Visualizations for Volumes of Solids in Calculus


Objective: The purpose of this demo collection is to help students to understand the concepts that motivate the elements of computation of volumes of solids of revolution.  Rather than introduce volumes of solids of revolution by a purely formula-driven approach, these demos provide the opportunity for visualization of the basic approximating elements that lead to the standard calculus expressions that, when computed, give the desired volumes.  The demos in this collection were developed to provide a toolbox of aids that instructors have found to be effective for teaching students about volumes of solids. We include a variety of approaches which can be easily adapted to different levels. In addition, a collection of animations are included that can be run on a number of platforms.

Level:  These demos can be presented in any course in which volumes of solids are introduced.  

Prerequisites: Students should be familiar with basic planar area computations such as areas of circles, triangles, and rectangles.  In addition, basic volume computations such as volumes of cylinders, volumes of rectangular  parallelepipeds, etc. are useful.  Students should also be familiar with areas of planar regions using approximations by Riemann sums and limits that lead to the definite integral.

Platform: Interactive MATLAB routines and Mathematica notebooks that illustrate volumes modeled using disks and shells are given.  A gallery of animations that run in web browsers is provided.  In addition, several physical props are suggested.  

Instructor's Notes:  We tell our calculus students that the "second" major topic is the "area problem". In examples and demos we stuff rectangles or trapezoids in or around a region and argue that by taking a special type of limit we obtain an integral expression that represents the area of the region. The demos below illustrate the development of areas of planar regions by approximations that lead to Riemann sums (and the definite integral representations of the areas).

A natural extension of the area of a planar region is the volume of a solid. We usually rely on a students visualization skills as we banter about terms like "cross sections of equal area", or "revolve the region about an axis to obtain a solid of revolution". In each of these cases we wave our hands, make (often) crude sketches, and claim that we dissect the solid in such a way that we can sum up the volumes of each piece to obtain an approximation of the volume of the whole solid. We tell students that we can cut the solid into pieces that have a known cross section or, in the case of solids of revolution, we dissect the surface of revolution into "disks", "washers," or "cylindrical shells".

The difficulty with our usual classroom approach is that the visualization skills of many students are not well-developed when it comes to three dimensional objects. To provide better opportunities for a student to improve such skills we present a collection of demos that provide a variety of instructional aids. These include physical objects, computer generated sketches, and computer generated animations. Combinations of such aids can provide students with an opportunity to sharpen visualization skills and then to make informed decisions on how to proceed with the calculations for volumes of solids encountered in an introductory calculus course.

Volumes by Section deals with solids that can be sliced into pieces with a known cross section.  While this broad class of solids includes solids of revolution that can be sliced into disks or washers, in this demo we focus on solids that are not necessarily solids of revolution.  The kth approximating element has volume computed by multiplying the area of the cross section, Ak,  by the thickness so the volume of the approximating element is

.

The Disk Method for Volumes of Solids of Revolution concerns solids generated when a planar region is revolved about the x-axis.  In particular, if a region bounded a curve y = f(x) and the domain interval [a,b] is revolved about the x-axis, the resulting solid may be sliced into disks.  The kth disk has radius rk and thickness so for a solid generated by revolving about the x-axis, the volume of the kth approximating element is

.

Solids of Revolution: The Method of Shells involves revolving a region bounded by a curve y = f(x) and the domain interval [a,b] about the y-axis.  The solid is then filled with cylindrical shells.  The kth approximating element is a cylindrical shell that has average radius rk, height hk, and thickness .  Thus,  for a solid generated by revolving about the y-axis, the kth approximating element has volume

.

This demo also includes method of shells visualizations for the solid of revolution formed by revolving a region bounded by two curves about the y-axis. 

The Washer Method for Volumes of Solids of Revolution involves revolving a region revolved about one of the coordinate axes.  In this demo, the resulting solid has a "hole."  The kth approximating element is a washer that has inside radius rin,  outside radius rout, and thickness .  Thus for a solid generated by revolving about a coordinate axis, the kth approximating element has volume 

.

The symbolic form of the volume element depends on whether the axis of revolution is the x-axis or y-axis.  Visualizations for revolution about both x and y axes are provided.

A gallery of animations has been developed to accompany the demos.  In addition to animated gifs that run in a browser, movies (mov format) are included. 


References

1.  Carol M. Critchlow,  "A Prop is Worth Ten Thousand Words," Mathematics Teacher, 92(1), Jan. 1999, pp 27-29.

2.  Theresa Reardon Offerman, "Foam Images," Mathematics Teacher, 92 (5), May 1999, pp 391-399.

3. James Rahn, "Giving Meaning to Volume in Calculus," Mathematics Teacher, 84 (2), Feb. 1991, pp 110-112.

4. Judith Schimmel, "A New Spin on Volumes of Solids of Revolution," Mathematics Teacher, 90 (9), Dec. 1997, pp 715-717.


Credits:  This demo collection was organized by Dr. David R. Hill and Dr. Lila F.  Roberts.  MATLAB files to accompany the demos were written by David Hill; Mathematica notebooks were written by Lila Roberts.

David R. Hill
Mathematics Department
Temple University
Philadelphia, PA

Lila F. Roberts
College of Information & Mathematical Sciences
Clayton State University
Morrow, GA 30260

The gallery of animations were generated by Drs. Hill and Roberts.

LFR 5/15/04     Last updated 9/15/2009 DRH