Introduction to integral calculus | Accumulation and Riemann sums | AP Calculus AB | Khan Academy

So I have a curve here that represents ( y ) is equal to ( f(x) ), and there's a classic problem that mathematicians have long thought about: how do we find the area under this curve, maybe under the curve and above the x-axis, and let's say between two boundaries, let's say between ( x ) is equal to ( a ) and ( x ) is equal to ( b )?

So let me draw these boundaries right over here. That's our left boundary; this is our right boundary, and we want to think about this area right over here. Well, without calculus, you can actually get better and better approximations for it. How would you do it? Well, you could divide this section into a bunch of ( \Delta x )'s that go from ( a ) to ( b ). They could be equal sections or not, but let's just say for the sake of visualizations, I'm going to draw roughly equal sections here.

So that's the first, that's the second, this is the third, this is the fourth, this is the fifth, and then we have the sixth right over here. And so, each of these, this is ( \Delta x ); let's just call that ( \Delta x_1 ). This is ( \Delta x_2 ); this width right over here is ( \Delta x_3 ), all the way to ( \Delta x_n ). I'll try to be general here.

And so, what we could do is let's try to sum up the area of the rectangles defined here. And we could make the height—maybe we make the height based on the value of the function at the right bound. It doesn't have to be; it could be the value of the function someplace in this ( \Delta x ), but that's one solution. We're going to go into a lot more depth into it in future videos.

And so, we do that, and so now we have an approximation. Or we could say, look, the area of each of these rectangles are going to be ( f(x_i) ) or maybe ( x_i ) is the right boundary, the way I've drawn it, times ( \Delta x_i )—that's each of these rectangles. And then we can sum them up, and that would give us an approximation for the area.

But as long as we use a finite number, we might say, well, we can always get better by making our ( \Delta x ) smaller and then by having more depth, more of these rectangles, or get to a situation here. We're going from ( i ) is equal to 1 to ( i ) is equal to ( n ).

But what happens is ( \Delta x ) gets thinner and thinner and thinner, and we have ( n ) gets larger and larger and larger as ( \Delta x ) gets infinitesimally small. And then, as ( n ) approaches infinity— and so you're probably sensing something—then maybe we could think about the limit. We could say, as ( n ) approaches infinity, or the limit as ( \Delta x ) becomes very, very, very, very small.

And this notion of getting better and better approximations as we take the limit as ( n ) approaches infinity, this is the core idea of integral calculus. And it's called integral calculus because the central operation we use—the summing up of an infinite number of infinitesimally thin things—is one way to visualize it.

This is going to be the integral—in this case, from ( a ) to ( b )—and we're going to learn a lot more depth. In this case, it is a definite integral of ( f(x) , dx ), but you can already see the parallels here. You can view the integral sign as like a sigma notation, as a summation sign, but instead of taking the sum of a discrete number of things, you're taking the sum of an infinite, an infinite number—infinitely thin things.

Instead of ( \Delta x ), you now have ( dx )—infinitesimally small things. And this is the notion of an integral. So this right over here is an integral. Now, what makes it interesting to calculus is it's using this notion of a limit. But what makes it even more powerful is it's connected to the notion of a derivative, which is one of these beautiful things in mathematics.

As we will see in the fundamental theorem of calculus, that integration—the notion of an integral—is closely tied, closely to the notion of a derivative. In fact, the notion of an antiderivative. In differential calculus, we looked at the problem of, hey, if I have some function, I can take its derivative, and I can get the derivative of the function.

In integral calculus, we're going to be doing a lot of, well, what if we start with a derivative? Can we figure out, through integration, can we figure out its antiderivative or the function whose derivative it is? As we will see, all of these are related: the idea of the area under a curve, the idea of a limit of summing an infinite number of infinitely thin things, and the notion of an antiderivative. They all come together in our journey in integral calculus.