1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,620 Commons license. 3 00:00:03,620 --> 00:00:05,770 Your support will help MIT OpenCourseWare 4 00:00:05,770 --> 00:00:09,900 continue to offer high quality educational resources for free. 5 00:00:09,900 --> 00:00:12,550 To make a donation or to view additional materials 6 00:00:12,550 --> 00:00:15,850 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,850 --> 00:00:22,190 at ocw.mit.edu. 8 00:00:22,190 --> 00:00:24,910 PROFESSOR: So we're going on to the third unit here. 9 00:00:24,910 --> 00:00:31,030 So we're getting started with Unit 3. 10 00:00:31,030 --> 00:00:38,330 And this is our intro to integration. 11 00:00:38,330 --> 00:00:44,820 It's basically the second half of calculus 12 00:00:44,820 --> 00:00:51,320 after differentiation. 13 00:00:51,320 --> 00:00:55,600 Today what I'll talk about is what are 14 00:00:55,600 --> 00:01:16,450 known as definite integrals. 15 00:01:16,450 --> 00:01:19,380 Actually, it looks like, are we missing 16 00:01:19,380 --> 00:01:21,580 a bunch of overhead lights? 17 00:01:21,580 --> 00:01:24,260 Is there a reason for that? 18 00:01:24,260 --> 00:01:27,320 Hmm. 19 00:01:27,320 --> 00:01:30,460 Let's see. 20 00:01:30,460 --> 00:01:32,650 Aah. 21 00:01:32,650 --> 00:01:34,360 All right. 22 00:01:34,360 --> 00:01:39,200 OK, that's a little brighter now. 23 00:01:39,200 --> 00:01:39,700 All right. 24 00:01:39,700 --> 00:01:47,420 So the idea of definite integrals 25 00:01:47,420 --> 00:01:49,740 can be presented in a number of ways. 26 00:01:49,740 --> 00:01:53,707 But I will be consistent with the rest of the presentation 27 00:01:53,707 --> 00:01:54,290 in the course. 28 00:01:54,290 --> 00:01:57,060 We're going to start with the geometric point of view. 29 00:01:57,060 --> 00:01:59,990 And the geometric point of view is, 30 00:01:59,990 --> 00:02:13,680 the problem we want to solve is to find the area under a curve. 31 00:02:13,680 --> 00:02:16,460 The other point of view that one can take, 32 00:02:16,460 --> 00:02:18,590 and we'll mention that at the end of this lecture, 33 00:02:18,590 --> 00:02:26,220 is the idea of a cumulative sum. 34 00:02:26,220 --> 00:02:30,687 So keep that in mind that there's a lot going on here. 35 00:02:30,687 --> 00:02:32,520 And there are many different interpretations 36 00:02:32,520 --> 00:02:37,930 of what the integral is. 37 00:02:37,930 --> 00:02:41,710 Now, so let's draw a picture here. 38 00:02:41,710 --> 00:02:44,670 I'll start at a place a and end at a place b. 39 00:02:44,670 --> 00:02:46,720 And I have some curve here. 40 00:02:46,720 --> 00:02:55,100 And what I have in mind is to find this area here. 41 00:02:55,100 --> 00:02:56,600 And, of course, in order to do that, 42 00:02:56,600 --> 00:02:58,985 I need more information than just where we start 43 00:02:58,985 --> 00:02:59,760 and where we end. 44 00:02:59,760 --> 00:03:01,860 I also need the bottom and the top. 45 00:03:01,860 --> 00:03:05,170 By convention, the bottom is the x axis and the top 46 00:03:05,170 --> 00:03:11,670 is the curve that we've specified, which is y = f(x). 47 00:03:11,670 --> 00:03:15,660 And we have a notation for this, which 48 00:03:15,660 --> 00:03:18,810 is the notation using calculus for this as opposed 49 00:03:18,810 --> 00:03:20,490 to some geometric notation. 50 00:03:20,490 --> 00:03:24,180 And that's the following expression. 51 00:03:24,180 --> 00:03:26,560 It's called an integral, but now it's 52 00:03:26,560 --> 00:03:29,290 going to have what are known as limits on it. 53 00:03:29,290 --> 00:03:31,970 It will start at a and end at b. 54 00:03:31,970 --> 00:03:35,720 And we write in the function f(x) dx. 55 00:03:35,720 --> 00:03:40,830 So this is what's known as a definite integral. 56 00:03:40,830 --> 00:03:43,620 And it's interpreted geometrically 57 00:03:43,620 --> 00:03:46,460 as the area under the curve. 58 00:03:46,460 --> 00:03:49,260 The only difference between this collection of symbols 59 00:03:49,260 --> 00:03:51,690 and what we had before with indefinite integrals 60 00:03:51,690 --> 00:03:54,960 is that before we didn't specify where it started 61 00:03:54,960 --> 00:04:03,140 and where it ended. 62 00:04:03,140 --> 00:04:08,670 Now, in order to understand what to do with this guy, 63 00:04:08,670 --> 00:04:12,330 I'm going to just describe very abstractly what we do. 64 00:04:12,330 --> 00:04:17,050 And then carry out one example in detail. 65 00:04:17,050 --> 00:04:24,310 So, to compute this area, we're going 66 00:04:24,310 --> 00:04:27,270 to follow initially three steps. 67 00:04:27,270 --> 00:04:36,870 First of all, we're going to divide into rectangles. 68 00:04:36,870 --> 00:04:42,080 And unfortunately, because it's impossible to divide 69 00:04:42,080 --> 00:04:45,510 a curvy region into rectangles, we're going to cheat. 70 00:04:45,510 --> 00:04:49,310 So they're only quote-unquote rectangles. 71 00:04:49,310 --> 00:04:52,720 They're almost rectangles. 72 00:04:52,720 --> 00:05:01,060 And the second thing we're going to do is to add up the areas. 73 00:05:01,060 --> 00:05:04,430 And the third thing we're going to do 74 00:05:04,430 --> 00:05:10,970 is to rectify this problem that we didn't actually 75 00:05:10,970 --> 00:05:12,870 hit the answer on the nose. 76 00:05:12,870 --> 00:05:15,310 That we were missing some pieces or were 77 00:05:15,310 --> 00:05:17,460 choosing some extra bits. 78 00:05:17,460 --> 00:05:19,570 And the way we'll rectify that is 79 00:05:19,570 --> 00:05:35,070 by taking the limit as the rectangles get thin. 80 00:05:35,070 --> 00:05:39,380 Infinitesimally thin, very thin. 81 00:05:39,380 --> 00:05:43,780 Pictorially, again, that looks like this. 82 00:05:43,780 --> 00:05:46,950 We have a and our b, and we have our guy here, 83 00:05:46,950 --> 00:05:48,390 this is our curve. 84 00:05:48,390 --> 00:05:51,850 And I'm going to chop it up. 85 00:05:51,850 --> 00:05:57,470 First I'm going to chop up the x axis into little increments. 86 00:05:57,470 --> 00:06:00,460 And then I'm going to chop things up here. 87 00:06:00,460 --> 00:06:03,060 And I'll decide on some rectangle, maybe 88 00:06:03,060 --> 00:06:05,450 some staircase pattern here. 89 00:06:05,450 --> 00:06:12,960 Like this. 90 00:06:12,960 --> 00:06:16,100 Now, I don't care so much. 91 00:06:16,100 --> 00:06:19,140 In some cases the rectangles overshoot; in some cases 92 00:06:19,140 --> 00:06:20,400 they're underneath. 93 00:06:20,400 --> 00:06:23,720 So the new area that I'm adding up is off. 94 00:06:23,720 --> 00:06:28,260 It's not quite the same as the area under the curve. 95 00:06:28,260 --> 00:06:32,720 It's this region here. 96 00:06:32,720 --> 00:06:38,110 But it includes these extra bits here. 97 00:06:38,110 --> 00:06:42,250 And then it's missing this little guy here. 98 00:06:42,250 --> 00:06:47,950 This little bit there is missing. 99 00:06:47,950 --> 00:06:51,310 And, as I say, these little pieces up here, 100 00:06:51,310 --> 00:06:55,740 this a little bit up here is extra. 101 00:06:55,740 --> 00:06:58,406 So that's why we're not really dividing up 102 00:06:58,406 --> 00:06:59,530 the region into rectangles. 103 00:06:59,530 --> 00:07:01,300 We're just taking rectangles. 104 00:07:01,300 --> 00:07:05,060 And then the idea is that as these get thinner and thinner, 105 00:07:05,060 --> 00:07:09,080 the little itty-bitty amounts that we miss by are going to 0. 106 00:07:09,080 --> 00:07:10,830 And they're going to be negligible. 107 00:07:10,830 --> 00:07:13,840 Already, you can see it's kind of a thin piece of area, 108 00:07:13,840 --> 00:07:15,840 so we're not missing by much. 109 00:07:15,840 --> 00:07:19,150 And as these get thinner and thinner, the problem goes away 110 00:07:19,150 --> 00:07:27,040 and we get the answer on the nose in the limit. 111 00:07:27,040 --> 00:07:35,050 So here's our first example. 112 00:07:35,050 --> 00:07:41,270 I'll take the first interesting curve, which is f(x) = x^2. 113 00:07:41,270 --> 00:07:45,150 I don't want to do anything more complicated than one example, 114 00:07:45,150 --> 00:07:48,030 because this is a real labor here, 115 00:07:48,030 --> 00:07:50,910 what we're going to go through. 116 00:07:50,910 --> 00:07:52,710 And to make things easier for myself, 117 00:07:52,710 --> 00:07:55,930 I'm going to start at a = 0. 118 00:07:55,930 --> 00:07:58,400 But in order to see what the pattern is, 119 00:07:58,400 --> 00:08:11,180 I'm going to allow b to be arbitrary. 120 00:08:11,180 --> 00:08:15,230 Let's draw the graph and start breaking things up. 121 00:08:15,230 --> 00:08:18,015 So here's the parabola, and there's 122 00:08:18,015 --> 00:08:19,640 this piece that we want, which is going 123 00:08:19,640 --> 00:08:24,170 to stop at this place, b, here. 124 00:08:24,170 --> 00:08:37,380 And the first step is to divide into n pieces. 125 00:08:37,380 --> 00:08:40,840 That means, well, graphically, I'll just mark the first three. 126 00:08:40,840 --> 00:08:44,450 And maybe there are going to be many of them. 127 00:08:44,450 --> 00:08:47,480 And then I'll draw some rectangles here, 128 00:08:47,480 --> 00:08:51,420 and I'm going to choose to make the rectangles all 129 00:08:51,420 --> 00:08:53,310 the way from the right. 130 00:08:53,310 --> 00:08:55,545 That is, I'll make us this staircase pattern 131 00:08:55,545 --> 00:08:58,760 here, like this. 132 00:08:58,760 --> 00:09:00,160 That's my choice. 133 00:09:00,160 --> 00:09:02,360 I get to choose whatever level I want, 134 00:09:02,360 --> 00:09:04,500 and I'm going to choose the right ends 135 00:09:04,500 --> 00:09:07,490 as the shape of the staircase. 136 00:09:07,490 --> 00:09:17,190 So I'm overshooting with each rectangle. 137 00:09:17,190 --> 00:09:19,910 And now I have to write down formulas 138 00:09:19,910 --> 00:09:23,800 for what these areas are. 139 00:09:23,800 --> 00:09:26,840 Now, there's one big advantage that rectangles have. 140 00:09:26,840 --> 00:09:28,890 And this is the starting place. 141 00:09:28,890 --> 00:09:33,130 Which is that it's easy to find their areas. 142 00:09:33,130 --> 00:09:35,260 All you need to know is the base and the height, 143 00:09:35,260 --> 00:09:37,230 and you multiply, and you get the area. 144 00:09:37,230 --> 00:09:40,820 That's the reason why we can get started with rectangles. 145 00:09:40,820 --> 00:09:43,750 And in this case, these distances, 146 00:09:43,750 --> 00:09:46,350 I'm assuming that they're all equal, equally 147 00:09:46,350 --> 00:09:48,350 spaced, intervals. 148 00:09:48,350 --> 00:09:50,260 And I'll always be doing that. 149 00:09:50,260 --> 00:10:01,360 And so the spacing, the bases, the base length, is always b/n. 150 00:10:01,360 --> 00:10:09,590 All equal intervals. 151 00:10:09,590 --> 00:10:11,260 So that's the base length. 152 00:10:11,260 --> 00:10:15,370 And next, I need the heights. 153 00:10:15,370 --> 00:10:17,400 And in order to keep track of the heights, 154 00:10:17,400 --> 00:10:21,310 I'm going to draw a little table here, with x and f(x), 155 00:10:21,310 --> 00:10:27,460 and plug in a few values just to see what the pattern is. 156 00:10:27,460 --> 00:10:34,100 The first place here, after 0, is b/n. 157 00:10:34,100 --> 00:10:36,810 So here's b/n, that's an x-value. 158 00:10:36,810 --> 00:10:40,590 And the f(x) value is the height there. 159 00:10:40,590 --> 00:10:44,980 And that's just, I evaluate f(x), f(x) is x^2. 160 00:10:44,980 --> 00:10:49,450 And that's (b/n)^2. 161 00:10:49,450 --> 00:10:53,350 And similarly, the next one is 2b/n. 162 00:10:56,230 --> 00:10:58,560 And the value here is (2b/n)^2. 163 00:11:01,530 --> 00:11:02,950 That's this. 164 00:11:02,950 --> 00:11:07,600 This height here is 2b/n. 165 00:11:07,600 --> 00:11:14,070 That's the second rectangle. 166 00:11:14,070 --> 00:11:16,960 And I'll write down one more. 167 00:11:16,960 --> 00:11:18,600 3b/n, that's the third one. 168 00:11:18,600 --> 00:11:20,090 And the height is (3b/n)^2. 169 00:11:23,250 --> 00:11:29,940 And so forth. 170 00:11:29,940 --> 00:11:34,537 Well, my next job is to add up these areas. 171 00:11:34,537 --> 00:11:36,620 And I've already prepared that by finding out what 172 00:11:36,620 --> 00:11:39,020 the base and the height is. 173 00:11:39,020 --> 00:11:50,030 So the total area, or the sum of the areas, let's say, 174 00:11:50,030 --> 00:12:01,240 of these rectangles, is-- Well, the first one is (b/n) (b/n)^2. 175 00:12:01,240 --> 00:12:06,380 The second one is 2b/n -- I'm sorry, is (b/n) (2b/n)^2. 176 00:12:08,920 --> 00:12:11,070 And it just keeps on going. 177 00:12:11,070 --> 00:12:13,540 And the last one is (b/n) (nb / n)^2. 178 00:12:17,810 --> 00:12:19,870 So it's very important to figure out 179 00:12:19,870 --> 00:12:22,720 what the general formula is. 180 00:12:22,720 --> 00:12:25,000 And here we have a base. 181 00:12:25,000 --> 00:12:27,170 And here we have a height, and here we 182 00:12:27,170 --> 00:12:31,930 have the same kind of base, but we have a new height. 183 00:12:31,930 --> 00:12:32,880 And so forth. 184 00:12:32,880 --> 00:12:36,390 And the pattern is that the coefficient here is 1, then 2, 185 00:12:36,390 --> 00:12:42,976 then 3, all the way up to n. 186 00:12:42,976 --> 00:12:44,850 The rectangles are getting taller and taller, 187 00:12:44,850 --> 00:12:50,940 and this one, the last one is the biggest. 188 00:12:50,940 --> 00:12:55,300 OK, this is a very complicated gadget. 189 00:12:55,300 --> 00:12:58,800 and the first thing I want to do is simplify it 190 00:12:58,800 --> 00:13:00,600 and then I'm actually going to evaluate it. 191 00:13:00,600 --> 00:13:03,129 But actually I'm not going to evaluate it exactly. 192 00:13:03,129 --> 00:13:04,670 I'm just going to evaluate the limit. 193 00:13:04,670 --> 00:13:07,060 Turns out, limits are always easier. 194 00:13:07,060 --> 00:13:10,960 The point about calculus here is that these rectangles are hard. 195 00:13:10,960 --> 00:13:13,330 But the limiting value is an easy value. 196 00:13:13,330 --> 00:13:16,440 So what we're heading for is the simple formula, as opposed 197 00:13:16,440 --> 00:13:19,550 to the complicated one. 198 00:13:19,550 --> 00:13:22,190 Alright, so the first thing I'm going to do 199 00:13:22,190 --> 00:13:25,130 is factor out all these b/n factors. 200 00:13:25,130 --> 00:13:27,970 There's a b/n, here, and there's a (b/n)^2, So all told, 201 00:13:27,970 --> 00:13:31,390 we have a (b/n)^3. 202 00:13:31,390 --> 00:13:33,730 As a common factor. 203 00:13:33,730 --> 00:13:36,830 And then the first term is 1, and the second term, 204 00:13:36,830 --> 00:13:39,310 what's left over, is 2^2. 205 00:13:39,310 --> 00:13:41,100 2^2. 206 00:13:41,100 --> 00:13:43,670 And then the third term would be 3^2, 207 00:13:43,670 --> 00:13:46,970 although I haven't written it. 208 00:13:46,970 --> 00:13:51,200 In the last term, there's an extra factor of n^2. 209 00:13:51,200 --> 00:14:05,050 In the numerator. 210 00:14:05,050 --> 00:14:09,820 OK, is everybody with me here? 211 00:14:09,820 --> 00:14:23,930 Now, what I'd like to do is to eventually take the limit 212 00:14:23,930 --> 00:14:26,520 as n goes to infinity here. 213 00:14:26,520 --> 00:14:29,010 And the quantity that's hard to understand 214 00:14:29,010 --> 00:14:33,540 is this massive quantity here. 215 00:14:33,540 --> 00:14:36,690 And there's one change that I'd like to make, 216 00:14:36,690 --> 00:14:40,160 but it's a very modest one. 217 00:14:40,160 --> 00:14:41,410 Extremely minuscule. 218 00:14:41,410 --> 00:14:43,350 Which is that I'm going to write 1, 219 00:14:43,350 --> 00:14:45,450 just to see that there's a general pattern here. 220 00:14:45,450 --> 00:14:46,500 Going to write 1 as 1^2. 221 00:14:52,830 --> 00:14:59,420 And let's put in the 3 here, why not. 222 00:14:59,420 --> 00:15:05,400 And now I want to use a trick. 223 00:15:05,400 --> 00:15:08,560 This trick is not completely recommended, 224 00:15:08,560 --> 00:15:12,380 but I will say a lot more about that 225 00:15:12,380 --> 00:15:13,780 when we get through to the end. 226 00:15:13,780 --> 00:15:16,110 I want to understand how big this quantity is. 227 00:15:16,110 --> 00:15:18,795 So I'm going to use a geometric trick to draw 228 00:15:18,795 --> 00:15:20,880 a picture of this quantity. 229 00:15:20,880 --> 00:15:23,660 Namely, I'm going to build a pyramid. 230 00:15:23,660 --> 00:15:29,510 And the base of the pyramid is going to be n by n blocks. 231 00:15:29,510 --> 00:15:32,410 So imagine we're in Egypt and we're building a pyramid. 232 00:15:32,410 --> 00:15:39,190 And the next layer is going to be n-1 by n-1. 233 00:15:39,190 --> 00:15:43,250 So this next layer in is n-1 by n-1. 234 00:15:43,250 --> 00:15:46,590 So the total number of blocks on the bottom is n squared. 235 00:15:46,590 --> 00:15:50,020 That's this rightmost term here. 236 00:15:50,020 --> 00:15:52,460 But the next term, which I didn't write in but maybe I 237 00:15:52,460 --> 00:15:57,280 should, the next-to-the-last term was this one. 238 00:15:57,280 --> 00:16:00,360 And that's the second layer that I've put on. 239 00:16:00,360 --> 00:16:05,500 Now, this is, if you like, the top view. 240 00:16:05,500 --> 00:16:08,950 But perhaps we should also think in terms of a side view. 241 00:16:08,950 --> 00:16:12,920 So here's the same picture, we're starting at n 242 00:16:12,920 --> 00:16:15,920 and we build up this layer here. 243 00:16:15,920 --> 00:16:18,507 And now we're going to put a layer on top of it, which 244 00:16:18,507 --> 00:16:19,340 is a little shorter. 245 00:16:19,340 --> 00:16:21,600 So the first layer is of length n. 246 00:16:21,600 --> 00:16:25,460 And the second layers is of length n-1, and then on top 247 00:16:25,460 --> 00:16:28,007 of that we have something of length n-2, and so forth. 248 00:16:28,007 --> 00:16:29,340 And we're going to pile them up. 249 00:16:29,340 --> 00:16:31,800 So we pile them up. 250 00:16:31,800 --> 00:16:37,630 All the way to the top, which is just one giant block of stone. 251 00:16:37,630 --> 00:16:39,490 And that's this last one, 1^2. 252 00:16:39,490 --> 00:16:43,290 So we're going backwards in the sum. 253 00:16:43,290 --> 00:16:46,140 And so I have to build this whole thing up. 254 00:16:46,140 --> 00:16:48,350 And I get all the way up in this staircase pattern 255 00:16:48,350 --> 00:16:57,720 to this top block, up there. 256 00:16:57,720 --> 00:17:00,140 So here's the trick that you can use 257 00:17:00,140 --> 00:17:02,870 to estimate the size of this, and it's 258 00:17:02,870 --> 00:17:06,410 sufficient in the limit as n goes to infinity. 259 00:17:06,410 --> 00:17:14,560 The trick is that I can imagine the solid thing 260 00:17:14,560 --> 00:17:19,560 underneath the staircase, like this. 261 00:17:19,560 --> 00:17:24,390 That's an ordinary pyramid, not a staircase pyramid. 262 00:17:24,390 --> 00:17:26,550 Which is inside. 263 00:17:26,550 --> 00:17:28,940 And this one is inside. 264 00:17:28,940 --> 00:17:32,150 And so, but it's an ordinary pyramid as opposed 265 00:17:32,150 --> 00:17:34,530 to a staircase pyramid. 266 00:17:34,530 --> 00:17:37,690 And so, we know the formula for the volume of that. 267 00:17:37,690 --> 00:17:40,580 Because we know the formula for volumes of cones. 268 00:17:40,580 --> 00:17:50,090 And the formula for the volume of this guy, of the inside, 269 00:17:50,090 --> 00:17:58,150 is 1/3 base times height. 270 00:17:58,150 --> 00:18:03,500 And in that case, the base here-- so that's 1/3, 271 00:18:03,500 --> 00:18:06,180 and the base is n by n, right? 272 00:18:06,180 --> 00:18:08,540 So the base is n^2. 273 00:18:08,540 --> 00:18:10,050 That's the base. 274 00:18:10,050 --> 00:18:13,340 And the height, it goes all the way to the top point. 275 00:18:13,340 --> 00:18:21,290 So the height is n. 276 00:18:21,290 --> 00:18:27,130 And what we've discovered here is that this whole sum is 277 00:18:27,130 --> 00:18:30,910 bigger than 1/3 n^3. 278 00:18:42,680 --> 00:18:46,050 Now, I claimed that - this line, by the way has slope 2. 279 00:18:46,050 --> 00:18:50,140 So you go 1/2 over each time you go up 1. 280 00:18:50,140 --> 00:18:52,960 And that's why you get to the top. 281 00:18:52,960 --> 00:18:56,950 On the other hand, I can trap it on the outside, 282 00:18:56,950 --> 00:19:01,840 too, by drawing a parallel line out here. 283 00:19:01,840 --> 00:19:07,100 And this will go down 1/2 more on this side and 1/2 more 284 00:19:07,100 --> 00:19:08,230 on the other side. 285 00:19:08,230 --> 00:19:14,870 So the base will be n+1 by n+1 of this bigger pyramid. 286 00:19:14,870 --> 00:19:18,120 And it'll go up 1 higher. 287 00:19:18,120 --> 00:19:22,790 So on the other end, we get that this is less than 1/3 (n+1)^3. 288 00:19:25,690 --> 00:19:34,290 Again, (n+1)^2 times n+1, again, this is a base times a height. 289 00:19:34,290 --> 00:19:36,920 Of this bigger pyramid. 290 00:19:36,920 --> 00:19:38,110 Yes, question. 291 00:19:38,110 --> 00:19:47,860 STUDENT: [INAUDIBLE] and then equating it to volume. 292 00:19:47,860 --> 00:19:49,360 PROFESSOR: The question is, it seems 293 00:19:49,360 --> 00:19:54,360 as if I'm adding up areas and equating it to volume. 294 00:19:54,360 --> 00:19:57,160 But I'm actually creating volumes 295 00:19:57,160 --> 00:20:00,060 by making these honest increments here. 296 00:20:00,060 --> 00:20:07,750 That is, the base is n but the height is 1. 297 00:20:07,750 --> 00:20:09,380 Thank you for pointing that out. 298 00:20:09,380 --> 00:20:11,220 Each one of these little staircases 299 00:20:11,220 --> 00:20:14,150 here has exactly height 1. 300 00:20:14,150 --> 00:20:15,950 So I'm honestly sticking blocks there. 301 00:20:15,950 --> 00:20:18,200 They're sort of square blocks, and I'm lining them up. 302 00:20:18,200 --> 00:20:21,450 And I'm thinking of n by n cubes, if you like. 303 00:20:21,450 --> 00:20:22,960 Honest cubes, there. 304 00:20:22,960 --> 00:20:25,447 And the height is 1. 305 00:20:25,447 --> 00:20:26,280 And the base is n^2. 306 00:20:33,360 --> 00:20:36,760 Alright, so I claim that I've trapped this guy in between two 307 00:20:36,760 --> 00:20:38,520 quantities here. 308 00:20:38,520 --> 00:20:52,510 And now I'm ready to take the limit. 309 00:20:52,510 --> 00:20:55,120 If you look at what our goal is, we 310 00:20:55,120 --> 00:20:57,060 want to have an expression like this. 311 00:20:57,060 --> 00:21:00,840 And I'm going to-- This was the massive expression that we had. 312 00:21:00,840 --> 00:21:03,310 And actually, I'm going to write it differently. 313 00:21:03,310 --> 00:21:10,230 I'll write it as b^3 times 1^2 plus 2^2 plus... plus n^2, 314 00:21:10,230 --> 00:21:12,070 divided by n^3. 315 00:21:12,070 --> 00:21:15,520 I'm going to combine all the n's together. 316 00:21:15,520 --> 00:21:17,420 Alright, so the right thing to do 317 00:21:17,420 --> 00:21:20,480 is to divide what I had up there. 318 00:21:20,480 --> 00:21:28,280 Divide by n^3 in this set of inequalities there. 319 00:21:28,280 --> 00:21:35,930 And what I get here is 1/3 is less than 1 plus 2^2 plus 3^2 320 00:21:35,930 --> 00:21:41,740 plus... plus n^2 divided by n^3 is less than 1/3 (n+1)^3 / n^3. 321 00:21:45,490 --> 00:21:49,230 And that's 1/3 + (1 + 1/n)^3. 322 00:21:56,230 --> 00:21:59,530 And now, I claim we're done. 323 00:21:59,530 --> 00:22:03,170 Because this is 1/3, and the limit, 324 00:22:03,170 --> 00:22:06,450 as n goes to infinity, of this quantity here, 325 00:22:06,450 --> 00:22:09,350 is easily seen to be, this is, as n goes to infinity, 326 00:22:09,350 --> 00:22:10,470 this goes to 0. 327 00:22:10,470 --> 00:22:14,790 So this also goes to 1/3. 328 00:22:14,790 --> 00:22:28,810 And so our total here, so our total area, under x^2, 329 00:22:28,810 --> 00:22:33,040 which we sometimes might write the integral from 0 to b x^2 330 00:22:33,040 --> 00:22:38,210 dx, is going to be equal to - well, 331 00:22:38,210 --> 00:22:40,350 it's this 1/3 that I've got. 332 00:22:40,350 --> 00:22:43,400 But then there was also a b^3 there. 333 00:22:43,400 --> 00:22:45,040 So there's this extra b cubed here. 334 00:22:45,040 --> 00:22:49,240 So it's 1/3 b^3. 335 00:22:49,240 --> 00:22:54,630 That's the result from this whole computation. 336 00:22:54,630 --> 00:22:55,620 Yes, question. 337 00:22:55,620 --> 00:22:57,060 STUDENT: [INAUDIBLE] 338 00:22:57,060 --> 00:23:05,370 PROFESSOR: So that was a very good question. 339 00:23:05,370 --> 00:23:08,960 The question is, why did we leave the b/n^3 out, 340 00:23:08,960 --> 00:23:11,190 for this step. 341 00:23:11,190 --> 00:23:16,290 And a part of the answer is malice aforethought. 342 00:23:16,290 --> 00:23:19,290 In other words, we know what we're heading for. 343 00:23:19,290 --> 00:23:21,670 We know, we understand, this quantity. 344 00:23:21,670 --> 00:23:23,570 It's all one thing. 345 00:23:23,570 --> 00:23:26,080 This thing is a sum, which is growing larger and larger. 346 00:23:26,080 --> 00:23:28,430 It's not what's called a closed form. 347 00:23:28,430 --> 00:23:31,440 So, the thing that's not known, or not well understood, 348 00:23:31,440 --> 00:23:33,320 is how big is this quantity here. 349 00:23:33,320 --> 00:23:37,050 1^2 + 2^2, the sum of the squares. 350 00:23:37,050 --> 00:23:38,620 Whereas, this is something that's 351 00:23:38,620 --> 00:23:40,740 quite easy to understand. 352 00:23:40,740 --> 00:23:42,570 So we factor it out. 353 00:23:42,570 --> 00:23:47,740 And we analyze carefully the piece which we don't know yet, 354 00:23:47,740 --> 00:23:48,920 how big it is. 355 00:23:48,920 --> 00:23:52,560 And we discovered that it's very, very similar to n^3. 356 00:23:52,560 --> 00:23:57,030 But it's more similar to 1/3 n^3. 357 00:23:57,030 --> 00:24:00,500 It's almost identical to 1/3 n^3. 358 00:24:00,500 --> 00:24:02,080 This extra piece here. 359 00:24:02,080 --> 00:24:04,120 So that's what's going on. 360 00:24:04,120 --> 00:24:05,110 And then we match that. 361 00:24:05,110 --> 00:24:09,230 Since this thing is very similar to 1/3 n^3 we cancel 362 00:24:09,230 --> 00:24:24,780 the n^3's and we have our result. 363 00:24:24,780 --> 00:24:28,230 Let me just mention that although this may seem odd, 364 00:24:28,230 --> 00:24:30,760 in fact this is what you always do if you 365 00:24:30,760 --> 00:24:32,630 analyze these kinds of sum. 366 00:24:32,630 --> 00:24:34,810 You always factor out whatever you can. 367 00:24:34,810 --> 00:24:37,250 And then you still are faced with a sum like this. 368 00:24:37,250 --> 00:24:40,100 So this will happen systematically, every time 369 00:24:40,100 --> 00:24:45,860 you're faced with such a sum. 370 00:24:45,860 --> 00:24:53,450 OK, now I want to say one more word about notation. 371 00:24:53,450 --> 00:25:00,000 Which is that this notation is an extreme nuisance here. 372 00:25:00,000 --> 00:25:04,170 And it's really sort of too large for us to deal with. 373 00:25:04,170 --> 00:25:08,440 And so, mathematicians have a shorthand for it. 374 00:25:08,440 --> 00:25:10,930 Unfortunately, when you actually do a computation, 375 00:25:10,930 --> 00:25:15,280 you're going to end up with this collection of stuff anyway. 376 00:25:15,280 --> 00:25:19,530 But I want to just show you this summation notation in order 377 00:25:19,530 --> 00:25:24,980 to compress it a little bit. 378 00:25:24,980 --> 00:25:31,600 The idea of summation notation is the following. 379 00:25:31,600 --> 00:25:35,380 So this thing tends-- The ideas are the following. 380 00:25:35,380 --> 00:25:37,920 I'll illustrate it with an example first. 381 00:25:37,920 --> 00:25:45,810 So, the general notation is the sum of a_i, i = 1 to n, 382 00:25:45,810 --> 00:25:50,510 is a_1 plus a_2 plus dot dot dot plus a_n. 383 00:25:50,510 --> 00:25:53,700 So this is the abbreviation. 384 00:25:53,700 --> 00:26:03,950 And this is a capital sigma. 385 00:26:03,950 --> 00:26:06,720 And so, this quantity here, for instance, 386 00:26:06,720 --> 00:26:15,380 is 1/n^3 times the sum i^2, i = 1 to n. 387 00:26:15,380 --> 00:26:17,350 So that's what this thing is equal to. 388 00:26:17,350 --> 00:26:20,590 And what we just showed is that that tends to 1/3 389 00:26:20,590 --> 00:26:23,910 as n goes to infinity. 390 00:26:23,910 --> 00:26:30,630 So this is the way the summation notation is used. 391 00:26:30,630 --> 00:26:34,330 There's a formula for each of these coefficients, 392 00:26:34,330 --> 00:26:37,360 each of these entries here, or summands. 393 00:26:37,360 --> 00:26:39,460 And then this is just an abbreviation 394 00:26:39,460 --> 00:26:40,710 for what the sum is. 395 00:26:40,710 --> 00:26:43,407 And this is the reason why I stuck in that 1^2 396 00:26:43,407 --> 00:26:45,740 at the beginning, so that you could see that the pattern 397 00:26:45,740 --> 00:26:47,505 worked all the way down to i = 1. 398 00:26:47,505 --> 00:26:50,600 It isn't an exception to the rule. 399 00:26:50,600 --> 00:26:54,160 It's the same as all of the others. 400 00:26:54,160 --> 00:26:59,020 Now, over here, in this board, we also 401 00:26:59,020 --> 00:27:02,680 had one of these extremely long sums. 402 00:27:02,680 --> 00:27:06,770 And this one can be written in the following way. 403 00:27:06,770 --> 00:27:10,830 And I hope you agree, this is rather hard to scan. 404 00:27:10,830 --> 00:27:16,385 But one way of writing it is, it's the sum from i = 1 to n of 405 00:27:16,385 --> 00:27:19,720 - now I have to write down the formula for the general term. 406 00:27:19,720 --> 00:27:23,130 Which is b/n (ib/n)^2. 407 00:27:29,550 --> 00:27:34,350 So that's a way of abbreviating this massive formula into one 408 00:27:34,350 --> 00:27:36,980 which is just a lot shorter. 409 00:27:36,980 --> 00:27:40,090 And now, the manipulation that I performed with it, 410 00:27:40,090 --> 00:27:43,940 which is to factor out this (b/n)^3, 411 00:27:43,940 --> 00:27:47,560 is something that I'm perfectly well allowed to do also over 412 00:27:47,560 --> 00:27:49,130 here. 413 00:27:49,130 --> 00:27:51,480 This is the distributive law. 414 00:27:51,480 --> 00:27:56,506 This, if I factor out b^3 / n^3, I'm left with the sum, 415 00:27:56,506 --> 00:28:00,880 i = 1 to n, of i^2, right? 416 00:28:00,880 --> 00:28:06,040 So these notations make it a little bit more compact. 417 00:28:06,040 --> 00:28:10,870 What we're dealing with. 418 00:28:10,870 --> 00:28:14,940 The conceptual phenomenon is still the same. 419 00:28:14,940 --> 00:28:18,160 And the mess is really still just hiding under the rug. 420 00:28:18,160 --> 00:28:23,960 But the notation is-- at least fits with fewer symbols, 421 00:28:23,960 --> 00:28:32,320 anyway. 422 00:28:32,320 --> 00:28:39,560 So let's continue here. 423 00:28:39,560 --> 00:28:41,460 I've given you one calculation. 424 00:28:41,460 --> 00:28:51,010 And now I want to fit it into a pattern. 425 00:28:51,010 --> 00:28:54,650 And here's the thing that I'd like to calculate. 426 00:28:54,650 --> 00:28:59,120 So, first of all let's try the case-- S I'm 427 00:28:59,120 --> 00:29:02,697 going to do two more examples. 428 00:29:02,697 --> 00:29:04,280 I'll do two more examples, but they're 429 00:29:04,280 --> 00:29:05,650 going to be much, much easier. 430 00:29:05,650 --> 00:29:09,160 And then things are going to get much easier from now on. 431 00:29:09,160 --> 00:29:19,580 So, the second example is going to be the function f(x) = x. 432 00:29:19,580 --> 00:29:23,520 If I draw that, that's this function here, 433 00:29:23,520 --> 00:29:26,500 that's the line with slope 1. 434 00:29:26,500 --> 00:29:29,280 And here's b. 435 00:29:29,280 --> 00:29:32,300 And so this area here is the same 436 00:29:32,300 --> 00:29:36,440 as the area of the triangle with base b and height b. 437 00:29:36,440 --> 00:29:44,360 So the area is equal to 1/2 b * b, so this is the base. 438 00:29:44,360 --> 00:29:45,660 And this is the height. 439 00:29:45,660 --> 00:29:49,970 We also know how to find the area of triangles. 440 00:29:49,970 --> 00:29:52,770 And so, the formula is 1/2 b^2. 441 00:29:57,270 --> 00:30:02,480 And the third example-- Notice, by the way, 442 00:30:02,480 --> 00:30:05,920 I didn't have to do this elaborate summing to do that, 443 00:30:05,920 --> 00:30:07,770 because we happen to know this area. 444 00:30:07,770 --> 00:30:13,370 The third example is going to be even easier. f(x) = 1. 445 00:30:13,370 --> 00:30:16,200 By far the most important example. 446 00:30:16,200 --> 00:30:20,230 Remarkably, when you get to 18.02, multivariable calculus, 447 00:30:20,230 --> 00:30:22,420 you will forget this calculation. 448 00:30:22,420 --> 00:30:23,160 Somehow. 449 00:30:23,160 --> 00:30:26,080 And I don't know why, but it happens to everybody. 450 00:30:26,080 --> 00:30:30,690 So, the function is just horizontal, like this. 451 00:30:30,690 --> 00:30:31,190 Right? 452 00:30:31,190 --> 00:30:32,610 It's the constant 1. 453 00:30:32,610 --> 00:30:37,390 And if we stop it at b, then the area we're interested in 454 00:30:37,390 --> 00:30:42,190 is just this, from 0 to b. 455 00:30:42,190 --> 00:30:47,850 And we know that this is height 1, so this is area 456 00:30:47,850 --> 00:30:51,600 is base, which is b, times 1. 457 00:30:51,600 --> 00:31:03,750 So it's b. 458 00:31:03,750 --> 00:31:13,300 Let's look now at the pattern. 459 00:31:13,300 --> 00:31:17,730 We're going to look at the pattern of the function, 460 00:31:17,730 --> 00:31:21,080 and it's the area under the curve, which 461 00:31:21,080 --> 00:31:26,330 is this-- has this elaborate formula in terms 462 00:31:26,330 --> 00:31:34,990 of-- so this is just the area under the curve. 463 00:31:34,990 --> 00:31:40,500 Between 0 and b. 464 00:31:40,500 --> 00:31:47,110 And we have x^2, which turned out to be b^3 / 3. 465 00:31:47,110 --> 00:31:49,824 And we have x, which turned out to be-- well, 466 00:31:49,824 --> 00:31:52,240 let me write them over just a bit more to give myself some 467 00:31:52,240 --> 00:31:57,200 room. x, which turns out to be b^2 / 2. 468 00:31:57,200 --> 00:32:07,090 And then we have 1, which turned out to be b. 469 00:32:07,090 --> 00:32:10,830 So this, I claim, is suggestive. 470 00:32:10,830 --> 00:32:14,750 If you can figure out the pattern, 471 00:32:14,750 --> 00:32:19,750 one way of making it a little clearer is to see that x is 472 00:32:19,750 --> 00:32:22,210 x^1. 473 00:32:22,210 --> 00:32:24,410 And 1 is x^0. 474 00:32:27,250 --> 00:32:30,070 So this is the case, 0, 1 and 2. 475 00:32:30,070 --> 00:32:32,660 And b is b^1 / 1. 476 00:32:40,010 --> 00:32:56,720 So, if you want to guess what happens when f(x) is x^3, 477 00:32:56,720 --> 00:33:01,030 well if it's 0, you do b^1 / 1; if it's 1, you do b^2 / 2; 478 00:33:01,030 --> 00:33:04,270 if it's 2, you do b^3 / 3. 479 00:33:04,270 --> 00:33:07,850 So it's reasonable to guess that this should be b^4 / 4. 480 00:33:11,110 --> 00:33:15,110 That's a reasonable guess, I would say. 481 00:33:15,110 --> 00:33:24,740 Now, the strange thing is that in history, Archimedes figured 482 00:33:24,740 --> 00:33:27,550 out the area under a parabola. 483 00:33:27,550 --> 00:33:29,600 So that was a long time ago. 484 00:33:29,600 --> 00:33:30,910 It was after the pyramids. 485 00:33:30,910 --> 00:33:34,340 And he used, actually, a much more complicated method 486 00:33:34,340 --> 00:33:36,500 than I just described here. 487 00:33:36,500 --> 00:33:40,700 And his method, which is just fantastically amazing, 488 00:33:40,700 --> 00:33:43,850 was so brilliant that it may have set back mathematics 489 00:33:43,850 --> 00:33:46,080 by 2,000 years. 490 00:33:46,080 --> 00:33:48,630 Because people were so-- it was so difficult 491 00:33:48,630 --> 00:33:51,070 that people couldn't see this pattern. 492 00:33:51,070 --> 00:33:54,312 And couldn't see that, actually, these kinds of calculations 493 00:33:54,312 --> 00:33:54,812 are easy. 494 00:33:54,812 --> 00:33:56,740 So they couldn't get to the cubic. 495 00:33:56,740 --> 00:33:58,240 And even when they got to the cubic, 496 00:33:58,240 --> 00:33:59,690 they were struggling with everything else. 497 00:33:59,690 --> 00:34:01,550 And it wasn't until calculus fit everything 498 00:34:01,550 --> 00:34:04,770 together that people were able to make serious progress 499 00:34:04,770 --> 00:34:06,640 on calculating these areas. 500 00:34:06,640 --> 00:34:09,940 Even though he was the expert on calculating areas and volumes, 501 00:34:09,940 --> 00:34:12,130 for his time. 502 00:34:12,130 --> 00:34:15,470 So this is really a great thing that we now can 503 00:34:15,470 --> 00:34:16,810 have easy methods of doing it. 504 00:34:16,810 --> 00:34:21,430 And the main thing that I want to tell you is that's we 505 00:34:21,430 --> 00:34:25,740 will not have to labor to build pyramids to calculate 506 00:34:25,740 --> 00:34:27,250 all of these quantities. 507 00:34:27,250 --> 00:34:29,620 We will have a way faster way of doing it. 508 00:34:29,620 --> 00:34:32,790 This is the slow, laborious way. 509 00:34:32,790 --> 00:34:37,080 And we will be able to do it so easily that it will happen 510 00:34:37,080 --> 00:34:39,590 as fast as you differentiate. 511 00:34:39,590 --> 00:34:42,360 So that's coming up tomorrow. 512 00:34:42,360 --> 00:34:45,920 But I want you to know that it's going to be-- However, 513 00:34:45,920 --> 00:34:52,550 we're going to go through just a little pain before we do it. 514 00:34:52,550 --> 00:34:59,400 And I'll just tell you one more piece of notation here. 515 00:34:59,400 --> 00:35:01,410 So you need to have a little practice just 516 00:35:01,410 --> 00:35:04,850 to recognize how much savings we're going to make. 517 00:35:04,850 --> 00:35:07,340 But never again will you have to face 518 00:35:07,340 --> 00:35:16,190 elaborate geometric arguments like this. 519 00:35:16,190 --> 00:35:21,110 So let me just add a little bit of notation 520 00:35:21,110 --> 00:35:27,910 for definite integrals. 521 00:35:27,910 --> 00:35:35,810 And this goes under the name of Riemann sums. 522 00:35:35,810 --> 00:35:44,140 Named after a mathematician from the 1800s. 523 00:35:44,140 --> 00:36:01,150 So this is the general procedure for definite integrals. 524 00:36:01,150 --> 00:36:04,890 We divide it up into pieces. 525 00:36:04,890 --> 00:36:07,430 And how do we do that? 526 00:36:07,430 --> 00:36:16,120 Well, so here's our a and here's our b. 527 00:36:16,120 --> 00:36:19,600 And what we're going to do is break it up into little pieces. 528 00:36:19,600 --> 00:36:22,620 And we're going to give a name to the increment. 529 00:36:22,620 --> 00:36:28,510 And we're going to call that delta x. 530 00:36:28,510 --> 00:36:30,380 So we divide up into these. 531 00:36:30,380 --> 00:36:32,110 So how many pieces are there? 532 00:36:32,110 --> 00:36:37,570 If there are n pieces, then the general formula 533 00:36:37,570 --> 00:36:43,390 is always the delta x is 1/n times the total length. 534 00:36:43,390 --> 00:36:44,880 So it has to be (b-a) / n. 535 00:36:48,170 --> 00:36:50,550 We will always use these equal increments, 536 00:36:50,550 --> 00:36:53,020 although you don't absolutely have to do it. 537 00:36:53,020 --> 00:37:01,080 We will, for these Riemann sums. 538 00:37:01,080 --> 00:37:07,610 And now there's only one bit of flexibility 539 00:37:07,610 --> 00:37:10,560 that we will allow ourselves. 540 00:37:10,560 --> 00:37:13,020 Which is this. 541 00:37:13,020 --> 00:37:29,720 We're going to pick any height of f between-- in the interval, 542 00:37:29,720 --> 00:37:34,610 in each interval. 543 00:37:34,610 --> 00:37:39,020 So what that means is, let me just show it 544 00:37:39,020 --> 00:37:43,870 to you on the picture here. 545 00:37:43,870 --> 00:37:47,200 Is, I just pick any value in between, 546 00:37:47,200 --> 00:37:49,770 I'll call it c_i, which is in there. 547 00:37:49,770 --> 00:37:51,420 And then I go up here. 548 00:37:51,420 --> 00:37:55,180 And I have the level, which is f(c_i). 549 00:37:55,180 --> 00:37:58,730 And that's the rectangle that I choose. 550 00:37:58,730 --> 00:38:01,530 In the case that we did, we always 551 00:38:01,530 --> 00:38:03,730 chose the right-hand, which turned out 552 00:38:03,730 --> 00:38:04,930 to be the largest one. 553 00:38:04,930 --> 00:38:07,800 But I could've chosen some level in between. 554 00:38:07,800 --> 00:38:09,140 Or even the left-hand end. 555 00:38:09,140 --> 00:38:11,223 Which would have meant that the staircase would've 556 00:38:11,223 --> 00:38:13,580 been quite a bit lower. 557 00:38:13,580 --> 00:38:17,950 So any of these staircases will work perfectly well. 558 00:38:17,950 --> 00:38:25,650 So that means were picking f(c_i), and that's a height. 559 00:38:25,650 --> 00:38:33,210 And now we're just going to add them all up. 560 00:38:33,210 --> 00:38:35,680 And this is the sum of the areas of the rectangles, 561 00:38:35,680 --> 00:38:37,350 because this is the height. 562 00:38:37,350 --> 00:38:43,700 And this is the base. 563 00:38:43,700 --> 00:38:46,160 This notation is supposed to be, now, 564 00:38:46,160 --> 00:38:54,640 very suggestive of the notation that Leibniz used. 565 00:38:54,640 --> 00:38:58,250 Which is that in the limit, this becomes an integral from a 566 00:38:58,250 --> 00:39:01,010 to b of f(x) dx. 567 00:39:01,010 --> 00:39:05,230 And notice that the delta x gets replaced by a dx. 568 00:39:05,230 --> 00:39:07,960 So this is what happens in the limit. 569 00:39:07,960 --> 00:39:10,600 As the rectangles get thin. 570 00:39:10,600 --> 00:39:17,170 So that's as delta x goes to 0. 571 00:39:17,170 --> 00:39:21,780 And these gadgets are called Riemann sums. 572 00:39:21,780 --> 00:39:29,740 This is called a Riemann sum. 573 00:39:29,740 --> 00:39:31,580 And we already worked out an example. 574 00:39:31,580 --> 00:39:40,680 This very complicated guy was an example of a Riemann sum. 575 00:39:40,680 --> 00:39:42,060 So that's a notation. 576 00:39:42,060 --> 00:39:44,182 And we'll give you a chance to get 577 00:39:44,182 --> 00:39:45,640 used to it a little more when we do 578 00:39:45,640 --> 00:39:51,680 some numerical work at the end. 579 00:39:51,680 --> 00:39:55,130 Now, the last thing for today is, 580 00:39:55,130 --> 00:40:05,240 I promised you an example which was not an area example. 581 00:40:05,240 --> 00:40:10,010 I want to be able to show you that integrals can be 582 00:40:10,010 --> 00:40:21,630 interpreted as cumulative sums. 583 00:40:21,630 --> 00:40:36,480 Integrals as cumulative sums. 584 00:40:36,480 --> 00:40:39,020 So this is just an example. 585 00:40:39,020 --> 00:40:48,650 And, so here's the way it goes. 586 00:40:48,650 --> 00:40:51,440 So we're going to consider a function f, 587 00:40:51,440 --> 00:40:55,460 we're going to consider a variable t, which is time. 588 00:40:55,460 --> 00:40:59,340 In years. 589 00:40:59,340 --> 00:41:02,170 And we'll consider a function f(t), 590 00:41:02,170 --> 00:41:06,560 which is in dollars per year. 591 00:41:06,560 --> 00:41:09,470 Right, this is a financial example here. 592 00:41:09,470 --> 00:41:13,250 That's the unit here, dollars per year. 593 00:41:13,250 --> 00:41:21,500 And this is going to be a borrowing rate. 594 00:41:21,500 --> 00:41:24,000 Now, the reason why I want to put units in here 595 00:41:24,000 --> 00:41:27,320 is to show you that there's a good reason 596 00:41:27,320 --> 00:41:33,920 for this strange dx, which we append on these integrals. 597 00:41:33,920 --> 00:41:34,890 This notation. 598 00:41:34,890 --> 00:41:36,520 It allows us to change variables, 599 00:41:36,520 --> 00:41:39,020 it allows this to be consistent with units. 600 00:41:39,020 --> 00:41:42,360 And allows us to develop meaningful formulas, which are 601 00:41:42,360 --> 00:41:44,130 consistent across the board. 602 00:41:44,130 --> 00:41:46,020 And so I want to emphasize the units 603 00:41:46,020 --> 00:41:51,620 in this when I set up this modeling problem here. 604 00:41:51,620 --> 00:41:59,660 Now, you're borrowing money, let's say, every day. 605 00:41:59,660 --> 00:42:06,160 So that means delta t = 1/365. 606 00:42:06,160 --> 00:42:08,450 That's almost 1 / infinity, from the point 607 00:42:08,450 --> 00:42:11,700 of view of various purposes. 608 00:42:11,700 --> 00:42:15,180 So this is how much you're borrowing. 609 00:42:15,180 --> 00:42:17,820 In each time increment you're borrowing. 610 00:42:17,820 --> 00:42:23,990 And let's say that you borrow-- your rate varies over the year. 611 00:42:23,990 --> 00:42:27,140 I mean, sometimes you need more money sometimes you need less. 612 00:42:27,140 --> 00:42:29,650 Certainly any business would be that way. 613 00:42:29,650 --> 00:42:32,440 And so here you are, you've got your money. 614 00:42:32,440 --> 00:42:35,070 And you're borrowing but the rate is varying. 615 00:42:35,070 --> 00:42:36,960 And so how much did you borrow? 616 00:42:36,960 --> 00:42:51,230 Well, in day 45, which corresponds to t is 45/365, 617 00:42:51,230 --> 00:42:55,210 you borrowed the following amount. 618 00:42:55,210 --> 00:43:00,770 Here was your borrowing rate times this quantity. 619 00:43:00,770 --> 00:43:02,900 So, dollars per year. 620 00:43:02,900 --> 00:43:05,500 And so this is, if you like-- I want 621 00:43:05,500 --> 00:43:11,170 to emphasize the scaling that comes about here. 622 00:43:11,170 --> 00:43:14,910 You have dollars per year. 623 00:43:14,910 --> 00:43:21,060 And this is this number of years. 624 00:43:21,060 --> 00:43:23,180 So that comes out to be in dollars. 625 00:43:23,180 --> 00:43:24,050 This final amount. 626 00:43:24,050 --> 00:43:25,883 This is the amount that you actually borrow. 627 00:43:25,883 --> 00:43:30,250 So you borrow this amount. 628 00:43:30,250 --> 00:43:37,880 And now, if I want to add up how much you get-- 629 00:43:37,880 --> 00:43:39,920 you've borrowed in the entire year. 630 00:43:39,920 --> 00:43:50,380 That's this sum. i = 1 to 365 of f of, well, it's (i / 365) 631 00:43:50,380 --> 00:43:50,880 delta t. 632 00:43:50,880 --> 00:43:53,220 Which I'll just leave as delta t here. 633 00:43:53,220 --> 00:44:01,620 This is total amount borrowed. 634 00:44:01,620 --> 00:44:02,830 This is kind of a messy sum. 635 00:44:02,830 --> 00:44:05,570 In fact, your bank probably will keep track of it 636 00:44:05,570 --> 00:44:06,830 and they know how to do that. 637 00:44:06,830 --> 00:44:09,679 But when we're modeling things with strategies, you know, 638 00:44:09,679 --> 00:44:11,220 trading strategies, of course, you're 639 00:44:11,220 --> 00:44:13,910 really some kind of financial engineer 640 00:44:13,910 --> 00:44:17,000 and you want to cleverly optimize how much you borrow. 641 00:44:17,000 --> 00:44:19,610 And how much you spend, and how much you invest. 642 00:44:19,610 --> 00:44:23,900 This is going to be very, very similar to the integral 643 00:44:23,900 --> 00:44:29,460 from 0 to 1 of f(t) dt. 644 00:44:29,460 --> 00:44:36,340 At the scale of 1/35, it's probably-- 365, 645 00:44:36,340 --> 00:44:39,800 it's probably enough for many purposes. 646 00:44:39,800 --> 00:44:44,942 Now, however, there's another thing 647 00:44:44,942 --> 00:44:46,150 that you would want to model. 648 00:44:46,150 --> 00:44:47,670 Which is equally important. 649 00:44:47,670 --> 00:44:49,810 This is how much you borrowed, but there's also 650 00:44:49,810 --> 00:44:53,380 how much you owe the back at the end of the year. 651 00:44:53,380 --> 00:44:56,680 And the amount that you owe the bank at the end of the year, 652 00:44:56,680 --> 00:44:58,680 I'm going to do it in a fancy way. 653 00:44:58,680 --> 00:45:04,950 It's, the interest, we'll say, is compounded continuously. 654 00:45:04,950 --> 00:45:07,780 So the interest rate, if you start out with P 655 00:45:07,780 --> 00:45:20,120 as your principal, then after time t you owe-- So borrow P, 656 00:45:20,120 --> 00:45:30,000 after time t, you owe P e^(rt), where r is your interest rate. 657 00:45:30,000 --> 00:45:36,070 Say .05 per year. 658 00:45:36,070 --> 00:45:40,320 That would be an example of an interest rate. 659 00:45:40,320 --> 00:45:45,740 And so, if you want to understand how much money 660 00:45:45,740 --> 00:45:52,330 you actually owe at the end of the year. 661 00:45:52,330 --> 00:45:54,380 At the end of the year what you owe is, 662 00:45:54,380 --> 00:46:02,690 well, you borrowed these amounts here. 663 00:46:02,690 --> 00:46:04,680 But now you owe more at the end of the year. 664 00:46:04,680 --> 00:46:10,050 You owe e^r times the amount of time left in the year. 665 00:46:10,050 --> 00:46:15,270 So the amount of time left in the year is 1 - i / 365. 666 00:46:15,270 --> 00:46:18,900 Or 365 - i days left. 667 00:46:18,900 --> 00:46:26,600 So this is 1 - i / 365. 668 00:46:26,600 --> 00:46:34,770 And this is what you have to add up, to see how much you owe. 669 00:46:34,770 --> 00:46:39,540 And that is essentially the integral from 0 to 1. 670 00:46:39,540 --> 00:46:41,310 The delta t comes out. 671 00:46:41,310 --> 00:46:49,940 And you have here e^(r(1-t)), so the t is replacing this i / 672 00:46:49,940 --> 00:46:54,880 365, f(t) dt. 673 00:46:54,880 --> 00:46:58,630 And so when you start computing and thinking about what's 674 00:46:58,630 --> 00:47:04,170 the right strategy, you're faced with integrals of this type. 675 00:47:04,170 --> 00:47:06,140 So that's just an example. 676 00:47:06,140 --> 00:47:08,930 And see you next time. 677 00:47:08,930 --> 00:47:10,640 Remember to think about questions 678 00:47:10,640 --> 00:47:12,667 that you'll ask next time.