The basic idea of Integral calculus is finding the area under a curve. To find it exactly, we can divide the area into infinite rectangles of infinitely small width and sum their areas—calculus is great for working with infinite things! This idea is actually quite rich, and it's also tightly related to Differential calculus, as you will see in the upcoming videos.
- [Instructor] So I have a curve here that represents y is equal to f of 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 could 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 gonna 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 one. This is delta x two. This width right over here, this is delta x three, 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 gonna go into a lot more depth into it in future videos. And so we do that. And so now we have an approximation, where we could say, look, the area of each of these rectangles are going to be f of x sub i, where maybe x sub 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's smaller and then by having more of these rectangles, or get to a situation here we're going from i is equal to one to i is equal to n. But what happens is delta x gets thinner and thinner and thinner, and 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, that maybe we could think about the limit as 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, is the integral, that this is going to be the integral, in this case, from a to b. And we're gonna learn in a lot more depth, in this case, it is a definite integral of f of x, f of 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 infinitely, an infinite number, infinitely thin things. Instead of delta x, you now have dx, infinitesimally small things. And this is a notion of an integral. So this right over here is an integral. Now what makes it interesting to calculus, it is 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. Integral calculus, we're going to be doing a lot of, well, what if we start with the 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 things, thin things, and the notion of an antiderivative, they all come together in our journey in integral calculus.