Volumes of Solids of Revolution:  The Disk Method

This Mathematica notebook provides code for visualization of the disk method for computing the volume of a solid of revolution.  The examples involve revolving the region bounded between y = f[x], x = a, and x = b revolved about the x-axis. Further, we assume that f[x] is nonnegative over the interval.

The function is plotted.  The region is sliced and rectangles that determine the approximating disks are drawn.  Finally, the surface is shown together with the approximating disks.  Volume approximations and computations are not included; this demo focuses only on the visualization.

You may need to change the PlotRange options to optimize the effectiveness of the display.

Three-dimensional graphics objects can take awhile to render, however after each frame has been rendered, the animations play nicely.

[Graphics:Images/disk_mma_gr_1.gif]
[Graphics:Images/disk_mma_gr_2.gif]

n is the number of disks.

[Graphics:Images/disk_mma_gr_3.gif]
[Graphics:Images/disk_mma_gr_4.gif]

[Graphics:Images/disk_mma_gr_5.gif]

[Graphics:Images/disk_mma_gr_6.gif]

Dissect the region into rectangles that determine the disks.  Radii are determined by the function evaluated at the midpoint of each subinterval.

[Graphics:Images/disk_mma_gr_7.gif]
[Graphics:Images/disk_mma_gr_8.gif]
[Graphics:Images/disk_mma_gr_9.gif]

Show the radii; then draw the rectangles.

[Graphics:Images/disk_mma_gr_10.gif]
[Graphics:Images/disk_mma_gr_11.gif]

Plot the surface.

[Graphics:Images/disk_mma_gr_33.gif]

[Graphics:Images/disk_mma_gr_34.gif]

[Graphics:Images/disk_mma_gr_35.gif]

Generate the cross sections.

[Graphics:Images/disk_mma_gr_36.gif]
[Graphics:Images/disk_mma_gr_37.gif]
[Graphics:Images/disk_mma_gr_38.gif]
[Graphics:Images/disk_mma_gr_39.gif]

Generate the cylinders' edges.

[Graphics:Images/disk_mma_gr_40.gif]

Show approximating disks together.

[Graphics:Images/disk_mma_gr_41.gif]

Show surface and disks.

[Graphics:Images/disk_mma_gr_53.gif]

[Graphics:Images/disk_mma_gr_54.gif]

[Graphics:Images/disk_mma_gr_55.gif]
DOWNLOAD THE MATHEMATICA NOTEBOOK:  diskmethod.nb

Mathematica notebook by

Lila F. Roberts
Mathematics  Department
Georgia College & State University
Milledgeville, GA  31061

lila.roberts@gcsu.edu


Converted by Mathematica      September 23, 2001   Last updated 5/19/2006  DRH