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 using the `piecewise` command.)

> 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 -> piecewise(x<0,-x+1/2,x^2-x+1/2);

> 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);

- Compute the volume of the solid generated by revolving the
region bounded by the axis and the graph of the function

for . Include a plot of the solid. - A maker of brass candlesticks plans to make a candlestick whose
shape is given by revolving the function

about the axis over the interval .- Use the
`revolve`command to obtain a plot of the proposed candlestick. - Find the volume of the candlestick.

- Use the
- The candlesticks from the previous exercise are to be machined
from a cast, dumbbell-shaped
blank that
consists of two cylinders connected by a smaller-diameter cylinder. (A
blank is just the name used in fabrication for the piece used as a
starting point for the machining process.) The shape of the blank is
given by revolving the function

about the axis over the interval . Here is a Maple command you can use to define .> g := x -> piecewise(x<1.5,6,x<3.8,2,5);

- Use the
`revolve`command to obtain a plot of the blank. - Compute the volume of the blank.
- When the blank is machined into a candlestick, material is removed from the blank. Find the volume of material that must be removed.

- Use the

2001-04-03