Definite integral properties
Definite integral over a single point
- [Voiceover] We've already taken definite integrals and we've seen how they represent or denote the area under a function between two points and above the X axis. But let's do something interesting. Let's think about a definite integral of F of X DX, It's the area under the curve, F of X, but instead of it being mean between two different X values, say A and B like we see in multiple times, let's say it's between the same one. Let's say it's between C and C. Let's say C is right over here. What do you think this thing right over here is going to be equal to? What does this represent? What is this equal to? I encourage you to pause the video and try to think about it. Well if you try to visualize it, you're thinking, well the area under the curve F of X, above the X axis, from X equals C to X equals C. So this region, I guess we could call it, that we think about it, does have a height. The height here is F of C. What's the width? Well there is no width, we're just at a single point. We're not going from C to C plus some delta X or C plus some even very small change in X or C plus some other very small a value. We're just, saying at the point C. When we're thinking about area we're thinking about how much two-dimensional space you're taking up. But this idea, this is just a one-dimensional, I think you could think of it as a line segment. What's the area of a line segment? Well a line segment has no area. So this thing right over here is going to be equal to zero. Now you might say, I get that. I see why that could make sense, why that makes intuitive sense. I'm trying to find the area of a rectangle where I know it's height, but it's width is zero. So that areas going to be zero is one way to think about it. But Sal, why are you even pointing this out to me? As well see, especially when we do more complex definite integration problems and solving things sometimes recognizing this will help you simplify an integration problem dramatically. Or you could work to be able to get to a point like this so that you can cancel things out. Or you can say, hey that thing right over there is just going to be equal to zero.