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,y=0..5); > 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
,
- Plot over the given interval.
- Plot the approximation of the solid of revolution using LeftDisk with 12 disks.
- Plot the solid formed by revolving about the -axis.
- Plot the solid formed by revolving about the line .
- Find the exact volume of the solid of revolution using the RevInt command and label your output exact.
- 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.

- What function's graph can be revolved about the -axis to obtain a sphere of radius ? Use this function and the RevInt command to prove that the volume of a sphere is .
- Several years ago, Kevin Nordberg and James Rush (both class of '98) were asked to design a drinking glass by revolving a suitable function about the axis. Here is the function they came up with.

They obtained the shape of their glass by revolving this function about the axis over the interval . The Maple command they used to define this function is given below.

> f := x -> piecewise(x<-3/4,-2*x-1/2,x<0,1/6,x^(2/3)+1/6);

Plot this function (without revolving it) over the interval and identify the formula for each part of the graph. Then, revolve this function about the axis over the same interval. Finally, compute the volume of the part of this glass that could be filled with liquid, assuming the stem is solid. (Hint - your integral will involve only one of the formulas used to define the function.)

2011-02-05