As a simple example, consider the graph of the function for , which appears below.

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 the next graph.

To help you in plotting surfaces of revolution, A Maple procedure
called `revolve` has been written. The commands used to produce
the graphs are shown below. The `revolve` procedure, as well as
the `RevInt`, `LeftInt`, and `LeftDisk` procedures
described below are all part of the `CalcP7` 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(CalcP7): > 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 piecewise defined function 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 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 -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.

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
command 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 the graph below.

> LeftDisk(f(x),x=-2..2,5); > LeftDisk(f(x),x=-2..2,10);

In order to calculate the volume of a solid of revolution, you can
either use the `int` command implementing the formula above or
use the Maple procedure `RevInt` which sets up the integral for
you. Try the examples below to see the different types of output.

> Pi*int(f(x)^2,x=-2..2); > evalf(Pi*int(f(x)^2,x=-2..2)); > RevInt(f(x),x=-2..2); > value(RevInt(f(x),x=-2..2)); > evalf(RevInt(f(x),x=-2..2));

- For the function
over the interval
,
**A)**- Plot over the given interval.
**B)**- Plot the approximation of the solid of revolution using
`LeftDisk`with 9 disks. **C)**- Plot the solid formed by revolving about the -axis.
**D)**- Plot the solid formed by revolving about the line .
**E)**- Find the exact volume of the solid of revolution about the -axis using the
`RevInt`command and label your output`exact`. **F)**- Find the number of subintervals needed to approximate the volume of the solid of revolution about the -axis using
`LeftInt`with error no greater than 0.1 and then again with error no greater than 0.01.

- Last week, you used a definite integral to prove that the area of a circle of radius is . This week, use the
`RevInt`command and the same function from last week to prove that the volume of a sphere is . - 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.) Plot the function revolved about the -axis over the given interval. 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.

2007-01-28