# Averages with Functions

### Sample Problem

Let *f*(*x*) be the function graphed below.

We can see that

Find a constant *m* such that

In other words, find a constant m so that we have this:

Answer. . In order to have this equal 16, *m* must be 2.

We call this constant *m* the *average value of f on* [*a, b*]. When we take the integral of *f* on [*a, b*], we get some number. This number is like the sum of all the test scores: it's the accumulation of all the stuff.

To average that accumulation we give every *x* the same function value as every other *x*. Therefore we end up with a constant function whose integral on [*a, b*] is the same as the integral of *f* on [*a, b*].

The average value of *f* on [*a, b*] is a *y*-value. It's the particular *y*-value for which the weighted area between that *y*-value and the *x*-axis is equal to the integral of *f* on [*a, b*]. The average value of *f* on [*a, b*] is the (weighted) height of the rectangle whose (weighted) area is equal to the integral of *f* on [*a, b*].

Let *f* be non-negative for the sake of the pictures and let *m* be the average value of *f* on [*a, b*]. The area under *m* is a rectangle. Whatever area is in that rectangle but not under *f* must make up for the area that is under *f* but not part of the rectangle.