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Solids of Revolution

Introduction

The purpose of this lab is to use Maple to study solids of revolution. Solids of revolution are created by rotating curves in the x-y plane about an axis, generating a three dimensional object. They are discussed in Chapter 6 of Calculus by Bradley and Smith (sections 1 and 2). The specific properties of them that we wish to study are their volume, surface area, and graph.

Background

So far we have used the integral mainly to to compute areas of plane regions. It turns out that the definite integral can also be used to calculate the volumes of certain types of three-dimensional solids. The class of solids we will consider in this lab are called Solids of Revolution because they can be obtained by revolving a plane region about an axis.

As a simple example, consider the graph of the function f(x) = x²+1 for $-2\leq x \leq 2$ , which appears in Figure 1.

**Figure 1:** Plot of f(x)=x²+1.
$\begin{figure} \centerline{ \psfig {file=volrev_fig1.ps,height=4.0in,width=2.0in,angle=-90} }\end{figure}$

If we take the region between the graph and the x-axis and revolve it about the x-axis, we obtain the solid pictured in Figure 2.

To help you in plotting surfaces of revolution, A Maple procedure called revolve has been written. The command used to produce the graphs in Figures 1 and 2 is shown below. The revolve procedure, as well as the RevInt, LeftInt, and LeftDisk procedures described below are all part of the CalcP package, which must be loaded first. The last line in the example below shows the optional argument for revolving the graph of f(x) about the line y=-2 instead of the default y=0.

  > with(CalcP):

  > f := x -> x^2+1;

$\begin{maplelatex} \begin{displaymath} {f} := {x} \rightarrow {x}^{2} + 1\end{displaymath}\end{maplelatex}$

  > plot(f(x),x=-2..2);

  > revolve(f(x),x=-2..2);

  > revolve(f(x),x=-2..2,y=-2)

**Figure 2:** Solid generated by rotating f(x)=x²+1 about the x-axis.
$\begin{figure} \centerline{ \psfig {file=volrev_fig2.ps,height=4.0in,width=2.5in,angle=-90} }\end{figure}$

The revolve command has other options that you should read about in the help screen. For example, you can speed the command up by only plotting the surface generated by revolving the curve with the nocap argument, and you can also plot a solid of revolution formed by revolving the area between two functions. Try the following examples. (Note: The last example shows how to use revolve with a function defined piecewise.)

  > revolve({f(x),0.5},x=-2..2,y=-1);

  > revolve(cos(x),x=0..4*Pi,y=-2,nocap);

  > revolve({5,x^2+1},x=-2..2);

It turns out that the volume of the solid obtained by revolving the region in Figure 1 between the graph and the x-axis about the x-axis can be determined from the integral

$\begin{displaymath} \pi \int_{-2}^2 (x^2+1)^2 \, dx\end{displaymath}$

to have the value $\frac{412}{15} \pi$ . More generally, if you revolve the area under the graph of g(x) for $a \leq x \leq b$ about the x-axis, the volume is given by

$\begin{displaymath} \pi \int_{a}^{b} (g(x))^2 \, dx.\end{displaymath}$

Where does this formula come from? To help you understand it, Two more Maple procedures, RevInt and LeftDisk, have been written. The procedure RevInt sets up the integral for the volume of a solid of revolution, as shown below. The Maple commands evalf and value can be used to obtain a numerical or analytical value.

  > RevInt(f(x),x=-2..2);

$\begin{maplelatex} \begin{displaymath} { \pi}\,{\displaystyle \int_{-2}^{2}} (\,{x}^{2} + 1\,)^{2}\,{d}{ x}\end{displaymath}\end{maplelatex}$

  > value(RevInt(f(x),x=-2..2));

$\begin{maplelatex} \begin{displaymath} {\displaystyle \frac {412}{15}}\,{ \pi}\end{displaymath}\end{maplelatex}$

  > evalf(RevInt(f(x),x=-2..2));

$\begin{maplelatex} \begin{displaymath} 86.28907824\end{displaymath}\end{maplelatex}$

Finding Volumes of Revolution

Since this involves straightforward integration, one may simply issue commands such as:

  > f:= x-> sqrt(x) +1;

$\begin{maplelatex} \begin{displaymath} f := x \rightarrow \sqrt(x) + 1\end{displaymath}\end{maplelatex}$

  > vol:= int(Pi*f(x)^2, x=0..9);

$\begin{maplelatex} \begin{displaymath} vol := \frac{171}{2}\;\pi\end{displaymath}\end{maplelatex}$

  > evalf(vol);

$\begin{maplelatex} \begin{displaymath} 268.6061719\end{displaymath}\end{maplelatex}$

Surface Area

Assuming the function has been defined, one might issue:

  > S:= int(2*Pi*f(x)*sqrt(1+D(f)(x)^2), x= 0..9);

$\begin{maplelatex} \begin{displaymath} S : = \frac{55}{6} \pi \sqrt{37} + \frac{1}{4} \pi ln(73+12\sqrt{37})-\frac{1}{6}\pi\end{displaymath}\end{maplelatex}$

  > evalf(S);

$\begin{maplelatex} \begin{displaymath} 178.5614656\end{displaymath}\end{maplelatex}$
to find the surface area of the solid obtained by rotating f(x) about the x-axis.

Exercises

Please note that all of the functions will be revolved around the x-axis.

1.

For the following function find the volume of the solid generated by the equation

$\begin{displaymath} f(x)= \frac{rx}{h}.\end{displaymath}$

Once you have generated the volume for this equation, please plot the graph of the solid, using a radius =2 ft and a height = 5 ft. What geometric object is this?

2.

For the following function find the volume of the solid generated by the equation

$\begin{displaymath} g(x)= \sqrt{r^2-x^2}.\end{displaymath}$

Once you have generated the volume for this equation, please plot the graph of the solid, centered around the origin, with r=12 ft.

3.

The equation for an ellipsoid is as follows

$\begin{displaymath} \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\end{displaymath}$

Find the volume of the solid generated by this equation. Plot the solid formed when a=3 and b=7. What geometric object is this?

4.

A torus is formed by revolving the equation

x² + (y-a)²=r²

about the x-axis. Find the volume of the torus generated by this equation. After finding the general form of the volume of a torus, plot this solid by substituting numerical values for a and r. Note: In order to create a torus, r<a. Why must this be true?

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Christine M Palmer
3/27/1998