# How to Find a Function’s Average Value with the Mean Value Theorem for Integrals

You can find the average value of a function over a closed interval by using the mean value theorem for integrals. The best way to understand the mean value theorem is with a diagram — check it out below.

The graph on the left shows a rectangle whose area is clearly *less* *than* the area under the curve between 2 and 5. This rectangle has a height equal to the lowest point on the curve in the interval from 2 to 5.

The middle graph shows a rectangle whose height equals the highest point on the curve between 2 and 5. Its area is clearly *greater** than* the area under the curve. By now you’re thinking, “Isn’t there a rectangle taller than the short one and shorter than the tall one whose area is *the same as* the area under the curve?” Of course. And this rectangle obviously crosses the curve somewhere in the interval. This so-called *mean value rectangle*, shown on the right, basically sums up the Mean Value Theorem for Integrals.

It’s really just common sense. But here’s the math mumbo jumbo.

**The mean value theorem for integrals:** If *f* (*x*) is a continuous function on the closed interval [*a, b*], then there exists a number *c* in the closed interval such that

The theorem basically just guarantees the existence of the mean value rectangle.

The area of the mean value rectangle — which is the same as the area under the curve — equals *length* times *width*, or *base* times *height*, right?

This height is the *average value* of the function over the interval in question.

Here’s an example. What’s the average speed of a car between *t* = 9 seconds and *t* = 16 seconds whose speed in *feet per se**cond* is given by the function,

According to the definition of average value, this average speed is given by

Determine the area under the curve between 9 and 16.

This area, by the way, is the total distance traveled from 9 to 16 seconds. Do you see why? Consider the mean value rectangle for this problem. Its height is a speed (because the function values, or heights, are speeds) and its base is an amount of time, so its area is

*speed*times*time*which equals*distance*. Alternatively, recall that the derivative of position is velocity. So, the antiderivative of velocity — what you just did in this step — is position, and the change of position from 9 to 16 seconds gives the total distance traveled.Divide this area, total distance, by the time interval from 9 to 16, namely 7.

≈ 105.7 feet per second

It makes more sense to think about these problems in terms of division:

*area*equals*base*times*height*, so the height of the mean value rectangle equals its area*divided*by its base._{ }_{}