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.