1 00:00:00,040 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,690 Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,990 offer high quality educational resources for free. 5 00:00:09,990 --> 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 00:00:31,940 --> 00:00:33,380 PROFESSOR: Hi. 9 00:00:33,380 --> 00:00:36,550 Having already studied area and volume and its 10 00:00:36,550 --> 00:00:39,950 relationship to calculus, today, we turn our attention 11 00:00:39,950 --> 00:00:41,330 to the study of length. 12 00:00:41,330 --> 00:00:44,860 And this may seem a bit strange because, intuitively, 13 00:00:44,860 --> 00:00:48,170 I think it's fair to assume that you would imagine that 14 00:00:48,170 --> 00:00:51,170 length would be simpler than area which, in turn, would be 15 00:00:51,170 --> 00:00:54,050 simpler than volume and, hence, that perhaps we should 16 00:00:54,050 --> 00:00:56,210 have started with length in the first place. 17 00:00:56,210 --> 00:00:59,060 The interesting thing is in terms of our structure, which 18 00:00:59,060 --> 00:01:01,920 we so far have called two-dimensional area, 19 00:01:01,920 --> 00:01:05,099 three-dimensional area, and which, today, we shall call 20 00:01:05,099 --> 00:01:09,220 one-dimensional area, a rather peculiar thing that causes a 21 00:01:09,220 --> 00:01:12,470 great deal of difficulty, intellectually speaking, 22 00:01:12,470 --> 00:01:16,850 occurs in the study of arc length that does not occur in 23 00:01:16,850 --> 00:01:19,990 either the study of area or volume. 24 00:01:19,990 --> 00:01:22,760 And I think that we'll start our investigation today 25 00:01:22,760 --> 00:01:25,040 leading up to what this really means. 26 00:01:25,040 --> 00:01:29,190 So, as I say, I call today's lesson 'One-dimensional Area', 27 00:01:29,190 --> 00:01:30,400 which is arc length. 28 00:01:30,400 --> 00:01:34,660 And let's show that there is a parallel, at least in part, 29 00:01:34,660 --> 00:01:37,780 between the structure of arc length and the structure of 30 00:01:37,780 --> 00:01:39,140 area and volume. 31 00:01:39,140 --> 00:01:43,240 You may recall that for area, our initial axiom was that the 32 00:01:43,240 --> 00:01:45,990 building block of area was a rectangle. 33 00:01:45,990 --> 00:01:49,490 And for volumes, the building block we saw was a cylinder. 34 00:01:49,490 --> 00:01:50,950 For arc length-- 35 00:01:50,950 --> 00:01:52,560 I think it's fairly obvious to guess what 36 00:01:52,560 --> 00:01:53,500 we're going to say-- 37 00:01:53,500 --> 00:01:56,650 the basic building block is a straight line segment. 38 00:01:56,650 --> 00:01:59,740 And so without further ado, that becomes our first rule, 39 00:01:59,740 --> 00:02:02,230 our first axiom, axiom number one. 40 00:02:02,230 --> 00:02:05,870 We assume that we can measure the length of any straight 41 00:02:05,870 --> 00:02:07,440 line segment. 42 00:02:07,440 --> 00:02:09,050 That's our building block. 43 00:02:09,050 --> 00:02:12,560 The second axiom that we assume is that the length of 44 00:02:12,560 --> 00:02:15,140 the whole equals the sum of the lengths of the parts. 45 00:02:15,140 --> 00:02:18,290 In other words, if an arc is broken down into constituent 46 00:02:18,290 --> 00:02:22,080 bases, the total arc length is equal to the sum of the arc 47 00:02:22,080 --> 00:02:23,890 lengths of the constituent parts. 48 00:02:23,890 --> 00:02:26,670 And at this stage, we can say, so far, so good. 49 00:02:26,670 --> 00:02:28,500 This still looks like it's going to be the 50 00:02:28,500 --> 00:02:30,390 same as area or volume. 51 00:02:30,390 --> 00:02:34,400 But now remember what one of the axioms for both area and 52 00:02:34,400 --> 00:02:36,080 volume were, namely, what? 53 00:02:36,080 --> 00:02:40,120 That if region 'R' was contained in region 'S', the 54 00:02:40,120 --> 00:02:45,530 area or the volume of 'R' was no greater than that of the 55 00:02:45,530 --> 00:02:47,410 area or volume of 'S'. 56 00:02:47,410 --> 00:02:50,120 However, for arc length, this is not true. 57 00:02:50,120 --> 00:02:51,330 It need not be true. 58 00:02:51,330 --> 00:02:52,410 I shouldn't say it's not true. 59 00:02:52,410 --> 00:02:56,580 It need not be true that if region 'R' is contained in 'S' 60 00:02:56,580 --> 00:03:00,680 that the perimeter of region 'R' is less than or equal to 61 00:03:00,680 --> 00:03:01,880 the perimeter of 'S'. 62 00:03:01,880 --> 00:03:05,850 In fact, this little diagram that I've drawn over here, I 63 00:03:05,850 --> 00:03:08,680 hope will show you what I'm driving at. 64 00:03:08,680 --> 00:03:11,290 Notice that it's rather clear that the region 'R' here, 65 00:03:11,290 --> 00:03:15,080 which is shaded, is contained inside the region 'S', which 66 00:03:15,080 --> 00:03:16,370 is my rectangle. 67 00:03:16,370 --> 00:03:18,930 And yet, if you look at the perimeter here, all these 68 00:03:18,930 --> 00:03:22,620 finger-shaped things in here, I think it's easy to see that 69 00:03:22,620 --> 00:03:27,600 the perimeter of 'R' exceeds the perimeter of 'S'. 70 00:03:27,600 --> 00:03:30,510 And if it's not that easy to see, heck, just make a few 71 00:03:30,510 --> 00:03:34,180 more loops inside here and keep wiggling this thing 72 00:03:34,180 --> 00:03:37,370 around until you're convinced that you have created this 73 00:03:37,370 --> 00:03:38,450 particular situation. 74 00:03:38,450 --> 00:03:42,790 All I want you to see here is that it's plausible to you 75 00:03:42,790 --> 00:03:48,120 that we cannot talk about lengths by squeezing them, as 76 00:03:48,120 --> 00:03:51,620 we did areas and volumes, between regions that we 77 00:03:51,620 --> 00:03:54,800 already knew contained the given region and were 78 00:03:54,800 --> 00:03:56,250 contained in the given region. 79 00:03:56,250 --> 00:03:58,910 Now let me just pause here for one moment to make sure that 80 00:03:58,910 --> 00:04:00,140 we keep one thing straight. 81 00:04:00,140 --> 00:04:04,080 We're talking now about an analytical approach to length. 82 00:04:04,080 --> 00:04:06,390 In other words, an approach that will allow us to bring to 83 00:04:06,390 --> 00:04:10,980 bear all of the power of calculus to the study. 84 00:04:10,980 --> 00:04:14,380 I don't want you to forget for a moment that intuitively, we 85 00:04:14,380 --> 00:04:17,130 certainly do know what arc length is, just as we 86 00:04:17,130 --> 00:04:21,100 intuitively had a feeling for what area and volume were. 87 00:04:21,100 --> 00:04:23,780 Just to freshen our memories on this, remember the 88 00:04:23,780 --> 00:04:24,960 intuitive approach. 89 00:04:24,960 --> 00:04:30,970 That if you have an arc from 'A' to 'B', the typical way of 90 00:04:30,970 --> 00:04:33,910 measuring the arc length is to take, for example, a piece of 91 00:04:33,910 --> 00:04:37,480 string, lay it off along the curve from 'A' to 'B'. 92 00:04:37,480 --> 00:04:40,430 After you've done this, pick the string up. 93 00:04:40,430 --> 00:04:43,610 And then straighten the string out, whatever that means, and 94 00:04:43,610 --> 00:04:45,420 measure its length with a ruler. 95 00:04:45,420 --> 00:04:48,020 And we won't worry about how you know whether you're 96 00:04:48,020 --> 00:04:50,490 stretching the string too taut or what have you. 97 00:04:50,490 --> 00:04:52,890 We'll leave out these philosophic questions. 98 00:04:52,890 --> 00:04:56,500 All we'll say is we would like a more objective method that 99 00:04:56,500 --> 00:04:59,250 will allow us to use mathematical analysis. 100 00:04:59,250 --> 00:05:02,420 And so what we're going to try to do next is to find an 101 00:05:02,420 --> 00:05:07,190 analytic way that will allow us to use calculus, but at the 102 00:05:07,190 --> 00:05:11,150 same time will give us a definition which agrees with 103 00:05:11,150 --> 00:05:12,360 our intuition. 104 00:05:12,360 --> 00:05:14,930 And the first question is how shall we begin. 105 00:05:14,930 --> 00:05:18,890 And as so often is the case in mathematics, we begin our new 106 00:05:18,890 --> 00:05:23,470 quest by going back to an old way that worked 107 00:05:23,470 --> 00:05:24,750 for a previous case. 108 00:05:24,750 --> 00:05:27,640 And hopefully, we'll find a way of extending the old 109 00:05:27,640 --> 00:05:29,640 situation to cover the new. 110 00:05:29,640 --> 00:05:33,280 Now what does this mean in this particular instance? 111 00:05:33,280 --> 00:05:35,540 Well, let me just call it this. 112 00:05:35,540 --> 00:05:38,110 I'll call it analytical approach, trial number one. 113 00:05:38,110 --> 00:05:42,320 What I'm going to do is try to imitate exactly what we did in 114 00:05:42,320 --> 00:05:43,620 the area case. 115 00:05:43,620 --> 00:05:47,290 For example, if I take the region 'R', which I'll draw 116 00:05:47,290 --> 00:05:51,160 this way here, if this is the region 'R', namely bounded 117 00:05:51,160 --> 00:05:55,430 above by the curve 'y' equals 'f of x', below by the x-axis, 118 00:05:55,430 --> 00:05:58,000 on the left, by the line 'x' equals 'a', and on the right, 119 00:05:58,000 --> 00:06:01,010 by the line 'x' equals 'b', how did we find the area of 120 00:06:01,010 --> 00:06:01,860 the region 'R'? 121 00:06:01,860 --> 00:06:06,020 Well, what we did is we inscribed and we circumscribed 122 00:06:06,020 --> 00:06:07,150 rectangles. 123 00:06:07,150 --> 00:06:10,132 And we took the limit of the circumscribed rectangles, et 124 00:06:10,132 --> 00:06:13,270 cetera, and put the squeeze on as 'n' went to infinity. 125 00:06:13,270 --> 00:06:16,990 Now the idea is we might get the idea that maybe we should 126 00:06:16,990 --> 00:06:18,470 do the same thing for arc length. 127 00:06:18,470 --> 00:06:21,770 In other words, let me call one of these little pieces of 128 00:06:21,770 --> 00:06:23,540 arc length 'delta w'. 129 00:06:23,540 --> 00:06:24,990 In other words, I'm just isolating part 130 00:06:24,990 --> 00:06:26,370 of the diagram here. 131 00:06:26,370 --> 00:06:27,640 Here's 'delta w'. 132 00:06:27,640 --> 00:06:28,770 Here's 'delta x'. 133 00:06:28,770 --> 00:06:30,420 Here's 'delta y'. 134 00:06:30,420 --> 00:06:34,810 The idea is in the same way that I approximated a piece of 135 00:06:34,810 --> 00:06:37,830 area by an inscribed and a circumscribed rectangle, why 136 00:06:37,830 --> 00:06:41,660 can't I say something like, well, let me let 'delta w' be 137 00:06:41,660 --> 00:06:44,410 approximately equal to 'delta x'? 138 00:06:44,410 --> 00:06:46,820 And just to make sure that our memories are refreshed over 139 00:06:46,820 --> 00:06:51,950 here, notice that 'delta x' is just the length of each piece 140 00:06:51,950 --> 00:06:55,860 if the segment from 'a' to 'b', namely of length 'b - a', 141 00:06:55,860 --> 00:06:58,670 is divided into 'n' equal parts. 142 00:06:58,670 --> 00:07:01,710 See, the idea is why can't we mimic the same approach. 143 00:07:01,710 --> 00:07:05,140 And let me point out what is so crucial here in terms of 144 00:07:05,140 --> 00:07:08,310 what I mentioned above, namely, notice that the reason 145 00:07:08,310 --> 00:07:11,710 that we can say that the area of the region 'R' is just the 146 00:07:11,710 --> 00:07:14,920 limit of 'U sub n' as 'n' approaches infinity, where 'U 147 00:07:14,920 --> 00:07:18,000 sub n' is the area of the circumscribed rectangles. 148 00:07:18,000 --> 00:07:21,950 The only reason we can say that is because we squeezed 'A 149 00:07:21,950 --> 00:07:27,540 sub r' between 'L sub n', the inscribed rectangles, and 'U 150 00:07:27,540 --> 00:07:29,990 sub n', the circumscribed rectangles. 151 00:07:29,990 --> 00:07:32,310 And the limits of these lower bounds and 152 00:07:32,310 --> 00:07:34,030 upper bounds were equal. 153 00:07:34,030 --> 00:07:36,830 'A sub 'r was squeezed between these two. 154 00:07:36,830 --> 00:07:39,170 Hence, it had to equal the common limit. 155 00:07:39,170 --> 00:07:41,370 That was the structure that we used. 156 00:07:41,370 --> 00:07:44,910 On the other hand, we can't use that when we're dealing 157 00:07:44,910 --> 00:07:46,060 with arc length. 158 00:07:46,060 --> 00:07:48,450 And I'll mention that in a few moments again. 159 00:07:48,450 --> 00:07:50,860 But let me just point out what I'm driving at this way. 160 00:07:50,860 --> 00:07:53,380 Suppose we mimic this as we did before. 161 00:07:53,380 --> 00:07:56,620 And we say, OK, let the element of arc length, 'delta 162 00:07:56,620 --> 00:07:59,720 w', be approximately equal to 'delta x'. 163 00:07:59,720 --> 00:08:02,410 And now what I will do is define script 164 00:08:02,410 --> 00:08:04,500 'L' from 'a' to 'b'. 165 00:08:04,500 --> 00:08:06,570 I don't want to call it arc length because it may not be. 166 00:08:06,570 --> 00:08:10,090 But as a first approximation, let me define this symbol to 167 00:08:10,090 --> 00:08:14,640 be the limit of the sum of all these 'delta x's when we 168 00:08:14,640 --> 00:08:18,270 divide this region into 'n' parts as 'n' goes to infinity. 169 00:08:18,270 --> 00:08:21,060 Now look, I have the right to make up this particular 170 00:08:21,060 --> 00:08:22,540 definition. 171 00:08:22,540 --> 00:08:25,080 Now if I compute this limit, what happens? 172 00:08:25,080 --> 00:08:28,280 Recall that we mentioned that 'delta x' was 173 00:08:28,280 --> 00:08:30,060 'b - a' over 'n'. 174 00:08:30,060 --> 00:08:33,620 Consequently, if I have 'n' of these pieces, the total sum 175 00:08:33,620 --> 00:08:34,490 would be what? 176 00:08:34,490 --> 00:08:37,380 'n' times b' - a' over 'n'. 177 00:08:37,380 --> 00:08:40,995 And 'n' times 'b - a' over 'n' is just 'b - a'. 178 00:08:40,995 --> 00:08:45,170 In other words, script 'L' from 'a' to 'b' is defined. 179 00:08:45,170 --> 00:08:47,880 And it's 'b - a', not 'w'. 180 00:08:47,880 --> 00:08:49,550 In other words, coming back to our 181 00:08:49,550 --> 00:08:51,460 diagram, notice what happened. 182 00:08:51,460 --> 00:08:54,370 What we wanted was a recipe that would give 183 00:08:54,370 --> 00:08:56,160 us this length here. 184 00:08:56,160 --> 00:09:00,600 What we found was a recipe that gave us the length from 185 00:09:00,600 --> 00:09:01,730 'a' to 'b'. 186 00:09:01,730 --> 00:09:05,610 Now intuitively, we know that the length from 'a' to 'b' is 187 00:09:05,610 --> 00:09:07,620 not the arc length that we're looking for. 188 00:09:07,620 --> 00:09:10,520 In other words, what we defined to be script 'L' 189 00:09:10,520 --> 00:09:14,290 existed as a limit, but it gave us an answer which did 190 00:09:14,290 --> 00:09:16,530 not coincide with our intuition. 191 00:09:16,530 --> 00:09:19,480 Since we intuitively know what the right answer is, we must 192 00:09:19,480 --> 00:09:22,250 discard this approach in the sense that it doesn't give us 193 00:09:22,250 --> 00:09:24,330 an answer that we have any faith in. 194 00:09:24,330 --> 00:09:27,550 And by the way, notice where we went wrong over here if you 195 00:09:27,550 --> 00:09:29,200 want to look at it from that point of view. 196 00:09:29,200 --> 00:09:31,850 Notice that when we approximated 'delta w' by 197 00:09:31,850 --> 00:09:36,450 'delta x', it's clear from this diagram that 'delta x' 198 00:09:36,450 --> 00:09:38,370 was certainly less than 'delta w'. 199 00:09:38,370 --> 00:09:41,570 But notice that we didn't have an upper bound here. 200 00:09:41,570 --> 00:09:45,270 Or we can make speculations like, maybe 'delta x + delta 201 00:09:45,270 --> 00:09:48,320 y' would be more than 'delta w', and things like this. 202 00:09:48,320 --> 00:09:50,310 We'll talk about that more later. 203 00:09:50,310 --> 00:09:53,720 But for now, all I want us to see is the degree of 204 00:09:53,720 --> 00:09:57,420 sophistication that enters into the arc length problem 205 00:09:57,420 --> 00:10:00,160 that didn't bother us in either the area or the volume 206 00:10:00,160 --> 00:10:04,670 problems, namely, we are missing now the all-important 207 00:10:04,670 --> 00:10:06,430 squeeze element. 208 00:10:06,430 --> 00:10:09,050 Well, no sense crying over spilt milk. 209 00:10:09,050 --> 00:10:12,560 We go on, and we try the next type of approach. 210 00:10:12,560 --> 00:10:16,510 In other words, what we sense now is why don't we do this. 211 00:10:16,510 --> 00:10:20,730 Instead of approximating 'delta w' by 'delta x', why 212 00:10:20,730 --> 00:10:24,730 don't we approximate 'delta w' by the cord that joins the two 213 00:10:24,730 --> 00:10:27,070 endpoints of the arc. 214 00:10:27,070 --> 00:10:29,070 In other words, I think that we began to suspect 215 00:10:29,070 --> 00:10:32,530 intuitively that, somehow or other, for a small change in 216 00:10:32,530 --> 00:10:36,200 'delta x', 'delta s' should be a better approximation to 217 00:10:36,200 --> 00:10:38,660 'delta w' than 'delta x' was. 218 00:10:38,660 --> 00:10:41,720 Of course, the wide open question is granted that it's 219 00:10:41,720 --> 00:10:43,710 better, is it good enough. 220 00:10:43,710 --> 00:10:46,360 Well, we'll worry about that in a little more detail later. 221 00:10:46,360 --> 00:10:49,670 All we're saying is let 'delta w' be approximately 222 00:10:49,670 --> 00:10:50,620 equal to 'delta s'. 223 00:10:50,620 --> 00:10:52,330 In other words, we'll approximate 'delta 224 00:10:52,330 --> 00:10:54,100 w' by 'delta s'. 225 00:10:54,100 --> 00:10:58,230 And we'll now define 'L' from 'a' to 'b', 'L' from 'a' to 226 00:10:58,230 --> 00:11:02,160 'b' to be the limit not of the sum of 'delta x's now, but the 227 00:11:02,160 --> 00:11:06,500 sum of the 'delta s's, as 'k' goes from 1 to 'n', taken in 228 00:11:06,500 --> 00:11:08,790 the limit as 'n' goes to infinity. 229 00:11:08,790 --> 00:11:11,480 And for those of us who are more familiar with 'delta x's 230 00:11:11,480 --> 00:11:15,060 and 'delta y's, and the symbol delta s bothers us, simply 231 00:11:15,060 --> 00:11:18,750 observe that by the Pythagorean theorem, 'delta s' 232 00:11:18,750 --> 00:11:22,000 is related to 'delta x' and 'delta y' by ''delta s' 233 00:11:22,000 --> 00:11:23,860 squared' equals ''delta x' squared' 234 00:11:23,860 --> 00:11:25,290 plus ''delta y' squared'. 235 00:11:25,290 --> 00:11:28,490 So we can rewrite this in this particular form. 236 00:11:28,490 --> 00:11:32,570 In other words, I will define capital 'L' hopefully to stand 237 00:11:32,570 --> 00:11:33,590 for length later on. 238 00:11:33,590 --> 00:11:35,490 But we'll worry about that later too. 239 00:11:35,490 --> 00:11:39,370 But 'L' from 'a' to 'b' to be this particular limit. 240 00:11:39,370 --> 00:11:43,200 And now I claim that there are three natural questions with 241 00:11:43,200 --> 00:11:45,370 which we must come to grips. 242 00:11:45,370 --> 00:11:48,830 The first question is does this limit even exist. 243 00:11:48,830 --> 00:11:50,500 Does this limit exist? 244 00:11:50,500 --> 00:11:51,920 And the answer is that, except for 245 00:11:51,920 --> 00:11:54,200 far-fetched curves, it does. 246 00:11:54,200 --> 00:12:00,040 You really have to get a curve that wiggles uncontrollably to 247 00:12:00,040 --> 00:12:02,930 break the possibility of this limit existing. 248 00:12:02,930 --> 00:12:06,520 Unfortunately, there are pathological cases, one of 249 00:12:06,520 --> 00:12:10,120 which is described in the text assignment for this lesson, of 250 00:12:10,120 --> 00:12:13,110 a curve that doesn't have a finite limit when you try to 251 00:12:13,110 --> 00:12:14,730 compute the arc length this way. 252 00:12:14,730 --> 00:12:15,920 Just a little idiosyncrasy. 253 00:12:15,920 --> 00:12:19,110 However, for any curve that comes up in real life, that 254 00:12:19,110 --> 00:12:22,250 doesn't oscillate too violently with infinite 255 00:12:22,250 --> 00:12:25,210 variations, et cetera, et cetera, which we won't, again, 256 00:12:25,210 --> 00:12:28,920 talk about right now, the idea is that this limit does exist. 257 00:12:28,920 --> 00:12:31,990 As far as this course is concerned, we shall assume the 258 00:12:31,990 --> 00:12:33,940 answer to question one is yes. 259 00:12:33,940 --> 00:12:36,520 In fact, the way we'll do it without being dictatorial is 260 00:12:36,520 --> 00:12:38,795 we'll say, look, if this limit doesn't exist, we just won't 261 00:12:38,795 --> 00:12:40,120 study that curve. 262 00:12:40,120 --> 00:12:42,840 In fact, we will call a curve rectifiable 263 00:12:42,840 --> 00:12:44,510 if this limit exists. 264 00:12:44,510 --> 00:12:47,460 And so we'll assume that we deal only with rectifiable 265 00:12:47,460 --> 00:12:50,040 curves, in other words, that this limit does exist. 266 00:12:50,040 --> 00:12:51,380 Question number two. 267 00:12:51,380 --> 00:12:52,970 OK, the limit exists. 268 00:12:52,970 --> 00:12:54,790 So how do we compute it? 269 00:12:54,790 --> 00:12:58,060 And that, in general, is not a very easy thing to answer. 270 00:12:58,060 --> 00:13:00,830 What's even worse though is that after you've answered 271 00:13:00,830 --> 00:13:03,870 this, you have to come to grips with a question that we 272 00:13:03,870 --> 00:13:07,480 were able to dodge when we studied both area and volume, 273 00:13:07,480 --> 00:13:11,370 namely, the question is once this limit does exist and you 274 00:13:11,370 --> 00:13:14,750 compute it, how do you know that it agrees with our 275 00:13:14,750 --> 00:13:16,900 intuitive definition of arc length. 276 00:13:16,900 --> 00:13:19,230 In other words, if you recall what we did just a few minutes 277 00:13:19,230 --> 00:13:23,330 ago, we defined script 'L' from 'a' to 'b' to be a 278 00:13:23,330 --> 00:13:24,290 certain limit. 279 00:13:24,290 --> 00:13:26,800 We showed that that limit existed. 280 00:13:26,800 --> 00:13:30,560 The problem was is that limit, even though it existed, did 281 00:13:30,560 --> 00:13:34,350 not give us an answer that agreed intuitively with what 282 00:13:34,350 --> 00:13:36,700 we believed arc length was supposed to mean. 283 00:13:36,700 --> 00:13:38,810 In other words, you see, we've assumed the answer to the 284 00:13:38,810 --> 00:13:40,820 first question is yes. 285 00:13:40,820 --> 00:13:42,640 Now we have two questions to answer. 286 00:13:42,640 --> 00:13:45,290 How do you compute this limit, which is a hard question in 287 00:13:45,290 --> 00:13:46,080 it's own right? 288 00:13:46,080 --> 00:13:49,110 Secondly, once you do compute this limit, how do you know 289 00:13:49,110 --> 00:13:52,340 that it's going to agree with the intuitive answer that you 290 00:13:52,340 --> 00:13:53,430 get for arc length? 291 00:13:53,430 --> 00:13:56,510 And this shall be what we have to answer in the remainder of 292 00:13:56,510 --> 00:13:58,140 our lesson today. 293 00:13:58,140 --> 00:13:59,680 Let's take these in order. 294 00:13:59,680 --> 00:14:02,370 And let's try to answer question number two first. 295 00:14:02,370 --> 00:14:05,470 The idea is we've defined capital 'L' from 'a' to 'b' to 296 00:14:05,470 --> 00:14:07,770 be this particular limit, and we'd like to know if this 297 00:14:07,770 --> 00:14:09,000 limit exists. 298 00:14:09,000 --> 00:14:12,370 Not only that, but we have a great command of calculus at 299 00:14:12,370 --> 00:14:13,590 our disposal now. 300 00:14:13,590 --> 00:14:17,490 All of the previous lessons can be brought to bear here to 301 00:14:17,490 --> 00:14:20,470 help us put this into the perspective of what calculus 302 00:14:20,470 --> 00:14:21,380 is all about. 303 00:14:21,380 --> 00:14:24,890 For example, when I see an expression like this, I like 304 00:14:24,890 --> 00:14:26,670 to think in terms of a derivative. 305 00:14:26,670 --> 00:14:29,660 A derivative reminds me of 'delta y' divided by 'delta 306 00:14:29,660 --> 00:14:30,810 x', et cetera. 307 00:14:30,810 --> 00:14:33,910 So what I do here is I factor out a ''delta x' squared'. 308 00:14:33,910 --> 00:14:36,750 In other words, I divide through by ''delta x' squared' 309 00:14:36,750 --> 00:14:39,390 inside the radical sign, which is really the same 310 00:14:39,390 --> 00:14:41,920 equivalently as dividing by 'delta x'. 311 00:14:41,920 --> 00:14:44,230 And I multiply by 'delta x' outside. 312 00:14:44,230 --> 00:14:46,840 In other words, factoring out with ''delta x' squared', the 313 00:14:46,840 --> 00:14:49,100 square root of ''delta x' squared' plus ''delta y' 314 00:14:49,100 --> 00:14:53,060 squared' can be written as the square root of '1 + ''delta y' 315 00:14:53,060 --> 00:14:56,240 over 'delta x'' squared' times 'delta x'. 316 00:14:56,240 --> 00:14:59,900 Now the idea is that 'delta y' over 'delta x' is the slope of 317 00:14:59,900 --> 00:15:03,910 that cord that joins the two endpoints of 'delta w'. 318 00:15:03,910 --> 00:15:05,490 It's not a derivative as we know it. 319 00:15:05,490 --> 00:15:07,750 It's the slope of a straight line cord, not 320 00:15:07,750 --> 00:15:09,220 the slope of a curve. 321 00:15:09,220 --> 00:15:10,750 Now the whole idea is this. 322 00:15:10,750 --> 00:15:14,040 We know from the mean value theorem that if our curve is 323 00:15:14,040 --> 00:15:17,820 smooth, there is a point in the interval at which the 324 00:15:17,820 --> 00:15:20,690 derivative at that point is equal to the 325 00:15:20,690 --> 00:15:22,020 slope of the cord. 326 00:15:22,020 --> 00:15:25,170 In other words, if 'f' is differentiable on [a, b], we 327 00:15:25,170 --> 00:15:27,170 may invoke the Mean Value Theorem-- 328 00:15:27,170 --> 00:15:30,280 here abbreviated as MVT, the Mean Value Theorem-- 329 00:15:30,280 --> 00:15:34,660 to conclude that there is some point 'c sub k' in our 'delta 330 00:15:34,660 --> 00:15:38,680 x' interval for which ''delta y' over 'delta x'' is 'f prime 331 00:15:38,680 --> 00:15:39,860 of 'c sub k''. 332 00:15:39,860 --> 00:15:42,420 And in order to help you facilitate what we're talking 333 00:15:42,420 --> 00:15:45,005 about in your minds, look at the following diagram. 334 00:15:45,005 --> 00:15:46,110 This is all we're saying. 335 00:15:46,110 --> 00:15:48,350 What we're saying is here's our 'delta x', 336 00:15:48,350 --> 00:15:49,520 here's our 'delta y'. 337 00:15:49,520 --> 00:15:52,870 We'll call this point 'x sub 'k - 1'', this 338 00:15:52,870 --> 00:15:54,240 point 'x sub k'. 339 00:15:54,240 --> 00:15:56,020 This is our k-th partition. 340 00:15:56,020 --> 00:15:59,270 'Delta y' divided by 'delta x' is just the 341 00:15:59,270 --> 00:16:00,850 slope of this line. 342 00:16:00,850 --> 00:16:03,630 See, that's just the slope of this line. 343 00:16:03,630 --> 00:16:05,825 And what the Mean Value Theorem says is if this curve 344 00:16:05,825 --> 00:16:10,280 is smooth, some place on this arc, there is a point where 345 00:16:10,280 --> 00:16:16,380 the line tangent to the curve is parallel to this cord. 346 00:16:16,380 --> 00:16:18,980 And that's what I'm calling the point 'c sub k'. 347 00:16:18,980 --> 00:16:23,170 'c sub k' is the point at which the slope of the curve 348 00:16:23,170 --> 00:16:25,430 is equal to the slope of the cord. 349 00:16:25,430 --> 00:16:29,290 In other words, if 'f' is continuous, I can conclude 350 00:16:29,290 --> 00:16:34,880 that 'L' from 'a' to 'b' is the limit as 'n' approaches 351 00:16:34,880 --> 00:16:39,500 infinity, summation 'k' goes from 1 to 'n', square root of 352 00:16:39,500 --> 00:16:44,830 '1 + ''f prime 'c sub k'' squared' times 'delta x'. 353 00:16:44,830 --> 00:16:49,650 And notice that this now starts to look like my 354 00:16:49,650 --> 00:16:52,530 definite integral according to the definition that we were 355 00:16:52,530 --> 00:16:55,720 talking about in our earlier lectures in this block. 356 00:16:55,720 --> 00:16:59,070 In fact, how can we invoke the first fundamental theorem of 357 00:16:59,070 --> 00:17:00,340 integral calculus? 358 00:17:00,340 --> 00:17:04,583 Remember, if this expression here-- it's 359 00:17:04,583 --> 00:17:05,829 not an integral yet-- 360 00:17:05,829 --> 00:17:08,589 happens to be a continuous function, then we're in pretty 361 00:17:08,589 --> 00:17:09,700 good shape. 362 00:17:09,700 --> 00:17:12,630 In other words, if I can assume that 'f prime' is 363 00:17:12,630 --> 00:17:13,240 continuous-- 364 00:17:13,240 --> 00:17:15,470 let's go over here and continue on here. 365 00:17:15,470 --> 00:17:20,660 See, what I'm saying is if I can assume that 'f prime' is 366 00:17:20,660 --> 00:17:24,200 continuous, well, look, the square of a continuous 367 00:17:24,200 --> 00:17:25,980 function is continuous. 368 00:17:25,980 --> 00:17:29,280 The sum of two continuous functions is continuous. 369 00:17:29,280 --> 00:17:31,900 And the square root of a continuous function is 370 00:17:31,900 --> 00:17:32,830 continuous. 371 00:17:32,830 --> 00:17:36,150 In other words, and this is a key point, if the derivative 372 00:17:36,150 --> 00:17:41,380 is continuous, I can conclude that the 'L' from 'a' to 'b' 373 00:17:41,380 --> 00:17:43,650 can be replaced by the definite integral from 'a' to 374 00:17:43,650 --> 00:17:48,690 'b' square root of '1 + ''dy/dx' squared'' times 'dx', 375 00:17:48,690 --> 00:17:54,150 which I quickly point out may be hard to evaluate. 376 00:17:54,150 --> 00:17:57,370 In other words, one thing I could try to do over here is 377 00:17:57,370 --> 00:18:01,110 to find the function g whose derivative with respect to 'x' 378 00:18:01,110 --> 00:18:04,000 is the square root of '1 + ''dy/dx' squared'' and 379 00:18:04,000 --> 00:18:05,890 evaluate that between 'a' and 'b'. 380 00:18:05,890 --> 00:18:10,850 I can put approximations on here, whatever I want. 381 00:18:10,850 --> 00:18:12,500 In fact, let's summarize it down here. 382 00:18:12,500 --> 00:18:16,220 If 'f' is differentiable on the closed interval from 'a' 383 00:18:16,220 --> 00:18:20,000 to 'b' and if 'f prime' is the derivative-- 384 00:18:20,000 --> 00:18:21,320 you see, 'f prime'-- 385 00:18:21,320 --> 00:18:23,750 is also continuous on the closed interval from 'a' to 386 00:18:23,750 --> 00:18:27,250 'b', then not only does capital 'L' from 'a' to 'b' 387 00:18:27,250 --> 00:18:30,410 exist, but it's given computationally by this 388 00:18:30,410 --> 00:18:32,150 particular integral. 389 00:18:32,150 --> 00:18:35,220 And that answers question number two, that the limit 390 00:18:35,220 --> 00:18:38,000 exists, and this is what it's equal to. 391 00:18:38,000 --> 00:18:40,110 The problem that we're faced with-- 392 00:18:40,110 --> 00:18:41,180 and I've written this out. 393 00:18:41,180 --> 00:18:44,010 I think it looks harder than what it says. 394 00:18:44,010 --> 00:18:46,825 But I've taken the trouble to write this whole thing out, so 395 00:18:46,825 --> 00:18:49,870 that if you have trouble following what I'm saying, 396 00:18:49,870 --> 00:18:52,530 that you can see this thing blocked out for you. 397 00:18:52,530 --> 00:18:53,590 The idea is this. 398 00:18:53,590 --> 00:18:57,060 What we have done is we have approximated 'delta 399 00:18:57,060 --> 00:18:59,170 w' by 'delta s'. 400 00:18:59,170 --> 00:19:02,390 Then what we said is 'w' is the sum of all 401 00:19:02,390 --> 00:19:03,900 these 'delta w's. 402 00:19:03,900 --> 00:19:08,460 And since each 'delta w' is approximately 'delta s', then 403 00:19:08,460 --> 00:19:12,340 what we can be sure of is that 'w' is approximated by this 404 00:19:12,340 --> 00:19:13,730 sum over here. 405 00:19:13,730 --> 00:19:14,740 Now here's what we did. 406 00:19:14,740 --> 00:19:17,050 We didn't work with 'w' at all after this. 407 00:19:17,050 --> 00:19:20,000 We turned our attention to this. 408 00:19:20,000 --> 00:19:21,440 This is what we did in our case here. 409 00:19:21,440 --> 00:19:24,750 And we showed that this limit existed. 410 00:19:24,750 --> 00:19:29,810 We showed that the limit, as 'k' went from 1 to 'n' and 411 00:19:29,810 --> 00:19:33,790 then went to infinity of these pieces here, was 'L' of 'ab'. 412 00:19:33,790 --> 00:19:34,770 And that existed. 413 00:19:34,770 --> 00:19:39,600 What we did not show is that this limit was w itself. 414 00:19:39,600 --> 00:19:42,940 Intuitively, you might say, if I put the squeeze on this, 415 00:19:42,940 --> 00:19:45,670 doesn't this get rid of all the error for me? 416 00:19:45,670 --> 00:19:48,930 We haven't shown that we've gotten rid of all the error. 417 00:19:48,930 --> 00:19:51,750 In essence, how do we know if all the error has been 418 00:19:51,750 --> 00:19:53,000 squeezed out? 419 00:19:53,000 --> 00:19:56,210 This is precisely what question three is all about. 420 00:19:56,210 --> 00:19:59,530 Again, going back to what we did earlier, remember, when we 421 00:19:59,530 --> 00:20:03,330 approximated 'delta w' by 'delta x', then we said, OK, 422 00:20:03,330 --> 00:20:06,380 add up all these 'delta x's, and take the limit as 'n' goes 423 00:20:06,380 --> 00:20:07,190 to infinity. 424 00:20:07,190 --> 00:20:11,090 We found that that limit was 'b - a', which was not the 425 00:20:11,090 --> 00:20:12,670 length of the curve. 426 00:20:12,670 --> 00:20:14,750 In other words, somehow or other, even though the limit 427 00:20:14,750 --> 00:20:17,760 existed, we did not squeeze out all the error. 428 00:20:17,760 --> 00:20:21,220 And this is why the study of arc length is so difficult. 429 00:20:21,220 --> 00:20:23,210 Because we don't have a sandwiching effect. 430 00:20:23,210 --> 00:20:27,210 It is very difficult for us to figure out when we've squeezed 431 00:20:27,210 --> 00:20:28,350 out all the error. 432 00:20:28,350 --> 00:20:30,290 So at any rate, let me generalize 433 00:20:30,290 --> 00:20:32,880 question number three. 434 00:20:32,880 --> 00:20:34,120 Remember what question number three is? 435 00:20:34,120 --> 00:20:36,510 How do we know that if the limit exists, 436 00:20:36,510 --> 00:20:37,610 it's equal to 'w'? 437 00:20:37,610 --> 00:20:40,860 All I'm saying is don't even worry about arc length. 438 00:20:40,860 --> 00:20:44,420 Just suppose that 'w' is any function defined on a closed 439 00:20:44,420 --> 00:20:47,470 interval from 'a' to 'b' and that we've approximated 'delta 440 00:20:47,470 --> 00:20:52,350 w' by something of the form 'g of 'c sub k'' times 'delta x', 441 00:20:52,350 --> 00:20:55,700 where 'g' is what I call some intuitive function 442 00:20:55,700 --> 00:20:57,140 defined on [a, b]. 443 00:20:57,140 --> 00:21:01,160 For example, in our earlier example, we started with 444 00:21:01,160 --> 00:21:02,820 'delta w' being arc length. 445 00:21:02,820 --> 00:21:06,460 And we approximated 'delta w' by 'delta x' in which case 'g' 446 00:21:06,460 --> 00:21:09,220 would've been the function which is identically 1. 447 00:21:09,220 --> 00:21:12,130 In the area situation, remember we approximated 448 00:21:12,130 --> 00:21:15,120 'delta A' by something times 'delta x'. 449 00:21:15,120 --> 00:21:16,860 Well, what times 'delta x'? 450 00:21:16,860 --> 00:21:18,570 Well, it was the height of a rectangle. 451 00:21:18,570 --> 00:21:21,370 In other words, we look at the thing we're trying to find, we 452 00:21:21,370 --> 00:21:22,370 use our intuition-- 453 00:21:22,370 --> 00:21:25,090 and this is difficult because intuition varies from person 454 00:21:25,090 --> 00:21:25,890 to person-- 455 00:21:25,890 --> 00:21:28,290 and we say, what would make a good approximation here. 456 00:21:28,290 --> 00:21:30,600 What would be an approximation? 457 00:21:30,600 --> 00:21:35,770 We say, OK, let's approximate 'delta w' by 'g of 'c sub k'' 458 00:21:35,770 --> 00:21:38,420 times 'delta x', where 'c' is some point in the 459 00:21:38,420 --> 00:21:39,570 interval, et cetera. 460 00:21:39,570 --> 00:21:42,290 Then we add up all of these 'delta w's as 'k' 461 00:21:42,290 --> 00:21:43,800 goes from 1 to 'n'. 462 00:21:43,800 --> 00:21:45,830 We say, OK, that's approximately 463 00:21:45,830 --> 00:21:47,420 this thing over here. 464 00:21:47,420 --> 00:21:51,750 Now what we have shown is that if 'g' is continuous on [a, b] 465 00:21:51,750 --> 00:21:57,200 then as 'n' goes to infinity, this particular limit exists 466 00:21:57,200 --> 00:21:59,210 and is denoted by the integral from 'a' to 467 00:21:59,210 --> 00:22:01,840 'b', ''g of x' dx'. 468 00:22:01,840 --> 00:22:03,900 This is what we've shown so far. 469 00:22:03,900 --> 00:22:07,320 What the big question is is, granted that this limit 470 00:22:07,320 --> 00:22:09,810 exists, does it equal 'w'? 471 00:22:09,810 --> 00:22:13,280 In other words, is 'w' equal to the integral from 'a' to 472 00:22:13,280 --> 00:22:15,870 b', ''g of x' dx'? 473 00:22:15,870 --> 00:22:19,060 That's what the remainder of today's lesson is about as far 474 00:22:19,060 --> 00:22:20,400 as arc length is concerned. 475 00:22:20,400 --> 00:22:23,170 And I'm going to solve this problem in general first and 476 00:22:23,170 --> 00:22:25,420 then make some applications about this 477 00:22:25,420 --> 00:22:27,590 to arc length itself. 478 00:22:27,590 --> 00:22:30,560 And by the way, what we're going to see next is you may 479 00:22:30,560 --> 00:22:33,610 remember that very, very early in our course, we came to 480 00:22:33,610 --> 00:22:36,220 grips with something called infinitesimals. 481 00:22:36,220 --> 00:22:40,630 We came to grips with this delta y tan infinitesimals of 482 00:22:40,630 --> 00:22:41,520 higher order. 483 00:22:41,520 --> 00:22:45,120 And now we're going to see how just as this came up in 484 00:22:45,120 --> 00:22:49,170 differential calculus, these same problems of approximation 485 00:22:49,170 --> 00:22:51,200 come up in integral calculus. 486 00:22:51,200 --> 00:22:54,020 The only difference, as we've mentioned before, is instead 487 00:22:54,020 --> 00:22:56,690 of having to come to grips with the indeterminate form 488 00:22:56,690 --> 00:22:59,890 0/0, we're going to have to come to grips with the 489 00:22:59,890 --> 00:23:02,700 indeterminate form infinity times 0. 490 00:23:02,700 --> 00:23:04,560 Let me show you what I mean by that. 491 00:23:04,560 --> 00:23:05,550 The idea is this. 492 00:23:05,550 --> 00:23:08,920 Let's suppose that our case 'delta w'-- we've broken up 493 00:23:08,920 --> 00:23:10,800 'w' now into increments-- 494 00:23:10,800 --> 00:23:13,960 and let's suppose that we're approximating 'delta w', as we 495 00:23:13,960 --> 00:23:18,500 said before, by 'g of 'c sub k'' times 'delta x'. 496 00:23:18,500 --> 00:23:20,800 Well, what do we mean by we're approximating this? 497 00:23:20,800 --> 00:23:23,400 What we mean is there's some error in here. 498 00:23:23,400 --> 00:23:27,110 Let's call the error 'alpha sub k' times 'delta x'. 499 00:23:27,110 --> 00:23:29,960 In other words, this is just a correction factor. 500 00:23:29,960 --> 00:23:32,260 This is what we have to add on to this to make 501 00:23:32,260 --> 00:23:34,130 this equality whole. 502 00:23:34,130 --> 00:23:37,510 Once I add on the error, I'm no longer working with an 503 00:23:37,510 --> 00:23:38,230 inequality. 504 00:23:38,230 --> 00:23:40,920 I'm working with an equality. 505 00:23:40,920 --> 00:23:43,130 And that allows me to use some theorems. 506 00:23:43,130 --> 00:23:47,140 What I can say now is by definition, w is the sum of 507 00:23:47,140 --> 00:23:48,720 all these 'delta w's. 508 00:23:48,720 --> 00:23:52,120 But 'delta w' being a sum, we can use theorems about the 509 00:23:52,120 --> 00:23:53,390 sigma notation. 510 00:23:53,390 --> 00:23:56,080 In other words, what is the sum of all these 'delta w's? 511 00:23:56,080 --> 00:23:59,200 It's the sum of all of these pieces plus the sum of all of 512 00:23:59,200 --> 00:24:01,700 these pieces, which I've written over here. 513 00:24:01,700 --> 00:24:05,130 And now you see, if I transpose, I get that 'w' 514 00:24:05,130 --> 00:24:09,330 minus this sum is equal to the 'sum k' goes from 1 to 'n', 515 00:24:09,330 --> 00:24:12,260 'alpha k' times 'delta x'. 516 00:24:12,260 --> 00:24:14,670 Now the next thing I do is take the limit 517 00:24:14,670 --> 00:24:16,600 as 'n' goes to infinity. 518 00:24:16,600 --> 00:24:20,810 By definition, since 'g' is a continuous function, this 519 00:24:20,810 --> 00:24:23,350 limit here is just the definite integral from 'a' to 520 00:24:23,350 --> 00:24:25,530 'b', ''g of x' dx'. 521 00:24:25,530 --> 00:24:28,490 On the other hand, this limit here is what we have to 522 00:24:28,490 --> 00:24:29,560 investigate. 523 00:24:29,560 --> 00:24:32,500 In other words, we would like to know whether 'w' is equal 524 00:24:32,500 --> 00:24:34,010 to the definite integral or not. 525 00:24:34,010 --> 00:24:36,910 If we look at this particular equation, what we have now 526 00:24:36,910 --> 00:24:40,350 shown is whatever the relationship is between these 527 00:24:40,350 --> 00:24:44,130 two terms, it's typified by the fact that this difference 528 00:24:44,130 --> 00:24:45,810 is this particular limit. 529 00:24:45,810 --> 00:24:48,840 In other words, if this limit happens to be 0, then the 530 00:24:48,840 --> 00:24:52,330 integral will equal what we're setting out to show it's equal 531 00:24:52,330 --> 00:24:53,850 to, namely, this function itself. 532 00:24:53,850 --> 00:24:57,450 On the other hand, what we're saying is we do not know that 533 00:24:57,450 --> 00:24:58,770 this limit is 0. 534 00:24:58,770 --> 00:25:00,470 By the way, notice what's happening over here. 535 00:25:00,470 --> 00:25:04,020 As 'n' goes to infinity, 'delta x' is going to 0. 536 00:25:04,020 --> 00:25:07,260 In other words, each individual term in the sum is 537 00:25:07,260 --> 00:25:10,540 going to 0, but the number of pieces is becoming infinite. 538 00:25:10,540 --> 00:25:13,340 There's your infinity times 0 form here. 539 00:25:13,340 --> 00:25:16,420 And let me show you a case where the pieces are growing 540 00:25:16,420 --> 00:25:20,170 too fast in number to be offset by the fact that their 541 00:25:20,170 --> 00:25:21,730 size is going to 0. 542 00:25:21,730 --> 00:25:25,110 For the sake of argument, let me suppose that 'alpha sub k' 543 00:25:25,110 --> 00:25:29,100 happens to be some non-0 constant for all 'k'. 544 00:25:29,100 --> 00:25:33,110 If I come back to this expression here, if 'alpha sub 545 00:25:33,110 --> 00:25:36,310 k' is equal to a constant, I'll replace 'alpha sub k' by 546 00:25:36,310 --> 00:25:38,300 that constant, which is 'c'. 547 00:25:38,300 --> 00:25:39,450 I now have what? 548 00:25:39,450 --> 00:25:42,340 That the limit that I'm looking for is the 'sum k' 549 00:25:42,340 --> 00:25:46,030 goes from 1 to 'n', 'c' times 'delta x', taking the limit as 550 00:25:46,030 --> 00:25:47,650 'n' goes to infinity. 551 00:25:47,650 --> 00:25:50,100 'c' is a constant, so I can take it outside 552 00:25:50,100 --> 00:25:51,550 the integral sign. 553 00:25:51,550 --> 00:25:54,280 Since 'c' is a constant and it's outside the integral 554 00:25:54,280 --> 00:25:56,410 sign, let's look at what 'delta x' is. 555 00:25:56,410 --> 00:25:59,800 'Delta x' is 'b - a' divided by 'n', same as we were 556 00:25:59,800 --> 00:26:01,390 talking about earlier in the lecture. 557 00:26:01,390 --> 00:26:03,070 I have 'n' of these pieces. 558 00:26:03,070 --> 00:26:06,370 The 'n' in the denominator cancels the 'n' in the 559 00:26:06,370 --> 00:26:07,930 numerator when I add these up. 560 00:26:07,930 --> 00:26:10,900 And notice that this particular sum here, no matter 561 00:26:10,900 --> 00:26:13,570 what 'n' is, is just 'b - a'. 562 00:26:13,570 --> 00:26:15,770 In other words, in the case that 'alpha sub k' is a 563 00:26:15,770 --> 00:26:20,810 constant, notice that this limit is 'c' times 'b - a'. 564 00:26:20,810 --> 00:26:22,370 'c' is not 0. 565 00:26:22,370 --> 00:26:23,570 'b' is not equal to 'a'. 566 00:26:23,570 --> 00:26:25,090 We have an interval here. 567 00:26:25,090 --> 00:26:26,820 Therefore, this will not be 0. 568 00:26:26,820 --> 00:26:31,510 And notice that if this is not 0, these two things 569 00:26:31,510 --> 00:26:33,210 here are not equal. 570 00:26:33,210 --> 00:26:37,150 And by the way, the aside that I would like to make here is 571 00:26:37,150 --> 00:26:43,710 that even though this error is not negligible, notice the 572 00:26:43,710 --> 00:26:48,280 fact that if 'alpha sub k' is a constant that as 'delta x' 573 00:26:48,280 --> 00:26:52,270 goes to 0, this whole term will go to 0. 574 00:26:52,270 --> 00:26:54,460 But it doesn't go to 0 fast enough. 575 00:26:54,460 --> 00:26:57,760 In other words, eventually, we're taking this sum as 'n' 576 00:26:57,760 --> 00:26:58,990 goes to infinity. 577 00:26:58,990 --> 00:27:00,460 And here's a case where, what? 578 00:27:00,460 --> 00:27:04,000 The pieces went to 0, but not fast enough to become 579 00:27:04,000 --> 00:27:05,800 negligible. 580 00:27:05,800 --> 00:27:07,990 Well, let me give you something in contrast to this. 581 00:27:07,990 --> 00:27:11,840 Situation number two is suppose instead 'alpha k' is a 582 00:27:11,840 --> 00:27:14,120 constant times 'delta x'. 583 00:27:14,120 --> 00:27:16,730 'B' times 'delta x', where 'B' is a constant. 584 00:27:16,730 --> 00:27:20,160 In that case, notice that summation 'k' goes from 1 to 585 00:27:20,160 --> 00:27:24,200 'n', 'alpha k' times 'delta x' is just summation 'k' goes 586 00:27:24,200 --> 00:27:27,840 from 1 to 'n', 'B' times ''delta x' squared'. 587 00:27:27,840 --> 00:27:30,040 Now keep in mind again that 'delta x' is still 588 00:27:30,040 --> 00:27:31,560 'b - a' over 'n'. 589 00:27:31,560 --> 00:27:33,300 So ''delta x' squared', of course, is 'b 590 00:27:33,300 --> 00:27:35,260 - a' over 'n squared'. 591 00:27:35,260 --> 00:27:39,070 Notice that what's inside the summation sign here does not 592 00:27:39,070 --> 00:27:40,280 depend on 'k'. 593 00:27:40,280 --> 00:27:41,450 It's a constant. 594 00:27:41,450 --> 00:27:44,340 I can take it outside the summation sign. 595 00:27:44,340 --> 00:27:46,580 How many terms of this size do I have? 596 00:27:46,580 --> 00:27:50,360 Well, 'k' goes from 1 to 'n', so I have 'n' of those pieces. 597 00:27:50,360 --> 00:27:53,350 Therefore, this sum is given by this. 598 00:27:53,350 --> 00:27:55,020 This is an 'n squared' term. 599 00:27:55,020 --> 00:27:57,930 One of the 'n's in the denominator cancels with my 600 00:27:57,930 --> 00:27:59,070 'n' in the numerator. 601 00:27:59,070 --> 00:28:02,550 And in this particular case, I find that the sum, as 'k' goes 602 00:28:02,550 --> 00:28:06,800 from 1 to 'n', 'alpha sub k' times 'delta x', is 'B', which 603 00:28:06,800 --> 00:28:10,480 is a constant, times ''b - a' squared', which is also a 604 00:28:10,480 --> 00:28:13,160 constant, divided by 'n'. 605 00:28:13,160 --> 00:28:17,270 Now look, if I now allow 'n' to go to infinity, my 606 00:28:17,270 --> 00:28:19,570 numerator is a constant. 607 00:28:19,570 --> 00:28:21,340 My denominator is 'n'. 608 00:28:21,340 --> 00:28:23,840 As 'n' goes to infinity, my denominator 609 00:28:23,840 --> 00:28:25,190 increases without bound. 610 00:28:25,190 --> 00:28:27,070 My numerator remains constant. 611 00:28:27,070 --> 00:28:29,650 So the limit is 0. 612 00:28:29,650 --> 00:28:33,210 In other words, in the case where 'alpha sub k' is a 613 00:28:33,210 --> 00:28:38,030 constant times 'delta x', this limit is 0, the error is 614 00:28:38,030 --> 00:28:42,070 squeezed out, and, in this particular case, 'w' is given 615 00:28:42,070 --> 00:28:46,450 by the integral from 'a' to 'b', ''g of x' dx' exactly in 616 00:28:46,450 --> 00:28:48,140 this particular situation. 617 00:28:48,140 --> 00:28:51,300 Well, the question is how many situations shall we go through 618 00:28:51,300 --> 00:28:52,710 before we generalize. 619 00:28:52,710 --> 00:28:55,030 And the answer is since this lecture is already becoming 620 00:28:55,030 --> 00:28:59,270 quite long, let's generalize now without any more details. 621 00:28:59,270 --> 00:29:01,190 And the generalization is this. 622 00:29:01,190 --> 00:29:05,720 In general, if you break down 'w' into increments, which 623 00:29:05,720 --> 00:29:09,030 we'll call 'delta 'w sub k'', and 'delta 'w sub 624 00:29:09,030 --> 00:29:11,360 k'' is equal to-- 625 00:29:11,360 --> 00:29:12,890 well, I've made a little slip here. 626 00:29:12,890 --> 00:29:13,850 That should be a 'g' in here. 627 00:29:13,850 --> 00:29:15,480 I'm using 'g's rather than 'f's. 628 00:29:15,480 --> 00:29:19,950 If 'delta 'w sub k'' is 'g of 'c sub k'' times 'delta x' 629 00:29:19,950 --> 00:29:24,100 plus the correction factor 'alpha k' times 'delta x', 630 00:29:24,100 --> 00:29:27,810 and, for each 'k', the limit of 'alpha k' as 'delta x' 631 00:29:27,810 --> 00:29:29,510 approaches 0 is 0. 632 00:29:29,510 --> 00:29:32,800 In other words, what we're saying is that 'alpha sub k' 633 00:29:32,800 --> 00:29:36,970 times 'delta x' must be a higher order infinitesimal. 634 00:29:36,970 --> 00:29:40,700 If this is a higher order infinitesimal, if 'alpha k' 635 00:29:40,700 --> 00:29:44,510 goes to 0 as 'delta x' goes to 0, that says, what? 636 00:29:44,510 --> 00:29:48,660 That 'alpha k' times 'delta x' is going to 0 much faster than 637 00:29:48,660 --> 00:29:50,100 'delta x' itself. 638 00:29:50,100 --> 00:29:53,570 So you compare this with our discussion on infinitesimals 639 00:29:53,570 --> 00:29:55,140 earlier in our course. 640 00:29:55,140 --> 00:29:57,790 I think that was in block two, but that's irrelevant here. 641 00:29:57,790 --> 00:30:01,370 But all I'm saying is if that is the case, in that 642 00:30:01,370 --> 00:30:05,560 particular case, the limit, that integral is exactly what 643 00:30:05,560 --> 00:30:06,470 we're looking for. 644 00:30:06,470 --> 00:30:08,350 The error has been squeezed out. 645 00:30:08,350 --> 00:30:12,570 In other words, now, in conclusion, what we must do in 646 00:30:12,570 --> 00:30:15,690 our present problem to answer question number three, 647 00:30:15,690 --> 00:30:21,900 remember, we have approximated delta wk by this intricate 648 00:30:21,900 --> 00:30:25,790 little thing, '1 + ''f prime 'c sub k' squared' 649 00:30:25,790 --> 00:30:26,920 times 'delta x'. 650 00:30:26,920 --> 00:30:29,360 In other words, in our particular illustration in 651 00:30:29,360 --> 00:30:33,230 this lecture, the role of 'g' is played by the square root 652 00:30:33,230 --> 00:30:35,260 of '1 + 'f prime squared''. 653 00:30:35,260 --> 00:30:39,750 What we must show is that this difference is a higher order 654 00:30:39,750 --> 00:30:41,140 differential. 655 00:30:41,140 --> 00:30:45,070 And this really requires much more advanced work than we 656 00:30:45,070 --> 00:30:46,430 really want to go into. 657 00:30:46,430 --> 00:30:49,810 The only trouble is, as a student, I always used to be 658 00:30:49,810 --> 00:30:55,410 upset when the instructor said, the proof is beyond our 659 00:30:55,410 --> 00:30:56,570 ability or knowledge. 660 00:30:56,570 --> 00:30:59,590 Whenever he used to say, the proof is beyond our knowledge 661 00:30:59,590 --> 00:31:02,100 at this stage of the game, I used to say to myself, ah, he 662 00:31:02,100 --> 00:31:03,190 doesn't know how to prove it. 663 00:31:03,190 --> 00:31:05,400 I think there's something upsetting about this. 664 00:31:05,400 --> 00:31:08,250 So what I'm going to try to do for a finale here is to at 665 00:31:08,250 --> 00:31:11,780 least give you a plausibility argument that we really do 666 00:31:11,780 --> 00:31:15,170 squeeze the error out in our approximation of 667 00:31:15,170 --> 00:31:16,410 'delta w' in this case. 668 00:31:16,410 --> 00:31:19,190 In other words, let me draw this little diagram to bring 669 00:31:19,190 --> 00:31:21,560 in the infinitesimal idea here. 670 00:31:21,560 --> 00:31:23,230 Here's my 'delta w'. 671 00:31:23,230 --> 00:31:24,890 Here's my 'delta s'. 672 00:31:24,890 --> 00:31:28,320 And what I'm doing now is I am going to take the tangent line 673 00:31:28,320 --> 00:31:33,400 to the curve at 'A', use that rather than 'delta x'. 674 00:31:33,400 --> 00:31:35,450 In other words, what I'm going to say is we're going to 675 00:31:35,450 --> 00:31:38,630 assume that our curve doesn't have infinite oscillations. 676 00:31:38,630 --> 00:31:42,200 So I can assume the special case of a monotonically 677 00:31:42,200 --> 00:31:46,350 increasing function, use the intuitive approach that in 678 00:31:46,350 --> 00:31:51,450 this diagram, 'delta w' is caught between 'delta s' and 679 00:31:51,450 --> 00:31:57,470 'AB' plus 'BC', observing that 'BC' is just what's called 680 00:31:57,470 --> 00:32:02,220 'delta y' minus 'delta y-tan'. 681 00:32:02,220 --> 00:32:06,110 And that, by the Pythagorean theorem, 'AB' is the square 682 00:32:06,110 --> 00:32:09,650 root of ''delta x' squared' plus ''delta 'y sub tan'' 683 00:32:09,650 --> 00:32:12,850 squared', which, of course, can be written this particular 684 00:32:12,850 --> 00:32:16,230 way, namely, notice that the slope here is the slope of 685 00:32:16,230 --> 00:32:21,560 this curve when 'x' is equal to 'x sub 'k - 1''. 686 00:32:21,560 --> 00:32:23,900 And again, this is written out, so I think you can fill 687 00:32:23,900 --> 00:32:27,490 in the details as part of your review of the lecture and your 688 00:32:27,490 --> 00:32:28,420 homework assignment. 689 00:32:28,420 --> 00:32:31,220 All I want to do here is present a plausibility 690 00:32:31,220 --> 00:32:36,170 argument using 'AB', 'AC', and 'delta s' as they occur in 691 00:32:36,170 --> 00:32:37,280 this diagram. 692 00:32:37,280 --> 00:32:40,370 All we're saying is, look, if we're willing to make the 693 00:32:40,370 --> 00:32:45,000 assumption that this curve has the right shape, 'delta w' is 694 00:32:45,000 --> 00:32:50,650 squeezed between 'delta s' and 'AB' plus 'BC'. 695 00:32:50,650 --> 00:32:54,140 As we showed on our little inset here, 'AB' is the square 696 00:32:54,140 --> 00:32:58,520 root of '1 + ''f prime' evaluated 'x sub 'k - 1'' 697 00:32:58,520 --> 00:33:00,600 squared' times 'delta x'. 698 00:33:00,600 --> 00:33:02,520 What is 'BC'? 699 00:33:02,520 --> 00:33:06,350 Remember, 'BC' was 'delta y' minus 'delta y-tan' . 700 00:33:06,350 --> 00:33:09,410 That's just your epsilon 'delta x' of your 701 00:33:09,410 --> 00:33:13,510 infinitesimal idea, where the limit of epsilon as 'delta x' 702 00:33:13,510 --> 00:33:15,760 approaches 0 is 0. 703 00:33:15,760 --> 00:33:17,860 In fact, let me just come over here and make sure we write 704 00:33:17,860 --> 00:33:19,080 that part again. 705 00:33:19,080 --> 00:33:24,130 Remember what we saw was that 'delta y-tan' was 'dy/dx' 706 00:33:24,130 --> 00:33:30,320 evaluated at the point in question plus what? 707 00:33:30,320 --> 00:33:33,940 An error term which was called epsilon 'delta x', where 708 00:33:33,940 --> 00:33:37,160 epsilon went to 0 as 'delta x' went to 0. 709 00:33:37,160 --> 00:33:39,520 And that's all I'm saying over here. 710 00:33:39,520 --> 00:33:44,420 In other words, where is delta s squeezed between right now? 711 00:33:44,420 --> 00:33:47,840 Well, let me put it this way, delta s itself, by definition, 712 00:33:47,840 --> 00:33:49,470 is the square root of ''delta x' squared' 713 00:33:49,470 --> 00:33:51,030 plus ''delta y' squared'. 714 00:33:51,030 --> 00:33:53,010 That we saw was this. 715 00:33:53,010 --> 00:33:55,770 That was our beginning definition in fact. 716 00:33:55,770 --> 00:33:59,600 Now if you look at our diagram once more, notice that since 717 00:33:59,600 --> 00:34:04,980 our curve is always holding water and rising, that the 718 00:34:04,980 --> 00:34:08,760 slope of the line 'delta s' is greater than the 719 00:34:08,760 --> 00:34:11,420 slope of the line 'AB'. 720 00:34:11,420 --> 00:34:13,969 Putting all of this together, we now 721 00:34:13,969 --> 00:34:16,050 have 'delta w' squeezed. 722 00:34:16,050 --> 00:34:18,440 And it was not at all trivial in putting the 723 00:34:18,440 --> 00:34:20,370 squeeze on 'delta w'. 724 00:34:20,370 --> 00:34:23,650 There was no self-evident way of saying just because one 725 00:34:23,650 --> 00:34:26,520 region was contained in another, it must have a 726 00:34:26,520 --> 00:34:28,159 smaller arc length. 727 00:34:28,159 --> 00:34:31,730 We really had to be ingenious in how we put the squeeze in 728 00:34:31,730 --> 00:34:33,110 to catch this thing. 729 00:34:33,110 --> 00:34:36,230 But in the long run, what we now have shown is what? 730 00:34:36,230 --> 00:34:40,300 That 'delta w' is equal to this. 731 00:34:40,300 --> 00:34:43,730 With an error of no greater than epsilon 'delta x'. 732 00:34:43,730 --> 00:34:46,290 In other words, the exact delta w is what? 733 00:34:46,290 --> 00:34:50,710 It's the square root of ''1 + 'f prime 'x sub 'k - 1'' 734 00:34:50,710 --> 00:34:55,429 squared' 'delta x' plus 'alpha delta x', where alpha can be 735 00:34:55,429 --> 00:34:56,489 no bigger than epsilon. 736 00:34:56,489 --> 00:34:58,450 In other words, this is the maximum error that we have 737 00:34:58,450 --> 00:35:00,050 here because it's caught between this. 738 00:35:00,050 --> 00:35:06,010 Well, look, as 'delta x' approaches 0, so does epsilon. 739 00:35:06,010 --> 00:35:09,570 And since alpha is no bigger than epsilon, it must be that 740 00:35:09,570 --> 00:35:13,520 as 'delta x' approaches 0, so does alpha approach 0. 741 00:35:13,520 --> 00:35:17,290 In other words, if we now write 'delta w' in this form, 742 00:35:17,290 --> 00:35:20,520 observe that, in line with what we're saying, this is a 743 00:35:20,520 --> 00:35:22,800 higher order infinitesimal. 744 00:35:22,800 --> 00:35:27,040 And as a result, the intuitive approach can be used as the 745 00:35:27,040 --> 00:35:28,710 correct answer. 746 00:35:28,710 --> 00:35:32,770 The idea is we could have said earlier, look, why don't we 747 00:35:32,770 --> 00:35:36,390 approximate the arc length by the straight line segment that 748 00:35:36,390 --> 00:35:38,490 joins the two endpoints of the arc. 749 00:35:38,490 --> 00:35:40,370 And the answer is you can do that. 750 00:35:40,370 --> 00:35:44,310 But you are really on shaky grounds if you say it's 751 00:35:44,310 --> 00:35:46,200 self-evident that all the error is 752 00:35:46,200 --> 00:35:48,020 squeezed out in the limit. 753 00:35:48,020 --> 00:35:49,890 This is a very, very touchy thing. 754 00:35:49,890 --> 00:35:54,280 In other words, in the same way that 0/0 is a very, very 755 00:35:54,280 --> 00:35:57,020 sensitive thing in the study of differential calculus, 756 00:35:57,020 --> 00:35:59,350 infinity times 0 is equally as 757 00:35:59,350 --> 00:36:02,110 sensitive in integral calculus. 758 00:36:02,110 --> 00:36:04,780 The whole upshot of today's lecture, however, is now that 759 00:36:04,780 --> 00:36:09,060 we've gone through this whole, hard approach, it turns out 760 00:36:09,060 --> 00:36:12,250 that we can justify our intuitive approach of 761 00:36:12,250 --> 00:36:15,700 approximating the arc length by straight line segments. 762 00:36:15,700 --> 00:36:18,820 At any rate, this concludes our lesson for today. 763 00:36:18,820 --> 00:36:20,620 And until next time, good-bye. 764 00:36:23,680 --> 00:36:26,880 Funding for the publication of this video was provided by the 765 00:36:26,880 --> 00:36:30,930 Gabriella and Paul Rosenbaum Foundation. 766 00:36:30,930 --> 00:36:35,110 Help OCW continue to provide free and open access to MIT 767 00:36:35,110 --> 00:36:39,310 courses by making a donation at ocw.mit.edu/donate.