Riemann sums, summation notation, and definite integral notation
Riemann sums in summation notation
In the last video, we attempted to approximate the area under a curve by constructing four rectangles of equal width and using the left boundary of each rectangle, the function evaluated at the left boundary, to determine the height, and we came up with an approximation. What I want to do in this video is generalize things a bit using the exact same method, but doing it for an arbitrary function with arbitrary boundaries and an arbitrary number of rectangles. So let's do it. So I'm going to draw the diagram as large as I can to make things as clear as possible. So that's my y-axis. And this right over here is my x-axis. Let me draw an arbitrary function. So let's say my function looks something like that. So that is y is equal to f of x. And let me define my boundaries. So let's say this right over here is x equals a. And this right over here is x equals b. So this is b. And I'm going to use n rectangles, and I'm going to use the function evaluated at the left boundary of the rectangle to determine its height. So, for example, this will be rectangle one. I'm going to evaluate what f of a is. So this right over here is f of a. And then I'm going to use that as the height of my first rectangle. So just like that. So rectangle number one looks like this. And I'll even number it. Rectangle one looks just like that. And just to have a convention here, because I'm going to want to label each of the x values at the left boundary, so we'll say a is equal to x0. a is equal to x0. So we could also call this point right over here x0, that x value. And then we go to the next rectangle. And we could call this one right over here, this x value, we'll call it x1. It's the left boundary of the next rectangle. If we evaluate f of x1, we get this value right over here. This right over here is f of x1, so it tells us our height. And we want an equal width to the previous one. We'll think about what the width is going to be in a second. So this right over here is our second rectangle that we're going to use to approximate the area under the curve. That's rectangle number two. Let's do rectangle number three. Well, rectangle number three, the left boundary, we're just going to call that x sub 2. And its height is going to be f of x sub 2. And its width is going to be the same width as the other ones. I'm just eyeballing it right over here. So this is rectangle number three. And we're going to continue this process all the way until we get to rectangle number n. So this is the n-th rectangle right over here, the n-th rectangle. And what am I going to label this point right over here? Well, we already see a pattern. The left boundary of the first rectangle is x sub 0. The left boundary of the second rectangle is x sub 1. The left boundary of the third rectangle is x sub 2. So the left boundary of the n-th rectangle is going to be x sub n minus 1. Whatever the rectangle number is, the left boundary is x sub that number minus 1. And this is just based on the convention that we've defined. Now, the next thing that we need to do in order to actually calculate this area is think about what is the width? So let's call the width of any of these rectangles-- and for these purposes, or the purpose of this example, I'm going to assume that it's constant, although you can do these sums where you actually vary the width of the rectangle. But then it gets a little bit fancier. So I want it equal width. So I want delta x to be equal width. And to think about what that has to be, we just have to think, what's the total width that we're covering? Well, the total distance here is going to be b minus a, and we're just going to divide by the number of rectangles that we want, the number of sections that we want. So we want to divide by n. So if we assume this is true, and then we assume that a is equal to x0, and then x1 is equal to x0 plus delta x, x2 is equal to x1 plus delta x, and we go all the way to xn is equal to xn minus 1 plus delta x, then we've essentially set up this diagram right over here. b is actually going to be equal xn. So this is xn. It's equal to xn minus 1 plus delta x. So now I think we've set up all of the notation and all the conventions in order to actually calculate the area, or our approximation of the area. So our approximation, approximate area, is going to be equal to what? Well, it's going to be the area of the first rectangle-- so let me write this down. So it's going to be rectangle one-- so the area of rectangle one-- so rectangle one plus the area of rectangle two plus the area of rectangle three-- I think you get the point here-- plus all the way to the area of rectangle n. And so what are these going to be? Rectangle one is going to be its height, which is f of x0 or f of a. Either way. x0 and a are the same thing. So it's f of a times our delta x, times our width, our height times our width. So times delta-- actually I can write as f of x0, I wanted to write-- f of x0 times delta x. What is our height of rectangle two? It's f of x1 times delta x. What's our area of rectangle three? It's f of x2 times delta x. And then we go all the way to our area. We're taking all the sums, all the way to rectangle n. What's its area? It's f of x sub n minus 1. Actually, that's a different shade of orange. I'll use that same shade. It is f of x sub n minus 1 times delta x. And we're done. We've written it in a very general way. But to really make us comfortable with the various forms of notation, especially the types of notation you might see when people are talking about approximating the areas or sums in general, I'm going to use the traditional sigma notation. So another way we could write this, as the sum, this is equal to the sum from-- and remember, this is just based on the conventions that I set up. I'll let i count which rectangle we're in, from i equals 1 to n. And then we're going to look at each rectangle. So the first rectangle, that's rectangle one. So it's going to be f of-- well, if we're in the i-th rectangle, then the left boundary is going to be x sub i minus 1 times delta x. And so here, right over here, is a general way of thinking about approximating the area under a curve using rectangles, where the height of the rectangles are defined by the left boundary. And this tells us it's the left boundary. And we see for each, if this is the i-th rectangle right over here, if this is rectangle i, then this right over here is x sub i minus 1, and this height right over here is f of x sub i minus 1. So that's all we did right over there times delta x. And then you sum of these from the first rectangle all the way to the end. So hopefully that makes you a little bit more comfortable with this notation. We're not doing anything different than we did in this first video, which was hopefully fairly straightforward for you. We have just generalized it using a little bit more mathy notation.
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