Volume of a Solid of Revolution 

Rotation about y=2 

? 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 the -axis is rotated about the line . Using the method of (a) disks and (b) shells, find , its volume. 

Solution 

Solution (a) 

In the method of disks when the rotation is about a horizontal axis, the volume of revolution is given by 

 

where and  

 

 

Step	Result
Launch and use the Volume of Revolution Tutor.  Click on Tools, select Tutors> Calculus- Single Variable>Volume of Revolution. Enter and and set a=0 and b=1. Select "Horizontal" for the Line of Revolution, and enter for the distance between the Line of Revolution and the coordinate axis. In plot options, select "Boxed" for axes and select "Use constrained scaling". See Figure 1 below.Press [Display].     The volume computed lies between the red and green surfaces.
Figure 1 Volume of Revolution Tutor used to compute the volume of the solid of revolution generated by rotating the region bound by and the x-axis about
For corroboration, form the integral representing the volume and evaluate.    Use the definite integral template  in the Expression palette to construct  the integral. Press [Enter] to evaluate.	(3.1.1)

 

Solution (b) 

In the method of shells when the rotation is about a horizontal axis, the volume is given by  

 

 

where and  

 

 

Step	Result
Form the integral and evaluate.     Use the definite integral template from the Expression palette to construct  the integral. Remember to change the variable of integration from x to y. Press [Enter].	(3.2.1)

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