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Integrate age article

- [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...