And if you take that mean value of the function and you multiply it times the base, you're going to get the area of the curve. Or another way of rewriting this, you could say that f of c, there exists a c in that interval where f of c is equal to 1 over delta x. I'm just dividing both sides by delta x, times the definite integral from x to x plus delta x of f of t dt. And this is often viewed as the mean value of the function over the interval. Why is that? Well, this part right over here gives you the area, and then you divide the area by the base, and you get the mean height. Or another way you could say it is, if you were to take the height right over here, multiply it times the base, you get a rectangle that has the exact same area as the area under the curve. Well, this is useful, because this is exactly what we got as the derivative of f prime of x.