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 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 `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(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 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 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 the graph below.

> 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);

- Consider the region bounded by the function
, and . Use the
`LeftDisk`command with to plot an approximation to the solid of revolution obtained by revolving this region about the axis. Then use the`LeftInt`command to find the volume of the disk approximation. - Using the same region as in the first exercise, use the
`revolve`command to plot the solid of revolution and then compute the volume of the solid you plotted. - The formula for the volume of a cone of radius and
height is
. Show that this formula is correct by
finding the volume of the solid generated when the triangle with
vertices at the points , , and is revolved
about the axis. (Hint - the equation of the line passing through
the points and is .)
- Several years ago, WPI students Kevin Nordberg and James Rush
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 and comment on the glass Kevin and Jim designed. 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.)

2002-01-26