Calculating the average value of a function over a interval requires using the definite integral. The exact calculation is the definite integral divided by the width of the interval. This calculates the average height of a rectangle which would cover the exact area as under the curve, which is the same as the average value of a function.
I want to talk about the average value of a function. Now you've probably learned how to take the average of 2 numbers. You add them and divide by 2 or 5 numbers. Add them all up and divide by 5. But when you have a function, a function has infinitely many values over any interval. And so how do you add those up and divide by infinity? That's not how we find average value. We find it the slightly different way.
Imagine this is the function we want to average y=f of x. And we want to average it over this interval from a to b. Think about the area under the curve, that area is going to be determined by the value of the integral of this function from a to b. Now imagine that this region is a body of water sitting in a tank, and when the water settles it'll find a level. The height if that level is going to be our average value f bar.
So we we call f bar the average value of f of x on the interval a, ab. And we define f so that the area of this rectangle is exactly equal to the area under the curve, this green area here. The area of the rectangle is whatever height this is, f bar times b-a this width, that's the rectangle.
On the right side we have the area of the curve which is the definite integral, right? The area of this green region, the definite integral of f from a to b. All we have to do to get a formula for the average value is divide both sides by this and that leads us to the average value of a function formula. The average value of a function f on the interval from a to b is 1 over b minus a, the width of the interval times the integral from a to b of f of x.