WASHER METHOD DEMO GALLERY

The following is a gallery of demos that can be used to illustrate the washer method for computing volumes of solids of revolution.  These animations are designed to be used by the instructor in a classroom setting or by students as they acquire a visual background relating to solids of revolution and the steps of the washer method.  

Two formats, animated gif and mov, are provided for the method of washers examples.  

1. The animated gif images can be viewed in virtually any browser or other application for displaying animated gifs (such as Quicktime Player).  It is more advantageous to replay the animations in an application that provides playback control buttons so that the animation can be stopped for discussion.

2. The mov files can be played in Quicktime Player (version 5 or later).  They may not play properly in older versions.

Each example in the gallery is organized the following way for maximum flexibility for your in-class demo.  A static picture of the region to be revolved about an axis identifies each example.   An animated gif shows how a region is partitioned and a typical washer is generated. Another animation shows a sequence of seven washers generated to approximate the volume.  A third animation illustrates the generation of the solid by revolving the region about the x- or y-axis.  These can be shown in any order.  Finally, movies are provided that contain all the frames of the previous animations.  

Because of the complexity of the graphics (generated by Mathematica) and the video compression, the mov files are fairly large (1 to 3 MB).  To download the files, right click on the link to see the download dialog box.

Regions Revolved about the x-axis

Notes: The sequence of images in the movies above follow a specific pattern that describes the method of washers. The pattern provides the basis for a script that can be used to narrate the action.  In addition to an in-class demo, this script can provide a guide for students who view the animations on their own.

Regions Revolved about the x-axis:

• Sketch the region R to be revolved about the x-axis. 

• Partition the region R into vertical strips (7 strips are used in the examples).

• Select a representative strip S and construct a rectangle using its width and height.  Note that the height is of the form f_top(x) - f_bottom(x) evaluated at some point in the subinterval. (The example uses midpoints to construct the height of the rectangles).

• Revolve the rectangle about the x-axis to form a washer.

• Construct washers that correspond to each strip.  Because some of the washers may become obscured if the complete washer is drawn, half-washers are drawn first.  Then the washers are completed.

The movies also show the solid generated by revolving R about the x-axis.

The volume of the solid is approximated by evaluating the sum of the volumes of the approximating washers.  By choosing more strips with smaller width, the approximation is refined.  Evaluating the limit of the Riemann Sums as the number of strips increases without bound leads to the integral expression for the volume of the solid of revolution using the method of washers. 

Regions Revolved about the y-axis

• Sketch the region R to be revolved about the y-axis.. 

• Partition the region R into strips (7 strips are used in the examples).

• Select a representative strip S and construct a rectangle using its width and length.  Note that the length is of the form g_right(y) - g_left(y) at some point in the subinterval. (The example uses midpoints to construct the length of the rectangles).

• Revolve the rectangle about the y-axis to form a washer.

• Construct washers that correspond to each strip.  Because some of the washers may become obscured if the complete washer is drawn, half-washers are drawn first.  Then the washers are completed.

The movies also show the solid generated by revolving R about the y-axis.

The volume of the solid is approximated by evaluating the sum of the volumes of the approximating washers.  By choosing more strips with smaller width, the approximation is refined.  Evaluating the limit of the Riemann Sums as the number of strips increases without bound leads to the integral expression for the volume of the solid of revolution using the method of washers.

In some cases, there is no single representative strip, illustrated by revolving the region bounded between x = 0, y = 0, y = x²+1, and x = 1 about the y-axis.  Situations such as these require dividing the region into parts and using appropriate approximating elements (and corresponding volume approximation) in each of the parts.  In this example, when y is between 0 and 1, the approximating elements are disks.  When y is between 1 and 2 the approximating elements are washers.

LFR 5/15/04     last updated DRH 5/25/2006