|
|
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
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
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
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
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
The gallery of animations were generated by Drs. Hill and Roberts. |
|
|
LFR 5/15/04 Last updated 5/19/2006 DRH
so the volume of the approximating element is 
.
.
.
.
Thus for a solid generated by revolving about a coordinate axis, the kth
approximating element has volume 
.
