We've already seen one definition of the definite integral, and many of them are closely related to this definition that we've already seen, is, well, look, the definite integral from a to b of f of x, d of x is this area shaded in blue, and we can approximate it by splitting it into n rectangles, so let's say that's the first rectangle, one, that's the second rectangle, two, and you're gonna go all the way to the nth rectangle, so this would be the n minus oneth rectangle, and for the sake of this argument I'm gonna make in this video, I'm gonna assume that they're all the same width, so this is the nth rectangle, and they all have the same width, and we see there are definitions of integration where you don't have to have the same width here, but let's say that each of those widths are delta x, and the way that we calculate delta x is we take b minus a, we take b minus a, and we divide it by n, we divide it by n, which is common sense, or this is what you learned in division, we're just taking this length and dividing it by n to get n equal spacings, which is n equal spacings of delta x, and so if you do this, you say, okay, we see this multiple times, you can approximate it, you can approximate this area using these rectangles as the sum from i equals one to n, so you're summing, you're summing n of these rectangles areas, where the height of each of these rectangles are gonna be f of x sub i, where x sub i is the point at which you're taking the function value to find out its height, so that could be x sub one, x sub two, x sub three, so on and so forth, and you're multiplying that times your delta x, times your delta x, so you take x sub two, f of x sub two is that height right there, f of x sub two is that height right there, you multiply it times delta x, you get the area, and we saw that when we looked at Riemann sums and using that to approximate, and we said, hey, the one definition of the definite integral is that since this is the area, this is going to be the limit as n approaches infinity of this, where delta x is defined as that, so let me just copy and paste that, so copy and paste, where that, so that's one way to think about it. Now, given this definition, what do you think, what do you think this, or maybe another way to think about it, how do you think this expression that I'm writing right over here, based on this definition, should relate to this expression? So notice, all I've done is I've, instead of going from a to b, I'm now going from b to a, I'm now going from b to a. How do you think these two things should relate? And I encourage you to look at all of this to come to that conclusion, and pause the video to do so. Well, let's just think about what's going to happen. This is going to be, if I were to literally just take this, if I were to literally just take this and copy and paste it, which is exactly what I'm going to do, if I just took this, by definition, since I swapped these two bounds, I am going to want to swap these two.