Volumes of Solids of Revolution--The Method of Shells

This Mathematica notebook provides the code to produce animations to illustrate the steps involved 
in the shell method for finding volumes of solids of revolution.  By making suitable modifications, it is 
possible to change the region to be revolved.  This notebook generates a region and revolves it 
about the y-axis.  

Documentation within the notebook illustrate what parts of the code should be changed so that different 
regions might be used.  This notebook is an example only: it is not intended to cover every possible region
you may wish to use.  You may have to make additional modifications to suit the particular region with
which you are working.
  By studying the code and through experimentation you should be able to modify
the code as your example requires.

Because it takes a significant time to render the graphics, it is NOT RECOMMENDED that you run the
program in class in real time unless you have a fast processor.  It is easier to generate the graphics ahead
of time and then select the frames to animate.  The graphics consume a large amount of space so the 
Mathematica
notebook is saved without the embedded graphics.

NOTE:  It is possible to modify the code so that the axis of revolution is the x-axis.

DOWNLOAD the Mathematica notebook HERE.

Required Packages:

[Graphics:Images/shells_yaxis_gr_1.gif]

Turn off spelling warning (merely a convenience).

[Graphics:Images/shells_yaxis_gr_2.gif]

Revolution axis is the y-axis.

[Graphics:Images/shells_yaxis_gr_3.gif]

Define the functions.  In this case the region is bounded by two curves.  Representation as a function of y
AND as a function of x are required.

[Graphics:Images/shells_yaxis_gr_4.gif]

x interval:  This needs to be modified specifically for the functions in your example.

[Graphics:Images/shells_yaxis_gr_5.gif]

Set the plotting range window and the viewpoint.  These statements need to be modified to suit the functions
in your example.

[Graphics:Images/shells_yaxis_gr_6.gif]

Plot x and y axes.

[Graphics:Images/shells_yaxis_gr_7.gif]

Establish the partition (7 in this example).

[Graphics:Images/shells_yaxis_gr_8.gif]

Define the "front" edges of the rectangles that generate the shells.

[Graphics:Images/shells_yaxis_gr_9.gif]

Define the surfaces of revolution.

[Graphics:Images/shells_yaxis_gr_10.gif]
[Graphics:Images/shells_yaxis_gr_11.gif]
[Graphics:Images/shells_yaxis_gr_12.gif]

The region is illustrated using an inscribed 50-vertex polygon between the curves. The vertices are generated
and stored as a list. The region is generated by constructing a polygon defined by the vertices.

[Graphics:Images/shells_yaxis_gr_13.gif]

Define the surfaces of revolution that construct the shells.

[Graphics:Images/shells_yaxis_gr_14.gif]
[Graphics:Images/shells_yaxis_gr_15.gif]
[Graphics:Images/shells_yaxis_gr_16.gif]
[Graphics:Images/shells_yaxis_gr_17.gif]

The surfaces will be displayed as Graphics3D primitives that have specific colorings.

[Graphics:Images/shells_yaxis_gr_18.gif]

The approximating shells are stored in a table.

[Graphics:Images/shells_yaxis_gr_19.gif]

The surfaces of revolution are generated by varying the value of theta in the SurfaceOfRevolution and
ListSurfaceOfRevolution statements.

[Graphics:Images/shells_yaxis_gr_20.gif]

The following code shows the steps (with pauses between the major steps) in the approximation process, 
1) a partition is generated, 2) a typical approximating element is generated by revolving a rectangle about
the y-axis, 3) washers are generated on each sub-interval with half-washers generated first, 4) the solid of
revolution is displayed.

[Graphics:Images/shells_yaxis_gr_21.gif]

Mathematica code produced for Demos with Positive Impact (NSF-DUE 9952306) by

Lila F. Roberts
Mathematics and Computer Science Department
Georgia Southern University
Statesboro, GA  30460
lroberts@gasou.edu.
Copyright 2002.  All Rights Reserved.


Converted by Mathematica      May 19, 2002