DISK METHOD DEMO GALLERY
The following is a small gallery of demos for illustrating the disk method for volumes of solids of revolution. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps of disk method. Three file formats, gif, mov, and avi are available.
The gif animations should run on most systems and the file sizes are relatively small.
The mov animations require the QuickTime Player (version 5 or newer) which is a free download available by clicking here; these file are also small. (The mov files may not execute properly in older versions of QuickTime.)
The avi files were saved with full frames uncompressed hence are quite large and thus are only offered in a zipped form.
The collection of animations in gif and mov format can be downloaded; see the 'bulk' zipped download category at the bottom of the following table.
|f(x)=x^2||click to see gif animation||click to see .mov file||click to download zipped avi file|
|f(x)=sqrt(x)||click to see gif animation||click to see .mov file||click to download zipped avi file|
|f(x) = sin(x)||click to see gif animation||click to see .mov file||click to download zipped avi file|
|a polynomial||click to see gif animation||click to see .mov file||click to download zipped avi file|
|f(x) = x^2|sin(2px)|||click to see gif animation||click to see .mov file||click to download zipped avi file|
|f(x) = exp(-x)|cos(2px)|||click to see gif animation||click to see .mov file||click to download zipped avi file|
|Zipped 'bulk' downloads ==>||click to download all the gif animations||click to download all the .mov files|
Notes: The sequence of images in the animations above follow a particular pattern. This pattern when viewed can be adapted to form a script which can be used to narrate the animation for a class or provide a guide for students to viewing the animations on their own.
- Sketch the curve y = f(x) over interval [a ,b]. Name the region to be revolved about the x-axis R.
- Restriction: To use the disk method to compute the volume of the solid obtained by revolving R about the x-axis the region must be enclosed from top-to-bottom (or bottom-to-top) by f(x) and the x-axis throughout the interval [a,b]. To see examples of regions that meet this requirement and others that do not, click here.
- Divide the region R into strips.
- Select a representative strip S and construct a rectangle using its width and a value of f(x) in the strip.
- Imagine that you revolve the rectangle about the x-axis to form a cylindrical slice.
- Repeat the preceding step for each strip.
- The sum of the volumes of the cylindrical slices approximate the volume of the solid of revolution.
- We refine the approximation by choosing more strips with smaller width. This performs a limiting process and constructs Riemann Sums.
- The limit of the Riemann Sums leads to the integral expression for the volume of the solid of revolution by the disk method.
DEMOS with POSITIVE IMPACT, NSF DUE 9952306
by David R. Hill, Temple University and Lila F. Roberts, Georgia Southern University.
DRH 2/24/02 last updated 5/19/2006 DRH