,820 --> 00:09:45,630 But I hope that it helps to give some perspective here. 168 00:09:45,630 --> 00:09:48,830 So I'm going to take the example y = m 169 00:09:48,830 --> 00:09:52,550 x, which is a linear function, a straight line. 170 00:09:52,550 --> 00:09:58,630 And then y' would be m, and so ds is going to be the square 171 00:09:58,630 --> 00:10:02,650 root of 1 + (y')^2, dx. 172 00:10:02,650 --> 00:10:10,840 Which is the square root of 1 + m^2, dx. 173 00:10:10,840 --> 00:10:17,700 And now, the length, say, if we go from, I don't know, 174 00:10:17,700 --> 00:10:27,300 let's say 0 to 10, let's say, of the graph is going to be 175 00:10:27,300 --> 00:10:33,990 the integral from 0 to 10 of the square root of 1 + m^2, dx. 176 00:10:33,990 --> 00:10:39,910 Which of course is just 10 square root of 1 + m^2. 177 00:10:39,910 --> 00:10:41,810 Not very surprising. 178 00:10:41,810 --> 00:10:43,360 This is a constant. 179 00:10:43,360 --> 00:10:46,830 It just factors out and the integral from 0 to 10 of dx 180 00:10:46,830 --> 00:10:51,210 is 10. 181 00:10:51,210 --> 00:10:54,340 Let's just draw a picture of this. 182 00:10:54,340 --> 00:10:57,530 This is something which has slope m here. 183 00:10:57,530 --> 00:10:59,020 And it's going to 10. 184 00:10:59,020 --> 00:11:02,340 So this horizontal is 10. 185 00:11:02,340 --> 00:11:05,930 And the vertical is 10m. 186 00:11:05,930 --> 00:11:08,030 Those are the dimensions of this. 187 00:11:08,030 --> 00:11:11,810 And the Pythagorean theorem says that the hypotenuse, 188 00:11:11,810 --> 00:11:15,457 not surprisingly, let's draw it in here in orange to remind 189 00:11:15,457 --> 00:11:18,040 ourselves that it was the same type of orange that we had over 190 00:11:18,040 --> 00:11:27,160 there, this length here is the square root of 10^2 + (10m)^2. 191 00:11:27,160 --> 00:11:31,400 That's the formula for the hypotenuse. 192 00:11:31,400 --> 00:11:38,520 And that's exactly the same as this. 193 00:11:38,520 --> 00:11:40,540 Maybe you're saying duh, this is obvious. 194 00:11:40,540 --> 00:11:44,160 But the point that I'm trying to make is this. 195 00:11:44,160 --> 00:11:48,220 If you can figure out these formulas for linear functions, 196 00:11:48,220 --> 00:11:52,510 calculus tells you how to do it for every function. 197 00:11:52,510 --> 00:11:56,370 The idea of calculus is that this easy calculation here, 198 00:11:56,370 --> 00:11:58,530 which you can do without any calculus 199 00:11:58,530 --> 00:12:04,360 at all, all of the tools, the notations of differentials 200 00:12:04,360 --> 00:12:06,115 and limits and integrals, is going 201 00:12:06,115 --> 00:12:10,040 to make you be able to do it for any curve. 202 00:12:10,040 --> 00:12:12,760 Because we can break things up into these little infinitesimal 203 00:12:12,760 --> 00:12:13,260 bits. 204 00:12:13,260 --> 00:12:15,820 This is the whole idea of all of the methods 205 00:12:15,820 --> 00:12:18,620 that we had to set up integrals here. 206 00:12:18,620 --> 00:12:25,770 This is the main point of these integrals. 207 00:12:25,770 --> 00:12:32,480 Now, so let's do something slightly more interesting. 208 00:12:32,480 --> 00:12:39,970 Our next example is going to be the circle, 209 00:12:39,970 --> 00:12:41,540 so y = square root of 1-x^2. 210 00:12:48,590 --> 00:12:51,820 If you like, that's the graph of a semicircle. 211 00:12:51,820 --> 00:12:57,040 And maybe we'll set it up here this way. 212 00:12:57,040 --> 00:13:00,030 So that the semicircle goes around like this. 213 00:13:00,030 --> 00:13:02,310 And we'll start it here at x = 0. 214 00:13:02,310 --> 00:13:04,250 And we'll go over to a. 215 00:13:04,250 --> 00:13:06,560 And we'll take this little piece of the circle. 216 00:13:06,560 --> 00:13:08,150 So down to here. 217 00:13:08,150 --> 00:13:11,850 If you like. 218 00:13:11,850 --> 00:13:14,970 So here's the portion of the circle 219 00:13:14,970 --> 00:13:17,660 that I'm going to measure the length of. 220 00:13:17,660 --> 00:13:19,110 Now, we know that length. 221 00:13:19,110 --> 00:13:20,160 It's called arc length. 222 00:13:20,160 --> 00:13:21,659 And I'm going to give it a name, I'm 223 00:13:21,659 --> 00:13:23,400 going to call it alpha here. 224 00:13:23,400 --> 00:13:39,640 So alpha's the arc length along the circle. 225 00:13:39,640 --> 00:13:42,660 Now, let's figure out what it is. 226 00:13:42,660 --> 00:13:45,970 First, in order to do this, I have to figure out what y' is. 227 00:13:45,970 --> 00:13:47,960 Or, if you like, dy/dx. 228 00:13:47,960 --> 00:13:50,680 Now, that's a calculation that we've done a number of times. 229 00:13:50,680 --> 00:13:52,830 And I'm going to do it slightly faster. 230 00:13:52,830 --> 00:13:57,160 But you remember it gives you a square root in the denominator. 231 00:13:57,160 --> 00:13:59,500 And then you have the derivative of what's 232 00:13:59,500 --> 00:14:01,180 inside the square root. 233 00:14:01,180 --> 00:14:02,260 Which is -2x. 234 00:14:02,260 --> 00:14:05,590 But then there's also 1/2, because in disguise it's really 235 00:14:05,590 --> 00:14:07,840 (1 - x^2)^(1/2). 236 00:14:07,840 --> 00:14:10,584 So we've done this calculation enough times 237 00:14:10,584 --> 00:14:12,500 that I'm not going to carry it out completely. 238 00:14:12,500 --> 00:14:14,290 I want you to think about what it is. 239 00:14:14,290 --> 00:14:17,690 It turns out to -x up here, because the 1/2 and the 2 240 00:14:17,690 --> 00:14:22,890 cancel. 241 00:14:22,890 --> 00:14:25,870 And now the thing that we have to integrate 242 00:14:25,870 --> 00:14:30,190 is this arc length element, as it's called, ds. 243 00:14:30,190 --> 00:14:38,350 And that's going to be the square root of 1 + (y')^2, dx. 244 00:14:38,350 --> 00:14:41,060 And so I'm going to have to carry out the calculation, 245 00:14:41,060 --> 00:14:42,970 some messy calculation here. 246 00:14:42,970 --> 00:14:47,020 Which is that this is 1 plus the quantity -x over square root 247 00:14:47,020 --> 00:14:49,274 of 1 - x^2, squared. 248 00:14:49,274 --> 00:14:51,440 So I have to figure out what's under the square root 249 00:14:51,440 --> 00:14:55,740 sign over here in order to carry out this calculation. 250 00:14:55,740 --> 00:14:58,570 Now let's do that. 251 00:14:58,570 --> 00:15:03,310 This is 1 + x^2 / (1 - x^2). 252 00:15:03,310 --> 00:15:06,140 That's what this simplifies to. 253 00:15:06,140 --> 00:15:11,190 And then that's equal to, over a common denominator, 1 - x^2. 254 00:15:11,190 --> 00:15:13,580 1 - x^2 + x^2. 255 00:15:13,580 --> 00:15:16,500 And there is a little bit of simplification now. 256 00:15:16,500 --> 00:15:19,190 Because the two x^2's cancel. 257 00:15:19,190 --> 00:15:20,390 And we get 1/(1-x^2). 258 00:15:28,640 --> 00:15:35,750 So now I get to finish off the calculation 259 00:15:35,750 --> 00:15:40,580 by actually figuring out what the arc length is. 260 00:15:40,580 --> 00:15:42,460 And what is it? 261 00:15:42,460 --> 00:15:51,834 Well, this alpha is equal to the integral from 0 to a of ds. 262 00:15:51,834 --> 00:15:53,750 Well, it's going to be the square root of what 263 00:15:53,750 --> 00:15:55,110 I have here. 264 00:15:55,110 --> 00:15:57,210 This was a square, this is just what 265 00:15:57,210 --> 00:15:58,710 was underneath the square root sign. 266 00:15:58,710 --> 00:16:01,500 This is 1 + (y')^2. 267 00:16:01,500 --> 00:16:03,270 Have to take the square root of that. 268 00:16:03,270 --> 00:16:08,380 So what I get here is dx over the square root of 1 - x^2. 269 00:16:13,350 --> 00:16:18,590 And now, we recognize this. 270 00:16:18,590 --> 00:16:21,100 The antiderivative of this is something that we know. 271 00:16:21,100 --> 00:16:23,790 This is the inverse sine. 272 00:16:23,790 --> 00:16:25,830 Evaluated at 0 and a. 273 00:16:25,830 --> 00:16:29,490 Which is just giving us the inverse sine 274 00:16:29,490 --> 00:16:35,860 of a, because the inverse sine of 0 is equal to 0. 275 00:16:35,860 --> 00:16:43,980 So alpha is equal to the inverse sine of a. 276 00:16:43,980 --> 00:16:51,320 That's a very fancy way of saying that sin(alpha) = a. 277 00:16:51,320 --> 00:16:54,870 That's the equivalent statement here. 278 00:16:54,870 --> 00:16:59,830 And what's going on here is something that's just 279 00:16:59,830 --> 00:17:01,950 a little deeper than it looks. 280 00:17:01,950 --> 00:17:03,120 Which is this. 281 00:17:03,120 --> 00:17:08,030 We've just figured out a geometric interpretation 282 00:17:08,030 --> 00:17:09,350 of what's going on here. 283 00:17:09,350 --> 00:17:13,680 That is, that we went a distance alpha along this arc. 284 00:17:13,680 --> 00:17:28,570 And now remember that the radius here is 1. 285 00:17:28,570 --> 00:17:34,430 And this horizontal distance here is a. 286 00:17:34,430 --> 00:17:37,450 This distance here is a. 287 00:17:37,450 --> 00:17:40,600 And so the geometric interpretation of this 288 00:17:40,600 --> 00:17:51,000 is that this angle is alpha radians. 289 00:17:51,000 --> 00:17:54,950 And sin(alpha) = a. 290 00:17:54,950 --> 00:17:57,570 So this is consistent with our definition 291 00:17:57,570 --> 00:18:00,430 previously, our previous geometric definition 292 00:18:00,430 --> 00:18:02,530 of radians. 293 00:18:02,530 --> 00:18:07,190 But this is really your first true definition of radians. 294 00:18:07,190 --> 00:18:09,850 We never actually-- People told you 295 00:18:09,850 --> 00:18:12,570 that radians were the arc length along this curve. 296 00:18:12,570 --> 00:18:14,770 This is the first time you're deriving it. 297 00:18:14,770 --> 00:18:18,420 This is the first time you're seeing it correctly done. 298 00:18:18,420 --> 00:18:20,134 And furthermore, this is the first time 299 00:18:20,134 --> 00:18:21,550 you're seeing a correct definition 300 00:18:21,550 --> 00:18:24,100 of the sine function. 301 00:18:24,100 --> 00:18:26,156 Remember we had this crazy way, we 302 00:18:26,156 --> 00:18:27,530 defined the exponential function, 303 00:18:27,530 --> 00:18:29,661 then we had another way of defining the log 304 00:18:29,661 --> 00:18:30,660 function as an integral. 305 00:18:30,660 --> 00:18:32,900 Then we defined the exponential in terms of it. 306 00:18:32,900 --> 00:18:34,690 Well, this is the same sort of thing. 307 00:18:34,690 --> 00:18:36,700 What's really happening here is that if you 308 00:18:36,700 --> 00:18:38,190 want to know what radians are, you 309 00:18:38,190 --> 00:18:40,847 have to calculate this number. 310 00:18:40,847 --> 00:18:42,430 If you've calculated this number, then 311 00:18:42,430 --> 00:18:47,930 by definition if sine is the thing whose alpha radian 312 00:18:47,930 --> 00:18:49,890 amount gives you a, then it must be 313 00:18:49,890 --> 00:18:52,520 that this is sine inverse of a. 314 00:18:52,520 --> 00:18:55,430 And so the first thing that gets defined is the arcsine. 315 00:18:55,430 --> 00:18:56,930 And the next thing that gets defined 316 00:18:56,930 --> 00:19:00,250 is the sine, afterwards. 317 00:19:00,250 --> 00:19:04,500 This is the way the foundational approach actually 318 00:19:04,500 --> 00:19:06,730 works when you start from first principles. 319 00:19:06,730 --> 00:19:10,200 This arc length being one of the first principles. 320 00:19:10,200 --> 00:19:13,720 So now we have a solid foundation for trig functions. 321 00:19:13,720 --> 00:19:15,760 And this is giving that connection. 322 00:19:15,760 --> 00:19:18,260 Of course, it's consistent with everything you already knew, 323 00:19:18,260 --> 00:19:22,079 so you don't have to make any transitional thinking here. 324 00:19:22,079 --> 00:19:23,620 It's just that this is the first time 325 00:19:23,620 --> 00:19:25,570 it's being done rigorously. 326 00:19:25,570 --> 00:19:36,380 Because you only now have arc length. 327 00:19:36,380 --> 00:19:41,220 So these are examples, as I say, that maybe you already know. 328 00:19:41,220 --> 00:19:44,820 And maybe we'll do one that we don't know quite as well. 329 00:19:44,820 --> 00:19:49,300 Let's find the length of a parabola. 330 00:19:49,300 --> 00:19:59,180 This is Example 3. 331 00:19:59,180 --> 00:20:00,660 Now, that was what I was suggesting 332 00:20:00,660 --> 00:20:03,140 we were going to do earlier. 333 00:20:03,140 --> 00:20:06,730 So this is the function y x^2. 334 00:20:06,730 --> 00:20:09,800 y' = 2x. 335 00:20:09,800 --> 00:20:20,120 And so ds is equal to the square root of 1 + (2x)^2, dx. 336 00:20:20,120 --> 00:20:24,600 And now I can figure out what a piece of a parabola is. 337 00:20:24,600 --> 00:20:28,220 So I'll draw the piece of parabola up to a, 338 00:20:28,220 --> 00:20:30,840 let's say, starting from 0. 339 00:20:30,840 --> 00:20:32,680 So that's the chunk. 340 00:20:32,680 --> 00:20:45,450 And then its arc length, between 0 and a of this curve, 341 00:20:45,450 --> 00:21:02,400 is the integral from 0 to a of square root of 1 + 4x^2, dx. 342 00:21:02,400 --> 00:21:08,490 OK, now if you like, this is the answer to the question. 343 00:21:08,490 --> 00:21:11,040 But people hate looking at answers 344 00:21:11,040 --> 00:21:13,800 when they're integrals if they can be evaluated. 345 00:21:13,800 --> 00:21:16,510 So one of the reasons why we went through all this rigmarole 346 00:21:16,510 --> 00:21:18,800 of calculating these things is to show you 347 00:21:18,800 --> 00:21:22,120 that we can actually evaluate a bunch of these functions 348 00:21:22,120 --> 00:21:23,340 here more explicitly. 349 00:21:23,340 --> 00:21:28,180 It doesn't help a lot, but there is an explicit calculation 350 00:21:28,180 --> 00:21:28,680 of this. 351 00:21:28,680 --> 00:21:30,560 So remember how you would do this. 352 00:21:30,560 --> 00:21:33,040 So this is just a little bit of review. 353 00:21:33,040 --> 00:21:35,420 What we did in techniques of integration. 354 00:21:35,420 --> 00:21:39,000 The first step is what? 355 00:21:39,000 --> 00:21:40,700 A substitution. 356 00:21:40,700 --> 00:21:43,950 It's a trig substitution. 357 00:21:43,950 --> 00:21:45,030 And what is it? 358 00:21:45,030 --> 00:21:47,250 STUDENT: [INAUDIBLE] 359 00:21:47,250 --> 00:21:50,270 PROFESSOR: So x equals something tan(theta). 360 00:21:50,270 --> 00:21:54,587 I claim that it's 1/2 tan, and I'm going to call it u. 361 00:21:54,587 --> 00:21:56,420 Because I'm going to use theta for something 362 00:21:56,420 --> 00:21:58,161 else in a couple of days. 363 00:21:58,161 --> 00:21:58,660 OK? 364 00:21:58,660 --> 00:22:01,420 So this is the substitution. 365 00:22:01,420 --> 00:22:10,620 And then of course dx = 1/2 sec^2 u du, etc. 366 00:22:10,620 --> 00:22:12,750 So what happens if you do this? 367 00:22:12,750 --> 00:22:15,160 I'll write down the answer, but I'm not 368 00:22:15,160 --> 00:22:16,330 going to carry this out. 369 00:22:16,330 --> 00:22:19,090 Because every one of these is horrendous. 370 00:22:19,090 --> 00:22:22,190 But I think I worked it out. 371 00:22:22,190 --> 00:22:23,370 Let's see if I'm lucky. 372 00:22:23,370 --> 00:22:24,300 Oh yeah. 373 00:22:24,300 --> 00:22:26,380 I think this is what it is. 374 00:22:26,380 --> 00:22:40,979 It's a 1/4 ln(2x + square root of (1+4x^2) + 1/2 x square root 375 00:22:40,979 --> 00:22:41,520 of (1+4x^2)). 376 00:22:46,540 --> 00:22:52,190 Evaluated at a and 0. 377 00:22:52,190 --> 00:22:53,090 So yick. 378 00:22:53,090 --> 00:22:53,810 I mean, you know. 379 00:22:53,810 --> 00:22:55,720 STUDENT: [INAUDIBLE] 380 00:22:55,720 --> 00:22:58,850 PROFESSOR: Why I did I make it 1/2? 381 00:22:58,850 --> 00:23:00,989 Because it turns out that when you differentiate. 382 00:23:00,989 --> 00:23:02,780 So the question is, why is there 1/2 there? 383 00:23:02,780 --> 00:23:05,734 If you differentiate it without the 1/2, you get this term 384 00:23:05,734 --> 00:23:07,650 and it looks like it's going to be just right. 385 00:23:07,650 --> 00:23:10,191 But then if you differentiate this one you get another thing. 386 00:23:10,191 --> 00:23:12,170 And it all mixes together. 387 00:23:12,170 --> 00:23:13,660 And it turns out that there's more. 388 00:23:13,660 --> 00:23:15,280 So it turns out that it's 1/2. 389 00:23:15,280 --> 00:23:18,750 Differentiate it and check. 390 00:23:18,750 --> 00:23:21,530 So this just an incredibly long calculation. 391 00:23:21,530 --> 00:23:24,220 It would take fifteen minutes or something like that. 392 00:23:24,220 --> 00:23:26,220 But the point is, you do know in principle 393 00:23:26,220 --> 00:23:27,840 how to do these things. 394 00:23:27,840 --> 00:23:43,455 STUDENT: [INAUDIBLE] 395 00:23:43,455 --> 00:23:45,330 PROFESSOR: Oh, he was talking about this 1/2, 396 00:23:45,330 --> 00:23:47,100 not this crazy 1/2 here. 397 00:23:47,100 --> 00:23:47,600 Sorry. 398 00:23:47,600 --> 00:23:48,637 STUDENT: [INAUDIBLE] 399 00:23:48,637 --> 00:23:49,470 PROFESSOR: Yeah, OK. 400 00:23:49,470 --> 00:23:50,740 So sorry about that. 401 00:23:50,740 --> 00:23:53,080 Thank you for helping. 402 00:23:53,080 --> 00:23:56,270 This factor of 1/2 here comes about because when you square 403 00:23:56,270 --> 00:23:58,690 x, you don't get tan^2. 404 00:23:58,690 --> 00:24:02,510 When you square 2x, you get (4x)^2 and that matches 405 00:24:02,510 --> 00:24:03,930 perfectly with this thing. 406 00:24:03,930 --> 00:24:07,300 And that's why you need this factor here. 407 00:24:07,300 --> 00:24:07,800 Yeah. 408 00:24:07,800 --> 00:24:09,216 Another question, way in the back. 409 00:24:09,216 --> 00:24:18,190 STUDENT: [INAUDIBLE] 410 00:24:18,190 --> 00:24:20,780 PROFESSOR: The question is, when you do this substitution, 411 00:24:20,780 --> 00:24:25,040 doesn't the limit from 0 to a change. 412 00:24:25,040 --> 00:24:27,020 And the answer is, absolutely yes. 413 00:24:27,020 --> 00:24:30,230 The limits in terms of u are not the same 414 00:24:30,230 --> 00:24:31,800 as the limits in terms of a. 415 00:24:31,800 --> 00:24:34,870 But if I then translate back to the x variables, which 416 00:24:34,870 --> 00:24:40,820 I've done here in this bottom formula, of x = 0 and x = a, 417 00:24:40,820 --> 00:24:44,740 it goes back to those in the original variables. 418 00:24:44,740 --> 00:24:46,840 So if I write things in the original variables, 419 00:24:46,840 --> 00:24:48,980 I have the original limits. 420 00:24:48,980 --> 00:24:51,962 If I use the u variables, I would have to change limits. 421 00:24:51,962 --> 00:24:53,670 But I'm not carrying out the integration, 422 00:24:53,670 --> 00:24:55,060 because I don't want to. 423 00:24:55,060 --> 00:25:00,630 So I brought it back to the x formula. 424 00:25:00,630 --> 00:25:07,080 Other questions. 425 00:25:07,080 --> 00:25:11,580 OK, so now we're ready to launch into three-space a little bit 426 00:25:11,580 --> 00:25:14,250 here. 427 00:25:14,250 --> 00:25:41,310 We're going to talk about surface area. 428 00:25:41,310 --> 00:25:43,750 You're going to be doing a lot with surface 429 00:25:43,750 --> 00:25:48,550 area in multivariable calculus. 430 00:25:48,550 --> 00:25:50,990 It's one of the really fun things. 431 00:25:50,990 --> 00:25:55,430 And just remember, when it gets complicated, 432 00:25:55,430 --> 00:25:57,926 that the simplest things are the most important. 433 00:25:57,926 --> 00:25:59,300 And the simple things are, if you 434 00:25:59,300 --> 00:26:01,820 can handle things for linear functions, 435 00:26:01,820 --> 00:26:02,920 you know all the rest. 436 00:26:02,920 --> 00:26:04,544 So there's going to be some complicated 437 00:26:04,544 --> 00:26:06,390 stuff but it'll really only involve 438 00:26:06,390 --> 00:26:09,480 what's happening on planes. 439 00:26:09,480 --> 00:26:11,680 So let's start with surface area. 440 00:26:11,680 --> 00:26:15,615 And the example that I'd like to give 441 00:26:15,615 --> 00:26:20,110 - this is the only type of example that we'll have - 442 00:26:20,110 --> 00:26:28,560 is the surface of rotation. 443 00:26:28,560 --> 00:26:31,990 And as long as we have our parabola there, 444 00:26:31,990 --> 00:26:33,300 we'll use that one. 445 00:26:33,300 --> 00:26:51,920 So we have y = x^2, rotated around the x-axis. 446 00:26:51,920 --> 00:26:54,480 So let's take a look at what this looks like. 447 00:26:54,480 --> 00:26:57,910 It's the parabola, which is going like that. 448 00:26:57,910 --> 00:27:01,850 And then it's being spun around the x-axis. 449 00:27:01,850 --> 00:27:08,390 So some kind of shape like this with little circles. 450 00:27:08,390 --> 00:27:17,880 It's some kind of trumpet shape, right? 451 00:27:17,880 --> 00:27:20,170 And that's the shape that we're-- Now, again, 452 00:27:20,170 --> 00:27:20,980 it's the surface. 453 00:27:20,980 --> 00:27:27,770 It's just the metal of the trumpet, not the insides. 454 00:27:27,770 --> 00:27:33,280 Now, the principle for figuring out what the formula for area 455 00:27:33,280 --> 00:27:36,470 is, is not that different from what we 456 00:27:36,470 --> 00:27:38,530 did for surfaces of revolution. 457 00:27:38,530 --> 00:27:42,610 But it just requires a little bit of thought and imagination. 458 00:27:42,610 --> 00:27:50,360 We have a little chunk of arc length along here. 459 00:27:50,360 --> 00:27:55,580 And we're going to spin that around this axis. 460 00:27:55,580 --> 00:28:01,610 Now, if this were a horizontal piece of arc length, 461 00:28:01,610 --> 00:28:04,500 then it would spin around just like a shell. 462 00:28:04,500 --> 00:28:07,550 It would just be a surface. 463 00:28:07,550 --> 00:28:11,670 But if it's tilted, if it's tilted, 464 00:28:11,670 --> 00:28:15,030 then there's more surface area proportional to the amount 465 00:28:15,030 --> 00:28:17,040 that it's tilted. 466 00:28:17,040 --> 00:28:19,310 So it's proportional to the length of the segment 467 00:28:19,310 --> 00:28:22,890 that you spin around. 468 00:28:22,890 --> 00:28:29,430 So the total is going to be ds, that's one of the factors here. 469 00:28:29,430 --> 00:28:32,220 Maybe I'll write that second. 470 00:28:32,220 --> 00:28:33,560 That's one of the dimensions. 471 00:28:33,560 --> 00:28:36,660 And then the other dimension is the circumference. 472 00:28:36,660 --> 00:28:43,180 Which is 2 pi, in this case, y. 473 00:28:43,180 --> 00:28:46,300 So that's the end of the calculation. 474 00:28:46,300 --> 00:28:55,990 This is the area element of surface area. 475 00:28:55,990 --> 00:28:59,860 Now, when you get to 18.02, and maybe even before that, 476 00:28:59,860 --> 00:29:03,000 you'll also see some people referring to this area element 477 00:29:03,000 --> 00:29:09,110 when it's a curvy surface like this with a notation dS. 478 00:29:09,110 --> 00:29:10,610 That's a little confusing because we 479 00:29:10,610 --> 00:29:12,070 have a lower case s here. 480 00:29:12,070 --> 00:29:15,450 We're not going to use it right now. 481 00:29:15,450 --> 00:29:17,620 But the lower case s is usually arc length. 482 00:29:17,620 --> 00:29:23,950 The upper case S is usually surface area. 483 00:29:23,950 --> 00:29:25,650 So. 484 00:29:25,650 --> 00:29:32,510 Also used for dA. 485 00:29:32,510 --> 00:29:33,730 The area element. 486 00:29:33,730 --> 00:29:39,780 Because this is a curved area element. 487 00:29:39,780 --> 00:29:47,630 So let's figure out this example. 488 00:29:47,630 --> 00:29:54,090 So in the example-- ...is equal to x ^2 then the situation is, 489 00:29:54,090 --> 00:30:01,050 we have the surface area is equal to the integral from, 490 00:30:01,050 --> 00:30:03,770 I don't know, 0 to a if those are the limits that we wanted 491 00:30:03,770 --> 00:30:05,040 to choose. 492 00:30:05,040 --> 00:30:10,470 Of 2 pi x^2, right? 493 00:30:10,470 --> 00:30:11,160 Because y = x^2. 494 00:30:11,160 --> 00:30:17,960 Times the square root of 1 + 4x^2, dx. 495 00:30:17,960 --> 00:30:20,300 Remember we had this from our previous example. 496 00:30:20,300 --> 00:30:32,850 This was ds from previous. 497 00:30:32,850 --> 00:30:41,930 And this, of course, is 2 pi y. 498 00:30:41,930 --> 00:30:49,150 Now again, the calculation of this integral is kind of long. 499 00:30:49,150 --> 00:30:52,060 And I'm going to omit it. 500 00:30:52,060 --> 00:30:54,300 But let me just point out that it follows 501 00:30:54,300 --> 00:30:56,350 from the same substitution. 502 00:30:56,350 --> 00:31:05,150 Namely, x = 1/2 tan u is going to work for this integral. 503 00:31:05,150 --> 00:31:06,200 It's kind of a mess. 504 00:31:06,200 --> 00:31:08,590 There's a tan squared here and the secant squared. 505 00:31:08,590 --> 00:31:10,170 There's another secant and so on. 506 00:31:10,170 --> 00:31:12,640 So it's one of these trig integrals 507 00:31:12,640 --> 00:31:19,670 that then takes a while to do. 508 00:31:19,670 --> 00:31:22,940 So that just is going to trail off into nothing. 509 00:31:22,940 --> 00:31:25,540 And the reason is that what's important here 510 00:31:25,540 --> 00:31:27,250 is more the method. 511 00:31:27,250 --> 00:31:29,650 And the setup of the integrals. 512 00:31:29,650 --> 00:31:32,902 The actual computation, in fact, you could go to a program 513 00:31:32,902 --> 00:31:34,610 and you could plug in something like this 514 00:31:34,610 --> 00:31:37,300 and you would spit out an answer immediately. 515 00:31:37,300 --> 00:31:41,100 So really what we just want is for you to have enough control 516 00:31:41,100 --> 00:31:43,640 to see that it's an integral that's a manageable one. 517 00:31:43,640 --> 00:31:45,630 And also to know that if you plugged it in, 518 00:31:45,630 --> 00:31:50,970 you would get an answer. 519 00:31:50,970 --> 00:31:53,320 When I actually do carry out a calculation, though, 520 00:31:53,320 --> 00:31:57,740 what I want to do is to do something 521 00:31:57,740 --> 00:32:00,190 that has an answer that you can remember. 522 00:32:00,190 --> 00:32:02,090 And that's a nice answer. 523 00:32:02,090 --> 00:32:05,940 So that turns out to be the example of the surface 524 00:32:05,940 --> 00:32:07,580 area of a sphere. 525 00:32:07,580 --> 00:32:10,110 So it's analogous to this 2 here. 526 00:32:10,110 --> 00:32:15,170 And maybe I should remember this result here. 527 00:32:15,170 --> 00:32:24,390 Which was that the arc length element was given by this. 528 00:32:24,390 --> 00:32:38,700 So we'll save that for a second. 529 00:32:38,700 --> 00:32:41,710 So we're going to do this surface area now. 530 00:32:41,710 --> 00:32:43,930 So if you like, this is another example. 531 00:32:43,930 --> 00:32:54,820 The surface area of a sphere. 532 00:32:54,820 --> 00:32:59,670 This is a good example, and one, as I say, 533 00:32:59,670 --> 00:33:01,160 that has a really nice answer. 534 00:33:01,160 --> 00:33:07,130 So it's worth doing. 535 00:33:07,130 --> 00:33:09,770 So first of all, I'm not going to set it up quite the way 536 00:33:09,770 --> 00:33:11,794 I did in Example 2. 537 00:33:11,794 --> 00:33:13,710 Instead, I'm going to take the general sphere, 538 00:33:13,710 --> 00:33:18,790 because I'd like to watch the dependence on the radius. 539 00:33:18,790 --> 00:33:22,160 So here this is going to be the radius. 540 00:33:22,160 --> 00:33:27,250 It's going to be radius a. 541 00:33:27,250 --> 00:33:30,440 And now, if I carry out the same calculations 542 00:33:30,440 --> 00:33:33,840 as before, if you think about it for a second, 543 00:33:33,840 --> 00:33:42,322 you're going to get this result. And then, the rest 544 00:33:42,322 --> 00:33:43,780 of the arithmetic, which is sitting 545 00:33:43,780 --> 00:33:47,750 up there in the case, a = 1, will give us 546 00:33:47,750 --> 00:33:53,120 that ds is equal to what? 547 00:33:53,120 --> 00:33:56,670 Well, maybe I'll just carry it out. 548 00:33:56,670 --> 00:33:58,510 Because that's always nice. 549 00:33:58,510 --> 00:34:03,630 So we have 1 + x^2 / (a^2 - x^2). 550 00:34:03,630 --> 00:34:07,060 That's 1 + (y')^2. 551 00:34:07,060 --> 00:34:09,630 And now I put this over a common denominator. 552 00:34:09,630 --> 00:34:11,800 And I get a^2 - x^2. 553 00:34:11,800 --> 00:34:15,030 And I have in the numerator a^2 - x^2 + x^2. 554 00:34:15,030 --> 00:34:17,590 So the same cancellation occurs. 555 00:34:17,590 --> 00:34:25,390 But now we get an a^2 in the numerator. 556 00:34:25,390 --> 00:34:28,070 So now I can set up the ds. 557 00:34:28,070 --> 00:34:30,200 And so here's what happens. 558 00:34:30,200 --> 00:34:35,370 The area of a section of the sphere, so let's see. 559 00:34:35,370 --> 00:34:38,920 We're going to start at some starting place x_1, 560 00:34:38,920 --> 00:34:40,760 and end at some place x_2. 561 00:34:40,760 --> 00:34:43,110 So what does that look like? 562 00:34:43,110 --> 00:34:45,140 Here's the sphere. 563 00:34:45,140 --> 00:34:47,490 And we're starting at a place x_1. 564 00:34:47,490 --> 00:34:49,810 And we're ending at a place x_2. 565 00:34:49,810 --> 00:34:53,830 And we're taking more or less the slice here, if you like. 566 00:34:53,830 --> 00:34:59,480 The section of this sphere. 567 00:34:59,480 --> 00:35:02,150 So the area's going to equal this. 568 00:35:02,150 --> 00:35:06,730 And what is it going to be? 569 00:35:06,730 --> 00:35:12,820 Well, so I have here 2 pi y. 570 00:35:12,820 --> 00:35:15,310 I'll write it out, just leave it as y for now. 571 00:35:15,310 --> 00:35:19,190 And then I have ds. 572 00:35:19,190 --> 00:35:22,110 So that's always what the formula is when you're 573 00:35:22,110 --> 00:35:25,880 revolving around the x-axis. 574 00:35:25,880 --> 00:35:29,920 And then I'll plug in for those things. 575 00:35:29,920 --> 00:35:35,990 So 2 pi, the formula for y is square root a^2 - x^2. 576 00:35:38,700 --> 00:35:42,220 And the formula for ds, well, it's 577 00:35:42,220 --> 00:35:44,970 the square root of this times dx. 578 00:35:44,970 --> 00:35:48,920 So it's the square root of a^2 / (a^2 - x^2), dx. 579 00:35:51,760 --> 00:35:54,880 So this part is ds. 580 00:35:54,880 --> 00:36:02,170 And this part is y. 581 00:36:02,170 --> 00:36:07,590 And now, I claim we have a nice cancellation that takes place. 582 00:36:07,590 --> 00:36:09,770 Square root of a^2 is a. 583 00:36:09,770 --> 00:36:12,740 And then there's another good cancellation. 584 00:36:12,740 --> 00:36:13,990 As you can see. 585 00:36:13,990 --> 00:36:17,140 Now, what we get here is the integral from x_1 to x_2, 586 00:36:17,140 --> 00:36:21,510 of 2 pi a dx, which is about the easiest integral 587 00:36:21,510 --> 00:36:23,340 you can imagine. 588 00:36:23,340 --> 00:36:24,940 It's just the integral of a constant. 589 00:36:24,940 --> 00:36:27,980 So it's 2 pi a (x_2 - x_1). 590 00:36:36,810 --> 00:36:40,780 Let's check this in a couple of examples. 591 00:36:40,780 --> 00:36:48,390 And then see what it's saying geometrically, a little bit. 592 00:36:48,390 --> 00:36:53,220 So what this is saying-- So special cases 593 00:36:53,220 --> 00:36:56,560 that you should always check, when 594 00:36:56,560 --> 00:36:59,220 you have a nice formula like this, at least. 595 00:36:59,220 --> 00:37:00,730 But really with anything in order 596 00:37:00,730 --> 00:37:03,250 to make sure that you've got the right answer. 597 00:37:03,250 --> 00:37:05,710 If you take, for example, the hemisphere. 598 00:37:05,710 --> 00:37:08,920 So you take 1/2 of this sphere. 599 00:37:08,920 --> 00:37:11,090 So that would be starting at 0, sorry. 600 00:37:11,090 --> 00:37:14,300 And ending at a. 601 00:37:14,300 --> 00:37:17,270 So that's the integral from 0 to a. 602 00:37:17,270 --> 00:37:21,890 So this is the case x_1 = 0. x_2 = a. 603 00:37:21,890 --> 00:37:29,190 And what you're going to get is a hemisphere. 604 00:37:29,190 --> 00:37:36,270 And the area is 2 pi a times a. 605 00:37:36,270 --> 00:37:38,150 Or in other words, 2 pi a^2. 606 00:37:42,410 --> 00:37:48,920 And if you take the whole sphere, 607 00:37:48,920 --> 00:37:55,900 that's starting at x_1 = -a, and x_2 = a, you're getting 2 pi 608 00:37:55,900 --> 00:37:58,330 a times (a - (-a)). 609 00:38:02,640 --> 00:38:06,380 Which is 4 pi a^2. 610 00:38:06,380 --> 00:38:09,230 That's the whole thing. 611 00:38:09,230 --> 00:38:10,140 Yeah, question. 612 00:38:10,140 --> 00:38:21,780 STUDENT: [INAUDIBLE] 613 00:38:21,780 --> 00:38:23,340 PROFESSOR: The question is, would it 614 00:38:23,340 --> 00:38:27,730 be possible to rotate around the y-axis? 615 00:38:27,730 --> 00:38:30,330 And the answer is yes. 616 00:38:30,330 --> 00:38:34,550 It's legal to rotate around the y-axis. 617 00:38:34,550 --> 00:38:43,170 And there is-- If you use vertical slices as we did here, 618 00:38:43,170 --> 00:38:45,490 that is, well they're sort of tips of slices, 619 00:38:45,490 --> 00:38:46,790 it's a different idea. 620 00:38:46,790 --> 00:38:49,780 But anyway, it's using dx as the integral 621 00:38:49,780 --> 00:38:52,670 of the variable of integration. 622 00:38:52,670 --> 00:38:55,310 So we're checking each little piece, 623 00:38:55,310 --> 00:38:58,750 each little strip of that type. 624 00:38:58,750 --> 00:39:00,540 If we use dx here, we get this. 625 00:39:00,540 --> 00:39:02,710 If you did the same thing rotated the other way, 626 00:39:02,710 --> 00:39:06,140 and use dy as the variable, you get exactly the same answer. 627 00:39:06,140 --> 00:39:08,120 And it would be the same calculation. 628 00:39:08,120 --> 00:39:14,080 Because they're parallel. 629 00:39:14,080 --> 00:39:14,870 So you're, yep. 630 00:39:14,870 --> 00:39:17,280 STUDENT: [INAUDIBLE] 631 00:39:17,280 --> 00:39:19,770 PROFESSOR: Can you do surface area with shells? 632 00:39:19,770 --> 00:39:26,540 Well, the shell shape-- The short answer is not quite. 633 00:39:26,540 --> 00:39:30,851 The shell shape is a vertical shell 634 00:39:30,851 --> 00:39:32,600 which is itself already three-dimensional, 635 00:39:32,600 --> 00:39:34,850 and it has a thickness. 636 00:39:34,850 --> 00:39:37,270 So this is just a matter of terminology, though. 637 00:39:37,270 --> 00:39:41,210 This thickness is this dx, when we do this rotation here. 638 00:39:41,210 --> 00:39:43,820 And then there are two other dimensions. 639 00:39:43,820 --> 00:39:46,190 If we have a curved surface, there's 640 00:39:46,190 --> 00:39:56,460 no other dimension left to form a shell. 641 00:39:56,460 --> 00:39:59,421 But basically, you can chop things up into any bits 642 00:39:59,421 --> 00:40:00,670 that you can actually measure. 643 00:40:00,670 --> 00:40:04,020 That you can figure out what the area is. 644 00:40:04,020 --> 00:40:08,270 That's the main point. 645 00:40:08,270 --> 00:40:10,160 Now, I said we were going to, we've 646 00:40:10,160 --> 00:40:12,910 just launched into three-dimensional space. 647 00:40:12,910 --> 00:40:21,650 And I want to now move on to other space-like phenomena. 648 00:40:21,650 --> 00:40:26,780 But we're going to do this. 649 00:40:26,780 --> 00:40:31,510 So this is also a preparation for 18.02, 650 00:40:31,510 --> 00:40:34,840 where you'll be doing this a tremendous amount. 651 00:40:34,840 --> 00:40:49,530 We're going to talk now about parametric equations. 652 00:40:49,530 --> 00:40:57,102 Really just parametric curves. 653 00:40:57,102 --> 00:40:58,810 So you're going to see this now and we're 654 00:40:58,810 --> 00:41:00,539 going to interpret it a couple of times, 655 00:41:00,539 --> 00:41:02,580 and we're going to think about polar coordinates. 656 00:41:02,580 --> 00:41:05,920 These are all preparation for thinking in more variables, 657 00:41:05,920 --> 00:41:08,570 and thinking in a different way than you've 658 00:41:08,570 --> 00:41:09,720 been thinking before. 659 00:41:09,720 --> 00:41:11,970 So I want you to prepare your brain 660 00:41:11,970 --> 00:41:13,975 to make a transition here. 661 00:41:13,975 --> 00:41:15,600 This is the beginning of the transition 662 00:41:15,600 --> 00:41:21,580 to multivariable thinking. 663 00:41:21,580 --> 00:41:26,350 We're going to consider curves like this. 664 00:41:26,350 --> 00:41:29,850 Which are described with x being a function of t and y 665 00:41:29,850 --> 00:41:31,410 being a function of t. 666 00:41:31,410 --> 00:41:35,630 And this letter t is called the parameter. 667 00:41:35,630 --> 00:41:38,080 In this case you should think of it-- the easiest way 668 00:41:38,080 --> 00:41:39,700 to think of it is as time. 669 00:41:39,700 --> 00:41:43,840 And what you have is what's called a trajectory. 670 00:41:43,840 --> 00:41:48,090 So this is also called a trajectory. 671 00:41:48,090 --> 00:41:54,590 And its location, let's say, at time 0, is this location here. 672 00:41:54,590 --> 00:41:58,310 (x(0), y(0)), that's a point in the plane. 673 00:41:58,310 --> 00:42:01,380 And then over here, for instance, maybe it's 674 00:42:01,380 --> 00:42:04,410 (x(1), y(1)). 675 00:42:04,410 --> 00:42:06,540 And I drew arrows along here to indicate 676 00:42:06,540 --> 00:42:10,110 that we're going from this place over to that place. 677 00:42:10,110 --> 00:42:16,190 These are later times. t = 1 is a later time than t = 0. 678 00:42:16,190 --> 00:42:20,170 So that's just a very casual, it's 679 00:42:20,170 --> 00:42:22,680 just the way we use these notations. 680 00:42:22,680 --> 00:42:31,670 Now let me give you the first example, which is x = a cos 681 00:42:31,670 --> 00:42:40,340 t, y = a sin t. 682 00:42:40,340 --> 00:42:43,630 And the first thing to figure out is what kind of curve 683 00:42:43,630 --> 00:42:45,490 this is. 684 00:42:45,490 --> 00:42:47,880 And to do that, we want to figure out 685 00:42:47,880 --> 00:42:52,000 what equation it satisfies in rectangular coordinates. 686 00:42:52,000 --> 00:42:53,650 So to figure out what curve this is, 687 00:42:53,650 --> 00:42:59,730 we recognize that if we square and add-- So we add x^2 to y^2, 688 00:42:59,730 --> 00:43:02,670 we're going to get something very nice and clean. 689 00:43:02,670 --> 00:43:08,700 We're going to get a^2 cos^2 t + a^2 sin^2 t. 690 00:43:11,990 --> 00:43:13,500 Yeah that's right, OK. 691 00:43:13,500 --> 00:43:19,230 Which is just a ^2 a^2 (cos^2 + sin^2), or in other words a^2. 692 00:43:19,230 --> 00:43:23,450 So lo and behold, what we have is a circle. 693 00:43:23,450 --> 00:43:27,840 And then we know what shape this is now. 694 00:43:27,840 --> 00:43:33,240 And the other thing I'd like to keep track of 695 00:43:33,240 --> 00:43:35,990 is which direction we're going on the circle. 696 00:43:35,990 --> 00:43:40,020 Because there's more to this parameter then just the shape. 697 00:43:40,020 --> 00:43:42,570 There's also where we are at what time. 698 00:43:42,570 --> 00:43:46,610 This would be, think of it like the trajectory of a planet. 699 00:43:46,610 --> 00:43:51,600 So here, I have to do this by plotting the picture 700 00:43:51,600 --> 00:43:53,250 and figuring out what happens. 701 00:43:53,250 --> 00:43:59,980 So at t = 0, we have (x, y) is equal to, 702 00:43:59,980 --> 00:44:06,190 plug in here (a cos 0, a sin 0). 703 00:44:06,190 --> 00:44:10,570 Which is just a * 1 + a * 0, so (a, 0). 704 00:44:10,570 --> 00:44:11,830 And that's here. 705 00:44:11,830 --> 00:44:14,660 That's the point (a, 0). 706 00:44:14,660 --> 00:44:18,474 We know that it's the circle of radius a. 707 00:44:18,474 --> 00:44:19,890 So we know that the curve is going 708 00:44:19,890 --> 00:44:22,140 to go around like this somehow. 709 00:44:22,140 --> 00:44:26,390 So let's see what happens at t = pi / 2. 710 00:44:26,390 --> 00:44:30,950 So at that point, we have (x, y) = (a cos(pi/2), a sin(pi/2)). 711 00:44:37,310 --> 00:44:41,580 Which is (0, a), because sine of pi / 2 is 1. 712 00:44:41,580 --> 00:44:43,190 So that's up here. 713 00:44:43,190 --> 00:44:46,100 So this is what happens at t = 0. 714 00:44:46,100 --> 00:44:49,220 This is what happens at t = pi / 2. 715 00:44:49,220 --> 00:44:51,390 And the trajectory clearly goes this way. 716 00:44:51,390 --> 00:44:54,800 In fact, this turns out to be t = pi, etc. 717 00:44:54,800 --> 00:44:58,330 And it repeats at t = 2pi. 718 00:44:58,330 --> 00:45:00,610 So the other feature that we have, 719 00:45:00,610 --> 00:45:11,560 which is qualitative feature, is that it's counterclockwise. 720 00:45:11,560 --> 00:45:17,020 Now the last little bit is going to be the arc length. 721 00:45:17,020 --> 00:45:19,030 Keeping track of the arc length. 722 00:45:19,030 --> 00:45:23,390 And we'll do that next time.