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Definite integral as the limit of a Riemann sum

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Definite integrals represent the exact area under a given curve, and Riemann sums are used to approximate those areas. However, if we take Riemann sums with infinite rectangles of infinitely small width (using limits), we get the exact area, i.e. the definite integral! Created by Sal Khan.

Video transcript

We've done several videos already where we're approximating the area under a curve by breaking up that area into rectangles and then finding the sum of the areas of those rectangles as an approximation. And this was actually the first example that we looked at where each of the rectangles had an equal width. So we equally partitioned the interval between our two boundaries between a and b. And the height of the rectangle was the function evaluated at the left endpoint of each rectangle. And we wanted to generalize it and write it in sigma notation. It looked something like this. And this was one case. Later on, we looked at a situation where you define the height by the function value at the right endpoint or at the midpoint. And then we even constructed trapezoids. And these are all particular instances of Riemann sums. So this right over here is a Riemann sum. And when people talk about Riemann sums, they're talking about the more general notion. You don't have to just do it this way. You could use trapezoids. You don't even have to have equally-spaced partitions. I used equally-spaced partitions because it made things a little bit conceptually simpler. And this right here is a picture of the person that Riemann sums was named after. This is Bernhard Riemann. And he made many contributions to mathematics. But what he is most known for, at least if you're taking a first-year calculus course, is the Riemann sum. And how this is used to define the Riemann integral. Both Newton and Leibniz had come up with the idea of the integral when they had formulated calculus, but the Riemann integral is kind of the most mainstream formal, or I would say rigorous, definition of what an integral is. So as you could imagine, this is one instance of a Riemann sum. We have n right over here. The larger n is, the better an approximation it's going to be. So his definition of an integral, which is the actual area under the curve, or his definition of a definite integral, which is the actual area under a curve between a and b is to take this Riemann sum, it doesn't have to be this one, take any Riemann sum, and take the limit as n approaches infinity. So just to be clear, what's happening when n approaches infinity? Let me draw another diagram here. So let's say that's my y-axis. This is my x-axis. This is my function. As n approaches infinity-- so this is a, this is b-- you're just going to have a ton of rectangles. You're just going to get a ton of rectangles over there. And there are going to become better and better approximations for the actual area. And the actual area under the curve is denoted by the integral from a to b of f of x times dx. And you see where this is coming from or how these notations are close. Or at least in my brain, how they're connected. Delta x was the width for each of these sections. This right here is delta x. So that is a delta x. This is another delta x. This is another delta x. A reasonable way to conceptualize what dx is, or what a differential is, is what delta x approaches, if it becomes infinitely small. So you can conceptualize this, and it's not a very rigorous way of thinking about it, is an infinitely small-- but not 0-- infinitely small delta x, is one way that you can conceptualize this. So once again, as you have your function times a little small change in delta x. And you are summing, although you're summing an infinite number of these things, from a to b. So I'm going to leave you there just so that you see the connection. You know the name for these things. And once again, this one over here, this isn't the only Riemann sum. In fact, this is often called the left Riemann sum if you're using it with rectangles. You can do a right Riemann sum. You could use the midpoint. You could use a trapezoid. But if you take the limit of any of those Riemann sums, as n approaches infinity, then that you get as a Riemann definition of the integral. Now so far, we haven't talked about how to actually evaluate this thing. This is just a definition right now. And for that we will do in future videos.