The Disk Method for Volumes of Solids of Revolution

 
Objective: To provide a toolbox of aids for teaching students about the volume of solids by the disk method. We include a variety of approaches which can be easily adapted to different levels of instruction. In addition, a collection of animations is included which can be run on a number of platforms.

Level: Calculus courses in high school or college.

Prerequisites: An introduction to Riemann sums and the use of integrals to determine the area of a planar region. Basic ideas about volume.

Platform: We reference demonstration routines in Java, MATLAB, Mathematica, and Maple. However, there is also a collection of animations which can be viewed independent of a particular software platform.

Instructor's Notes: For this demo we consider a region R bounded by a (continuous) curve y = f(x) over an interval [a, b], the x-axis, and possibly one or two vertical lines x = a and x = b. We further require that f(x) be nonnegative over [a, b]. Figure 1 displays several regions of this type.

                                                 Figure 1.

The region R will be revolved about the x-axis to obtain a solid of revolution. A major objective is to provide visualization demonstrations to enable students to more clearly understand the procedure for the computation of the volume of the resulting solid by the disk method. 

Using Props   ('A prop is worth ten thousand words', see [x].)

A variety of props, objects you can handle, are quite useful for helping students visualize surfaces of revolution and the components of the disk method.

A sheet of paper and a length of dowel. Take a sheet of paper, fold it in half. Now cut a shape of a curve along the open end through both layers. Staple the folded sheet in several places along the curve you cut. Insert a length of dowel wood along the fold to act as an axis of revolution. See Figure 2.

Figure 2.

Now you are set to have your students see you create a solid of revolution. (Of course they will need to use their imagination a bit.) In class you point out the planar region which is the paper cut out. Next rotate the region about the dowel (or better yet have a student do it). Ask for a description of the solid generated by rotating the region. You may want to lead them to indicate a vase shaped object. A vase created on a potter's wheel is a nice example of solid of revolution generated by rotation about a vertical axis.

Make a vase.

                                             Figure 3. 

 

Wedding bell.

The common paper wedding bell (see Figure 4) nicely illustrates a solid of revolution. It is easily transported to the classroom and can be used both by the instructor and the student. 

Figure 4.

Figures 5 - 7 illustrate the generation of a solid of revolution about a horizontal axis. Here we held the bell against a vertical board and let it open by itself. The paper folds uncompress illustrating the rotation of the planar region shown in Figure 5 into the 'solid bell'.

Figure 5.

Figure 6.

Figure 7.

Fresh fruits and vegetables.

The produce section of your favorite market is a good source of solids of revolution. Cucumbers and squash illustrate solids of revolution of the type discussed in this demo. For simple illustrations we often draw curves like those in Figures 8 and 9 on a set of axes and ask students to imagine the surface generated as we revolve the region R about the horizontal axis.

  Figure 8.

Figure 9.

The solid generated by revolution by the region R in Figure 8 looks like a cucumber; see Figure 10. The solid produced  from the region R in Figure 9 looks like a squash; see Figure 11.

Figure 10.

 Figure 11.

Other produce like oranges and melons also provide nice models for solids of revolution. See Figures 12 and 13.

Figure 12.

Figure 13.

The Disk Method.

For the regions R in this demo we "slice" the planar region into strips of area and revolve the strips around the x-axis to generate "disks". The disks are cylinders which approximate the volume of the actual slices of the solid generated by the revolution of region R. Figure 14 shows a curve and then one strip of area which generates the cylindrical disk displayed in Figure 15. The circular faces of the cylindrical disk are called the cross sections of the solid.

Figure 14.

Figure 15.

Using our cucumber prop shown in Figure 10 a collection of disks is shown in Figure 16. The actual volume of a slice of cucumber is approximated by the volume of a cylindrical disk as in Figure 15. The volume of the disk in 

Figure 16.

Figure 15 is computed as the area of the circular face of the disk times it thickness. The area of the circular face is p times (radius)2 . In Figure 17 we show how this relates to the curve from Figure14. We see that the radius is a value of the function f(x) describing the curve and the thickness is an increment along the x-axis.

Figure 17.

In the general case we point out that the solid is sliced into approximating cylindrical disks. The kth disk has radius rk and thickness Dxk so for a solid generated by revolving about the x-axis, the volume of the kth approximating disk is 

.

Summing the volumes of the disks as we require the thickness to get smaller and smaller leads to the standard integral formula for the disk method:

.

We illustrate the steps of the process described above with the following animation which generates a solid like the wedding bell in Figure 7. 

A small gallery of demos for illustrating the disk method for volumes of solids of revolution is available by clicking on DISK-METHOD-GALLERY. These animations can be used by instructors in a class room setting or by students to aid in acquiring a visualization background relating to the steps of disk method. The demos provide a variety of animations for some common examples. Also included is a step-by-step narrative script of the displays in the animations with pictures that illustrate a step.

Classroom Activities:

  • For a glimpse of the disk method in the classroom go to Jim Rahn's calculus activities page at

 http://www.jamesrahn.com/CalculusI/Activities/disk_method.htm

Here you will see Jim's approach to the disk method using apples and oranges. You will also see his class having "tasteful fun" finding volumes by the disk method.

  • An informative discussion about using props as teaching aids is in  Carol Critchlow's paper 'A prop is worth ten thousand words', Mathematics Teacher, Vol. 92, No. 1, January 1999. 

  • For a description of a good hands-on project involving volumes of revolution see Judith Schimmel's paper 'A New Spin on Volumes of Solids of Revolution', Mathematics Teacher, Vol. 90, No.9, December 1997. This work incorporates modeling and employs both a calculator and computer software.

Technology Resources:

There are a variety of resources that employ calculators or software for illustrating and computing volumes of solids of revolution. Following is a sample of such resources which can be located using a search engine. We have chosen ones that relate to the disk method.

  • A MATLAB routine, diskmethod.m, for illustrating the basic steps of the disk method is available for download by clicking here. To see a description of the routine click here. Animations used in this demo were generated with diskmethod.m by capturing the evolving pictures and then editing the video files into various formats. The routine was written by David R. Hill and Lila F. Roberts.

  • A Mathematica notebook for illustrating the basic steps of the disk method is available in html by clicking here. The notebook can be downloaded from within the html version. The notebook was constructed by Lila F. Roberts.

The development in this project leads students to compute the volume of a cup (or vase) as illustrated in Figure 3 above.


Credits:  This demo was constructed by David R. Hill for Demos with Positive Impact. 

Dr. David R. Hill
Department of Mathematics 
Temple University

 


DRH 2/04/02     last updated 5/19/2006 DRH

Since 3/1/2002