We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Definite Integrals

Definite Integrals

At a Glance - 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.

Example 1

Let f (x) = 4 – x2. Is the average value of f on [-2, 2]

  • strictly between 0 and 2?
  • equal to 2?
  • strictly between 2 and 4?
  • equal to 4?

Example 2

Calculate the average value of  on [0, 3].


Exercise 1

Let f (x) = 3x. Find a constant m such that


Exercise 2

Let f (x) = x. Find a constant m such that


Exercise 3

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  •  f (x) = x on [-10, 0]

Exercise 4

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  •  on [-2, 2]

Exercise 5

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  • f (x) = sin x on [-π, π]

Exercise 6

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  • f (x) = 3x + 2 on [1, 4]

Exercise 7

Find the average value of the function on the specified interval. Check and make sure your answer has the sign you would expect.

  •  f (x) = -2x – 1 on [-4, 2]

People who Shmooped this also Shmooped...