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 for , which appears in Figure 1.

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 about the line
instead of the default .

> with(CalcP):

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

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

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

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

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 -axis
about the -axis can
be determined from the integral

to have the value . More generally, if you revolve the area under the graph of for about the x-axis, the volume is given by

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);

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

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

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 of the rectangle and thickness is , the width of the rectangle. The volume of this disk is . 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);

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

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

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

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

> evalf(vol);

- For each function below,
- i.
- Plot its surface of revolution using the command
`LeftDisk`for and and then plot the surface of revolution using`revolve`. - ii.
- Approximate the volume of the solid of revolution about the
-axis using
`LeftInt`for each of the -values given above and then find the exact value. Explain why it would be difficult to do these calculations by hand.

- A brass finial is to be made in the shape of the solid obtained
by revolving the function
about the axis
over the interval
. (A finial is a decorative cap
or projection often seen on top of fence posts or staircase posts.) If
the dimensions of the solid are all in inches, determine how many
cubic inches of brass will be needed to make of these finials.
- A doughnut is to be made by revolving a circle of radius
centered at the point about the axis.
Find the volume of this doughnut and plot your doughnut. (Hint - the
equation for the
circle is
. You will need to solve this equation
for , which will give you the two functions you need.
- A solid wood ball of diameter 15 cm floats in water with 4 cm
extending above the surface. Find the density of the ball, given that
the density of water is
. You might
find the following facts useful.
- The density of an object is defined as the ratio of its mass to its volume.
- Archimedes' principle says that the weight of the ball is equal to the weight of the water it displaces.

2000-09-19