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 reexecute 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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