1 00:00:00,012 --> 00:00:07,682 [music] We've already got an idea of how we can approximate the area of a curved 2 00:00:07,682 --> 00:00:13,420 region by replacing that curved region with rectangles. 3 00:00:13,420 --> 00:00:16,748 We've already done this. If I want to approximate the area under 4 00:00:16,748 --> 00:00:20,100 the curve, I'll just add up the areas of these rectangles. 5 00:00:20,100 --> 00:00:23,319 And that's a pretty cut approximation of if these rectangles are thin enough. 6 00:00:23,320 --> 00:00:28,768 If I make those rectangles thinner and thinner, that gives me better and better 7 00:00:28,768 --> 00:00:31,700 approximations to the area under the curve. 8 00:00:31,700 --> 00:00:36,257 In fact, more than just approximating the area under the curve. 9 00:00:36,258 --> 00:00:41,563 By making those rectangles thinner and thinner, I can write down a definition of 10 00:00:41,563 --> 00:00:45,240 the area under the curve. So by taking those triangles to be thinner 11 00:00:45,240 --> 00:00:48,148 and thinner. I'm going to get a better and better 12 00:00:48,148 --> 00:00:50,560 approximation to the area under this curve. 13 00:00:50,560 --> 00:00:56,111 And I'm going to denote that limit by this, the integral, which is this long, 14 00:00:56,111 --> 00:00:59,425 thin s, from a to b of the function dx, right? 15 00:00:59,425 --> 00:01:05,525 That's how I'm going to write down the area Under the graph of this function, 16 00:01:05,525 --> 00:01:10,730 above the x-axis, and between the line y equals a and y equals b. 17 00:01:10,730 --> 00:01:16,263 It's going to be by this notation. Here is the precise definition of the 18 00:01:16,263 --> 00:01:19,621 integral. To precisely, the integral of this 19 00:01:19,621 --> 00:01:25,222 function from a to b is a limit over partitions of the Riemann sum, alright? 20 00:01:25,222 --> 00:01:29,650 And it's a limit over partitions where I've also chosen some sample points for 21 00:01:29,650 --> 00:01:33,330 those partitions. And I'm taking the limit as a maximum 22 00:01:33,330 --> 00:01:37,027 width of a sub-interval in that partition goes to 0. 23 00:01:37,028 --> 00:01:43,394 In order to ensure that all of the widths of the sub-intervals are going to 0, I 24 00:01:43,394 --> 00:01:49,569 just demand that the maximum one, the biggest one goes to 0, and that forces all 25 00:01:49,569 --> 00:01:54,892 the other ones to be small as well. I had to say that the maximum width goes 26 00:01:54,892 --> 00:01:58,139 to 0 to prevent some sort of bizarre scenario. 27 00:01:58,140 --> 00:02:02,650 Well I've got one really big sub-interval and then lots and lots of small 28 00:02:02,650 --> 00:02:05,986 sub-intervals right? I want to make sure that all of those 29 00:02:05,986 --> 00:02:10,609 sub-intervals are getting very narrow so that I'm getting a very fine partition in 30 00:02:10,609 --> 00:02:14,814 the limit. There's no guarantee whatsoever that, that 31 00:02:14,814 --> 00:02:18,292 limit actually exists. On the other hand, here's a theorem. 32 00:02:18,292 --> 00:02:23,646 If a function f is continuous then it's integrable, meaning that the integral, 33 00:02:23,646 --> 00:02:28,204 right, the integral of the function from say a to b actually exists. 34 00:02:28,204 --> 00:02:32,765 So when I say integraable, what do I even mean by integrable? 35 00:02:32,765 --> 00:02:38,504 If I were to claim that the integral of f from a to b is equal to some number, big 36 00:02:38,504 --> 00:02:44,136 I, what that means is that no matter how I partition the interval from a to b, as 37 00:02:44,136 --> 00:02:50,414 long as that partition is fine enough. Alright, as long as the maximum width of 38 00:02:50,414 --> 00:02:57,860 any rectangle in my skyscraper picture is small enough, right, as long as the widest 39 00:02:57,860 --> 00:03:02,912 one Is thin enough. Then no matter how I choose those sample 40 00:03:02,912 --> 00:03:09,352 points within each of the sub-intervals in my partition, the resulting Reimann sum 41 00:03:09,352 --> 00:03:12,880 that I calculate is close to I. How close? 42 00:03:12,880 --> 00:03:15,443 Well it's as close as I want it to be. Right? 43 00:03:15,444 --> 00:03:18,754 To say that this is equal to something is really to say something about a limit. 44 00:03:18,755 --> 00:03:23,881 So that means that if I'm asserting that this is equal to I, it means that I can 45 00:03:23,881 --> 00:03:29,040 get the Reimann sum as close as I want to I by simply demanding that my partition is 46 00:03:29,040 --> 00:03:32,570 fine enough. Regardless of how I exactly choose those 47 00:03:32,570 --> 00:03:36,108 sample points. And all this talk about Reimann sums is 48 00:03:36,108 --> 00:03:40,929 reflected in the very notation in that we've chosen for the integral. 49 00:03:40,930 --> 00:03:44,210 So here, I've got the integral, the integral of f from a to b. 50 00:03:44,210 --> 00:03:49,704 And here, I've got a Riemann sum, right, and this integral is defined to be a limit 51 00:03:49,704 --> 00:03:53,251 of these Reimann sums. And the notation really reflects their 52 00:03:53,251 --> 00:03:57,402 common origin, right? Here, I've got a summation symbol, a Greek 53 00:03:57,402 --> 00:04:01,914 sigma, Greek letter s. And here, I've got a long s. 54 00:04:01,915 --> 00:04:06,848 Alright, these are both a kind of sum. And the thing I'm summing is the function 55 00:04:06,848 --> 00:04:11,734 evaluated somewhere times a change in x. And the thing I'm summing here is the same 56 00:04:11,734 --> 00:04:15,340 thing it's the function evaluated somewhere times a change in x. 57 00:04:15,340 --> 00:04:20,605 So I hope that even the way I write down integrals really reminds you that there 58 00:04:20,605 --> 00:04:23,780 are a limit of things that look like this right. 59 00:04:23,780 --> 00:04:36,282 The function evaluated somewhere times some change in x being added up.