Moments and Centroids
? 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.
The steps in the document can be repeated to solve similar problems.
Problem Statement
Obtain the centroid (geometric center) of the planar region 
bounded by the graphs of 
and 
, for 
.
Solution
The coordinates of the centroid of 
are given by
,
where 
is the area of 
and
and
Step
|
Result
|
Visualize the region  .
Enter the expressions for  and  separated by a comma, and press [Enter]. Right-click on the result. Plots>Plot builder and select 2-D plots and change the default domain to  to  , using the syntax Pi/4 for the right-hand bound.
|
 |
(3.1) |
|
Construct the integrals for  ,  and 
Using the definite integral template in the Expression palette, enter the expressions for  ,  , and  . Press [Enter] to evaluate each integral.
|
 |
(3.2) |
 |
(3.3) |
 |
(3.4) |
|
Compute the coordinates of the centroid.
Using the equation labels ([Ctrl]+L, and enter the equation number), enter the expressions for  and  . Simplify (via the right-click menu) the expression for  .
|
 |
(3.5) |
 |
(3.6) |
 |
(3.7) |
|
Thus, the coordinates for the centroid are 
.
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