Volumes of Solids of RevolutionThe Washer Method
This Mathematica notebook provides the code to produce animations to illustrate the steps involved in the
washer method for finding volumes of solids of revolution. By making suitable modifications, it is possible
to change the region to be revolved. This particular notebook generates a region and revolves it about the
xaxis.
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.
Download the Mathematica notebook HERE.
Required Packages:
Turn off spelling warning (merely a convenience).
Revolution axis is the xaxis.
Define the functions. In this case the region is bounded by a portion of a parabola and a line.
Representation as a function of x is required.
x interval: This needs to be modified specifically for the functions in your example.
Set the plotting range window and the viewpoint. These statements need to be modified to
suit the functions in your example.
Plot the x and y axes. Options should be changed depending on your functions.
Establish the partition (7 in this example).
Define the "front" edges of the rectangles that generate the washers.
Define the surfaces of revolution. They consist of 1) curve y = f(x), 2) y = g(x) 3) inside edge.
The region is illustrated using an inscribed 50vertex 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.
Define the surfaces of revolution that construct the washer.
The surfaces will be displayed as Graphics3D primitives that have specific colorings.
The approximating washers are stored in a table.
The surfaces of revolution are generated by varying the value of theta in the SurfaceOfRevolution and
ListSurfaceOfRevolution statements.
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 xaxis, 3) washers are generated on each subinterval with halfwashers generated first, 4) the solid of
revolution is displayed.
Mathematica code produced for Demos with Positive Impact (NSFDUE 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 16, 2002
