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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);

  > g := x -> if x < 0 then -x +1/2 else x^2-x+1/2 fi ;

g := proc(x) options operator,arrow; if x < 0 then -x+1/2 else 
x^2-x+1/2 fi end

  > revolve('g(x)',x=-1..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));

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

The integral formula given above for the volume of a solid of revolution comes, as usual, from a limit process. Recall the rectangular approximations we used for plane regions. If you think of taking one of the rectangles and revolving it about the x-axis, you get a disk whose radius is the height h of the rectangle and thickness is $\Delta x$ , the width of the rectangle. The volume of this disk is $\pi h^2 \Delta x$ . If you revolve all of the rectangles in the rectangular approximation about the x-axis, you get a solid made up of disks that approximates the volume of the solid of revolution obtained by revolving the plane region about the x-axis.

To help you visualize this approximation of the volume by disks, the LeftDisk procedure has been written. The syntax for this procedure is similar to that for revolve, except that the number of subintervals must be specified. The examples below produce approximations with five and ten disks. The latter approximation is shown in Figure 3.

  > LeftDisk(f(x),x=-2..2,5);

  > LeftDisk(f(x),x=-2..2,10);

  > LeftInt(f(x),x=-2..2,5);

$\begin{maplelatex} \begin{displaymath} 99.52163404\end{displaymath}\end{maplelatex}$

  > LeftInt(f(x),x=-2..2,10);

$\begin{maplelatex} \begin{displaymath} 89.62938708\end{displaymath}\end{maplelatex}$

The two LeftInt commands above add up the volumes in the disk approximations of the solid of revolution.

**Figure 3:** Cylindrical approximation with n=10 to the solid generated by rotating f(x)=x²+1 about the x-axis.
$\begin{figure} \centerline{ \psfig {file=volrev_fig3.ps,height=4in,width=2.5in,angle=-90} }\end{figure}$

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

1.

For each function below,

(a): Plot its surface of revolution.
(b): Find the volume of the solid of revolution about the x-axis.
(c): Find the surface area of the surface of revolution about the x-axis.

$\bullet \;\;f(x) = 2^x,\;\;\;0 \leq x \leq 2$
$\bullet\;\; g(x) = \sin 2x,\;\;\; -2\pi \leq x \leq 2\pi$
$\bullet \;\; h(x) = \sqrt{x^2+1},\;\;\; 0 \leq x \leq 1$

2.

Now generalize the results from Exercise 1 by taking advantage of Maple's symbolic capabilities and find the volume and surface area for:

(a): $f(x) = a^x,\;\;\;0 \leq x \leq a$ about the x-axis
(b): $g(x) = \sin(ax),\;\;\; - 2\pi \leq x \leq 2\pi$ about the x-axis
(c): $h(x) = \sqrt{x^2 + a^2},\;\;\;0 \leq x \leq a;\;\;a \geq 0$ about the x-axis.

(Your answers in each case may have parameters a and/or h in them.)

3.

Design a funky hat by revolving a suitable function about the x-axis. In your report, give the function, a three dimensional plot of your hat, and determine the amount of material needed to make your hat. You will be graded on the utility and originality of your design.

Note: The hat will be lying on its side.

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Christine M Palmer
12/11/1997