SHELL METHOD DEMO GALLERY

The following is a gallery of demos that can be used to illustrate the shell 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 shell method.

Two formats, animated gif and mov are provided for the method of shells 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 the y-axis identifies each example.   An animated gif shows how a region is partitioned and a typical shell is generated. Another animation shows a sequence of seven shells generated to approximate the volume.  A third animation illustrates the generation of the solid by revolving the region about the 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 Bounded Between a Curve and an Axis on an Interval [a,b]:

 Region Partition/Shell Shells Surface Movie Downloads Region bounded between y = 0, x = 0, y = 1,  y = x2 +1. View the animation. View the animation. View the animation. Formats mov  Quicktime Region bounded between y = 0, y = sin(x), x = 0, x = pi. View the animation. View the animation View the animation. Formats mov  Quicktime gif Region bounded between y = 0, y = -12+8x-x2, x = 2, x = 6. View the animation. View the animation. View the animation. Formats mov  Quicktime Region bounded between y = 0, x = 0, y = -2x+3. View the animation. View the animation. View the animation. Formats mov  Quicktime Region bounded between y = 0,  y = sin(x),x = pi/2, x = pi. View the animation. View the animation. View the animation. Formats mov  Quicktime

Regions Bounded Between Two Curves
 Region Partition/Shell Shells Surface Movie Downloads Region bounded between , y = x2. View the animation. View the animation. View the animation. Formats mov  Quicktime Region bounded  between y = 1/2 x, y = (x-1)2. View the animation. View the animation. View the animation. Formats mov  Quicktime gif

Notes: The sequence of images in the movies above follow a specific pattern that describes the method of shells. 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 Bounded Between a Curve and an Axis on an Interval [a,b]:

• Sketch the curve y = f(x) over interval [a ,b]. Name the region to be revolved about the x-axis R.
• Partition the region R into strips (7 strips of width (b-a)/7 are used in the examples).
• Select a representative strip S and construct a rectangle using its width and height computed by evaluating the function at some point in the subinterval. (The example uses midpoints to construct the height).
• Revolve the rectangle about the y-axis to form a cylindrical shell.
• Construct shells that correspond to each strip.  Because some of the shells may become obscured if the complete shell is drawn, half-shells are drawn first.  Then the shells 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 shells.  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 shells.

Regions Bounded Between Two Curves

• Sketch the curves y = f(x) and y = g(x), where f(x) > = g(x). Name the region bounded by the curves to be revolved about the x-axis R
• Partition the region R into strips (7 strips of width (b-a)/7 are used in the examples).
• Select a representative strip S and construct a rectangle using its width and height computed by evaluating f(x) - g(x) at some point in the subinterval. (The example uses midpoints to construct the height).
• Revolve the rectangle about the y-axis to form a cylindrical shell.
• Construct shells that correspond to each strip.  Because some of the shells may become obscured if the complete shell is drawn, half-shells are drawn first.  Then the shells 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 shells.  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 shells.

LFR 5/15/02         last updated DRH 5/24/2006