Applying properties of definite integrals
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Integrating scaled version of function
- [Voiceover] We've already seen and you're probably getting tired of me pointing it out repeatedly, that this yellow area right over here, this area under the curve y is equal to f of x and above the positive x-axis or I guess I can say just above the x-axis between x equals a and x equals b, that we can denote this area right over here as the definite integral of from a to b of f of x dx. Now what I want to explore in this video and it'll come up with kind of an answer that you probably could have guessed on your own, but at least get an intuition for it, is that I want to start thinking about the area under the curve that's a scaled version of f of x. Let's say it's y is equal to c times f of x. Y is equal to some number times f of x, so it's scaling f of x. And so I want this to be kind of some arbitrary number, but just to help me visualize, you know I have to draw something so I'm just gonna kind of in my head let's just pretend the c is a three for visualization purposes. So it's going to be three times, so instead of one, instead of this far right over here it's going to be about this far. For right over here, instead of this far right over here it's going to be that and another right over there. And then instead of it's going to be about there. And then instead of it being like that it's going to be one, two and then three, right around there. So I'm starting to get a sense of what this curve is going to look like, a scaled version of f of x. And at least what I'm drawing is pretty close to three times f of x, but just to give you an idea is going to look something like, and let's see over here if this distance, do a second one, a third one, is gonna be up here. It's gonna look something like this. It's gonna look something like that. So this is a scaled version and the scale I did right here I assumed a positive c greater than zero, but this is just for visualization purposes. Now what do we think the area under this curve is going to be between a and b? So what do we think this area right over here is going to be? Now we already know how we can denote it. That area right over there is equal to the definite integral from a to b of the function we're integrating is c f of x dx. I guess to make the question a little bit clearer, how does this relate to this? How does this green area relate to this yellow area? Well one way to think about it is we just scaled the vertical dimension up by c, so one way that you could reason it is if I'm finding the area of something, if I have the area of a rectangle and I have the vertical dimension is let's say I don't want to use those same letters over and over again. Well let's say the vertical dimension is alpha and the horizontal dimension is beta. We know that the area is going to be alpha times beta. Now if I scale up the vertical dimension by c, so instead of alpha this is c times alpha and this is, the width is beta, if I scale up the vertical dimension by c so this is now c times alpha, what's the area going to be? Well it's going to be c alpha times beta, or another way to think of it, when I scale one of the dimensions by c I take my old area and I scale up my old area up by c. And that's what we're doing, we're scaling up the vertical dimension by c. When you multiply c times f of x, f of x is giving us the vertical height. Now obviously that changes as our x changes, but when you think back to the Reimann sums the f of x was what gave us the height of our rectangles. We're now scaling up the height or scaling I should say because we might be scaling down depending on the c. We're scaling it, we're scaling one dimension by c. If you scale one dimension by c you're gonna scale the area by c. So this right over here, the integral, let me just rewrite it. The integral from a to b of c f of x dx, that's just going to be the scaled, we're just going to take the area of f of x, so let me do that in the same color. We're going to take the area under the curve f of x from a to b f of x dx and we're just going to scale it up by this c. So you might say, "Okay maybe I could have felt "that was, you know, if I have a c inside the integral "now I can take the c out of the integral", and once again this is not a rigorous proof based on the definition of the definite integral, but it hopefully gives you a little bit of intuition why you can do this. If you scale up the function, you're essentially scaling up the vertical dimension, so the area under this is going to just be a scaled up version of the area under the original function f of x. And once again really, really, really useful property of definite integrals that's going to help us solve a bunch of definite integrals. And kind of clarify what we're even doing with them.
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