Volume of a Solid of Revolution 

Rotation about y=0 

? 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. 

Problem Statement 

A solid of revolution is formed when the region bounded by the curves and is rotated about the x-axis.  Using the method of (a) disks, and (b) shells, find , its volume.  

Solution 

Solution (a) 

The volume of revolution is given by  

 

 

where is the radius of rotation for the solid.  Here, . 

 

 

Step	Result
Launch and use the Volume of Revolution Tutor to compute the volume.  Tools>Tutors> Calculus- Single Variable>Volume of Revolution.The default axis of rotation is horizontal with there no displacement from the coordinate axis. Enter , set a=0 and b=1. In plot options, select "Boxed" for axes and select "Use constrained scaling". Press [Display]. See Figure 1 below.
Figure 1 Volume of Revolution Tutor used to compute the volume of the solid of revolution generated by rotating the region bounded by and about the x-axis
Investigate the method of disks using the Volume of Revolution Tutor.     Tools>Tutors> Calculus- Single Variable>Volume of Revolution.Enter the information as before. Select "disks" for the Display option. Press [Display]. See Figure 2 below.
Figure 2 Volume of Revolution Tutor used to compute the volume of the solid of revolution generated by  rotating the region bound by and about the x-axis
Form the integral and evaluate for corroboration.     Use the definite integral template in the Expression palette to form the integral. Press [Enter] to evaluate the integral.	(3.1.1)

Solution (b) 

Using shells, the volume is given by  

 

where and . 

 

Step	Result
Form the integral and evaluate.    Use the definite integral template in the Expression palette to form the integral. Press [Enter] to evaluate the integral.	(3.2.1)

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