Mean Value Theorem for Integrals 

Copyright Maplesoft, a division of Waterloo Maple Inc., 2007 

Introduction 

This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  Click on the buttons to watch the videos. 

 

This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem. 

 

Problem Statement 

In essence, the Mean Value Theorem for Integrals states that a continuous function on a closed interval attains its average value on the interval.  Thus, if is the average value, then 

 

for some in .  This is generally written as .   

Verify this last equality for , . 

Solution 

 

Step	Result
Enter the function.    Use the function template from the Expression palette to construct the function.Press [Enter].	(3.1)
Using the definition in the Problem Statement section, find     Use the definite integral template from the Expression palette to write the integral, remembering that f has been defined as a function.	(3.2)
Construct the equation and solve for the variable .    Enter the equation on a new line and execute.  To solve, right click and select Solve>Obtain Solutions for> c	(3.3)     (3.4)
For the values of obtained, compute .  Use the Expression palette to construct the integral; press [Enter] to compute its value.	(3.5)
Compare the value of the integral to     Evaluate at by referencing through the equation label to which the indexing notation is appended.  Press [Enter], then  Right click and select Simplify.        Repeat for the other value of replacing by .	(3.6)     (3.7)     (3.8)

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