Volumes of Solids of Revolution—Graphing Guide

To produce rotatable 3-D graphs of solids of revolution, please use the Mathematics Visualization Toolkit (MVT) (http://amath.colorado.edu/java/) and take the following steps:

•	Open the MVT software (you may run this directly on their server or download it to your personal computer and run it from there)
•	Use the access path:  Applications/Solids of Revolution

When the Graphing Application opens, 

1)	Enter f(x) = 4x – x^2 
2)	Enter x = 0 to 4
3)	Enter a desired axis of rotation, either f(x)=c or x=c for real c
4)	Select a volume method, either Disks or Shells
5)	Click on Plot to see the graph, axis of rotation and a representative rectangle
6)	Click on Rotate to see the resulting solid of revolution (actually only the outer surface is displayed)
7)	Select the Rotate tool in the Toolbox on the right and then click and drag on the 3-D graph to rotate it.
8)	Repeat steps 3 through 7 for each of the activity exercises

Note:  The MVT software revolves only curves but not regions as presented in the work-pair-share activity.  Consequently, only the graphs for activity exercises #1, 4 and 5 accurately represent the rotated regions.  

In exercise #2, the graph is an accurate depiction of the desired solid provided that the cylinder of radius 2 at the center of the graph is not included.  The MVT software uses a representative rectangle that extends below the x-axis and hence calculates the volume of the entire space inside the resulting graph; the volume of the desired region, however, does not include the cylinder of radius 2 in the middle of the solid. To compensate for this, the volume of this cylinder (50.27) must be subtracted from the total MVT volume (291.54) in order to obtain the volume of the desired solid (241.27).

In exercise #3, the graph is a depiction of the solid generated by the inner radius and hence represents the volume (157.50) that would be subtracted from the volume generated by the outer radius (452.39) in order to obtain the volume of the desired solid (294.89); the outer radius in this case is 6.  The Gallery of Volume Visualizations provides some examples of similar solids of revolution and is a helpful resource (see References and Resources; http://mathdemos.gcsu.edu/mathdemos/solids/gallery.html) 

Error Notice:  The MVT volume calculations for exercises #4 and #5 are incorrect; there appears to be an error in the computation algorithm for the shell method.