WEBVTT 00:00:15.060 --> 00:00:19.800 Here, I want to discuss one common type of problem where integration comes up, 00:00:19.800 --> 00:00:22.500 finding the average of a continuous variable. 00:00:23.620 --> 00:00:26.297 This is a perfectly useful thing to know in its own right, 00:00:26.297 --> 00:00:29.426 but what's really neat is that it can give us a completely different 00:00:29.426 --> 00:00:32.740 perspective for why integrals and derivatives are inverses of each other. 00:00:33.460 --> 00:00:39.539 To start, take a look at the graph of sinx between 0 and pi, which is half of its period. 00:00:40.200 --> 00:00:43.800 What is the average height of this graph on that interval? 00:00:44.700 --> 00:00:46.120 It's not a useless question. 00:00:46.520 --> 00:00:50.380 All sorts of cyclic phenomena in the world are modeled using sine waves. 00:00:50.920 --> 00:00:54.349 For example, the number of hours the sun is up per day as a 00:00:54.348 --> 00:00:58.179 function of what day of the year it is follows a sine wave pattern. 00:00:58.820 --> 00:01:03.899 So if you wanted to predict the average effectiveness of solar panels in summer months vs. 00:01:04.159 --> 00:01:08.319 winter months, you'd want to be able to answer a question like this, 00:01:08.319 --> 00:01:12.659 what is the average value of that sine function over half of its period? 00:01:13.879 --> 00:01:18.247 Where as a case like this is going to have all sorts of constants mucking up the 00:01:18.248 --> 00:01:22.454 function, you and I are going to focus on a pure, unencumbered sinx function, 00:01:22.454 --> 00:01:27.039 but the substance of the approach would be totally the same in any other application. 00:01:28.260 --> 00:01:31.020 It's kind of a weird question to think about though, isn't it? 00:01:31.260 --> 00:01:33.320 The average of a continuous variable. 00:01:33.840 --> 00:01:37.594 Usually with averages we think of a finite number of variables, 00:01:37.594 --> 00:01:41.819 where you can add them all up and divide that sum by how many there are. 00:01:44.200 --> 00:01:48.221 But there are infinitely many values of sinx between 0 and pi, 00:01:48.221 --> 00:01:53.200 and it's not like we can just add up all those numbers and divide by infinity. 00:01:54.140 --> 00:01:58.215 This sensation comes up a lot in math, and it's worth remembering, 00:01:58.215 --> 00:02:02.656 where you have this vague sense that you want to add together infinitely 00:02:02.656 --> 00:02:07.340 many values associated with a continuum, even though that doesn't make sense. 00:02:08.060 --> 00:02:13.199 And almost always, when you get that sense, the key is to use an integral somehow. 00:02:13.840 --> 00:02:17.222 And to think through exactly how, a good first step is to 00:02:17.222 --> 00:02:20.780 just approximate your situation with some kind of finite sum. 00:02:20.780 --> 00:02:27.060 In this case, imagine sampling a finite number of points evenly spaced along this range. 00:02:27.919 --> 00:02:32.834 Since it's a finite sample, you can find the average by just adding up all the heights 00:02:32.835 --> 00:02:37.920 sinx at each one of these, and then dividing that sum by the number of points you sampled. 00:02:39.319 --> 00:02:43.594 And presumably, if the idea of an average height among all infinitely many 00:02:43.594 --> 00:02:47.527 points is going to make any sense at all, the more points we sample, 00:02:47.527 --> 00:02:50.549 which would involve adding up more and more heights, 00:02:50.550 --> 00:02:54.711 the closer the average of that sample should be to the actual average of 00:02:54.711 --> 00:02:56.080 the continuous variable. 00:02:57.159 --> 00:03:01.799 And this should feel at least somewhat related to taking an integral of sinx 00:03:01.799 --> 00:03:06.800 between 0 and pi, even if it might not be exactly clear how the two ideas match up. 00:03:07.460 --> 00:03:13.103 For that integral, remember, you also think of a sample of inputs on this continuum, 00:03:13.103 --> 00:03:18.812 but instead of adding the height sinx at each one and dividing by how many there are, 00:03:18.812 --> 00:03:23.460 you add up sinx times dx, where dx is the spacing between the samples. 00:03:24.400 --> 00:03:27.200 That is, you're adding up little areas, not heights. 00:03:28.300 --> 00:03:31.550 And technically, the integral is not quite this sum, 00:03:31.550 --> 00:03:34.800 it's whatever that sum approaches as dx approaches 0. 00:03:35.500 --> 00:03:39.764 But it is actually quite helpful to reason with respect to one of these finite 00:03:39.764 --> 00:03:44.406 iterations, where we're looking at a concrete size for dx and some specific number of 00:03:44.406 --> 00:03:45.000 rectangles. 00:03:45.960 --> 00:03:50.526 So what you want to do here is reframe this expression for the average, 00:03:50.526 --> 00:03:54.649 this sum of the heights divided by the number of sampled points, 00:03:54.649 --> 00:03:57.439 in terms of dx, the spacing between samples. 00:03:59.039 --> 00:04:04.117 And now, if I tell you that the spacing between these points is, say, 0.1, 00:04:04.117 --> 00:04:09.400 and you know that they range from 0 to pi, can you tell me how many there are? 00:04:11.099 --> 00:04:14.430 Well, you can take the length of that interval, pi, 00:04:14.431 --> 00:04:18.338 and divide it by the length of the space between each sample. 00:04:19.360 --> 00:04:23.812 If it doesn't go in perfectly evenly, you'd have to round down to the nearest integer, 00:04:23.812 --> 00:04:26.319 but as an approximation, this is completely fine. 00:04:27.240 --> 00:04:31.023 So if we write that spacing between samples as dx, 00:04:31.023 --> 00:04:34.140 the number of samples is pi divided by dx. 00:04:34.699 --> 00:04:38.017 And when we substitute that into our expression up here, 00:04:38.017 --> 00:04:42.559 you can rearrange it, putting that dx up top and distributing it into the sum. 00:04:43.759 --> 00:04:47.139 But think about what it means to distribute that dx up top. 00:04:48.120 --> 00:04:51.586 It means that the terms you're adding up will look like 00:04:51.586 --> 00:04:55.300 sinx times dx for the various inputs x that you're sampling. 00:04:56.079 --> 00:04:59.079 So that numerator looks exactly like an integral expression. 00:04:59.819 --> 00:05:02.944 And so for larger and larger samples of points, 00:05:02.944 --> 00:05:07.699 this average will approach the actual integral of sinx between 0 and pi, 00:05:07.699 --> 00:05:10.759 all divided by the length of that interval, pi. 00:05:11.939 --> 00:05:17.139 In other words, the average height of this graph is this area divided by its width. 00:05:18.079 --> 00:05:21.038 On an intuitive level, and just thinking in terms of units, 00:05:21.038 --> 00:05:23.060 that feels pretty reasonable, doesn't it? 00:05:23.459 --> 00:05:26.039 Area divided by width gives you an average height. 00:05:26.939 --> 00:05:30.399 So with this expression in hand, let's actually solve it. 00:05:31.180 --> 00:05:36.069 As we saw last video, to compute an integral, you need to find an antiderivative 00:05:36.069 --> 00:05:41.020 of the function inside the integral, some other function whose derivative is sinx. 00:05:42.000 --> 00:05:45.550 And if you're comfortable with derivatives of trig functions, 00:05:45.550 --> 00:05:48.759 you know that the derivative of cosine is negative sine. 00:05:49.439 --> 00:05:53.449 So if you just negate that, negative cosine is the function we want, 00:05:53.449 --> 00:05:55.019 the antiderivative of sine. 00:05:55.639 --> 00:05:59.620 And to gut-check yourself on that, look at this graph of negative cosine. 00:06:00.019 --> 00:06:06.335 At 0, the slope is 0, and then it increases up to some maximum slope at pi halves, 00:06:06.336 --> 00:06:09.000 and then goes back down to 0 at pi. 00:06:09.879 --> 00:06:12.735 And in general, its slope does indeed seem to 00:06:12.735 --> 00:06:15.840 match the height of the sine graph at every point. 00:06:17.060 --> 00:06:21.120 So what do we have to do to evaluate the integral of sine between 0 and pi? 00:06:22.079 --> 00:06:25.570 We evaluate this antiderivative at the upper bound, 00:06:25.571 --> 00:06:28.660 and subtract off its value at the lower bound. 00:06:29.560 --> 00:06:32.838 More visually, that's the difference in the height of 00:06:32.838 --> 00:06:36.240 this negative cosine graph above pi and its height at 0. 00:06:37.259 --> 00:06:40.779 And as you can see, that change in height is exactly 2. 00:06:41.920 --> 00:06:43.400 That's kind of interesting, isn't it? 00:06:43.540 --> 00:06:47.460 That the area under this sine graph turns out to be exactly 2? 00:06:48.220 --> 00:06:53.845 So the answer to our average height problem, this integral divided by the width 00:06:53.845 --> 00:06:59.400 of the region, evidently turns out to be 2 divided by pi, which is around 0.64. 00:07:01.300 --> 00:07:06.341 I promised at the start that this question of finding the average of a function offers 00:07:06.341 --> 00:07:11.324 an alternate perspective on why integrals and derivatives are inverses of each other, 00:07:11.324 --> 00:07:15.960 why the area under one graph has anything to do with the slope of another graph. 00:07:16.980 --> 00:07:20.662 Notice how finding this average value, 2 divided by pi, 00:07:20.661 --> 00:07:24.475 came down to looking at the change in the antiderivative, 00:07:24.475 --> 00:07:29.540 negative cosine x, over the input range, divided by the length of that range. 00:07:30.600 --> 00:07:35.820 And another way to think about that fraction is as the rise over run slope between 00:07:35.819 --> 00:07:41.040 the point of the antiderivative graph below 0 and the point of that graph above pi. 00:07:41.939 --> 00:07:45.552 Think about why it might make sense that this slope would 00:07:45.552 --> 00:07:48.980 represent an average value of sine of x on that region. 00:07:50.459 --> 00:07:55.298 By definition, sine of x is the derivative of this antiderivative graph, 00:07:55.298 --> 00:07:58.879 giving us the slope of negative cosine at every point. 00:07:59.759 --> 00:08:03.879 Another way to think about the average value of sine of x is 00:08:03.879 --> 00:08:08.000 as the average slope over all tangent lines between 0 and pi. 00:08:08.899 --> 00:08:12.620 And when you view things like that, doesn't it make a lot of sense 00:08:12.620 --> 00:08:16.341 that the average slope of a graph over all its points in a certain 00:08:16.341 --> 00:08:20.120 range should equal the total slope between the start and end points? 00:08:23.220 --> 00:08:27.520 To digest this idea, it helps to think about what it looks like for a general function. 00:08:28.319 --> 00:08:33.706 For any function f of x, if you want to find its average value on some interval, 00:08:33.706 --> 00:08:38.162 say between a and b, what you do is take the integral of f on that 00:08:38.162 --> 00:08:42.019 interval divided by the width of that interval, b minus a. 00:08:43.080 --> 00:08:47.321 You can think of this as the area under the graph divided by its width, 00:08:47.321 --> 00:08:50.678 or more accurately, it is the signed area of that graph, 00:08:50.678 --> 00:08:53.919 since any area below the x-axis is counted as negative. 00:08:55.500 --> 00:09:00.154 And it's worth taking a moment to remember what this area has to do with the usual notion 00:09:00.154 --> 00:09:04.500 of a finite average, where you add up many numbers and divide by how many there are. 00:09:05.059 --> 00:09:08.331 When you take some sample of points spaced out by dx, 00:09:08.331 --> 00:09:13.240 the number of samples is about equal to the length of the interval divided by dx. 00:09:14.220 --> 00:09:18.595 So if you add up the values of f of x at each sample and divide by 00:09:18.595 --> 00:09:23.035 the total number of samples, it's the same as adding up the product 00:09:23.034 --> 00:09:27.279 f of x times dx and dividing by the width of the entire interval. 00:09:27.919 --> 00:09:32.269 The only difference between that and the integral is that the integral asks 00:09:32.269 --> 00:09:36.562 what happens as dx approaches 0, but that just corresponds with samples of 00:09:36.562 --> 00:09:40.740 more and more points that approximate the true average increasingly well. 00:09:42.240 --> 00:09:48.365 Now for any integral, evaluating it comes down to finding an antiderivative of f of x, 00:09:48.365 --> 00:09:50.620 commonly denoted capital F of x. 00:09:51.500 --> 00:09:56.315 What we want is the change to this antiderivative between a and b, 00:09:56.315 --> 00:10:00.843 capital F of b minus capital F of a, which you can think of as 00:10:00.842 --> 00:10:05.299 the change in height of this new graph between the two bounds. 00:10:06.559 --> 00:10:11.958 I've conveniently chosen an antiderivative that passes through 0 at the lower bound here, 00:10:11.958 --> 00:10:16.219 but keep in mind you can freely shift this up and down adding whatever 00:10:16.220 --> 00:10:20.000 constant you want and it would still be a valid antiderivative. 00:10:21.320 --> 00:10:25.986 So the solution to the average problem is the change in the height of 00:10:25.985 --> 00:10:30.519 this new graph divided by the change to the x value between a and b. 00:10:31.399 --> 00:10:36.139 In other words, it is the slope of the antiderivative graph between the two endpoints. 00:10:37.120 --> 00:10:41.604 And again, when you stop to think about it, that should make a lot of sense, 00:10:41.604 --> 00:10:46.379 because little gives us the slope of the tangent line to this graph at each point. 00:10:47.120 --> 00:10:51.060 After all, it is by definition the derivative of capital F. 00:10:52.980 --> 00:10:56.500 So why are antiderivatives the key to solving integrals? 00:10:57.600 --> 00:11:01.267 My favorite intuition is still the one I showed last video, 00:11:01.267 --> 00:11:06.646 but a second perspective is that when you reframe the question of finding an average of 00:11:06.645 --> 00:11:11.841 a continuous value as instead finding the average slope of a bunch of tangent lines, 00:11:11.841 --> 00:11:15.264 it lets you see the answer just by comparing endpoints, 00:11:15.264 --> 00:11:19.299 rather than having to actually tally up all the points in between. 00:11:23.120 --> 00:11:27.658 In the last video I described a sensation that should bring integrals to your mind, 00:11:27.658 --> 00:11:31.764 namely if you feel like the problem you're solving could be approximated by 00:11:31.764 --> 00:11:35.439 breaking it up somehow and adding up a large number of small things. 00:11:36.100 --> 00:11:38.835 Here I want you to come away recognizing a second 00:11:38.835 --> 00:11:41.900 sensation that should also bring integrals to your mind. 00:11:42.879 --> 00:11:46.878 If ever there's some idea that you understand in a finite context, 00:11:46.878 --> 00:11:51.354 and which involves adding up multiple values, like taking the average of a 00:11:51.354 --> 00:11:56.248 bunch of numbers, and if you want to generalize that idea to apply to an infinite 00:11:56.248 --> 00:12:01.500 continuous range of values, try seeing if you can phrase things in terms of an integral. 00:12:02.139 --> 00:12:05.794 It's a feeling that comes up all the time, especially in probability, 00:12:05.794 --> 00:12:07.779 and it's definitely worth remembering. 00:12:09.039 --> 00:12:12.039 My thanks, as always, go to those making these videos possible. 00:12:31.500 --> 00:12:38.840 Thank you.