** Next:** About this document ...
**Up:** Labs and Projects for
** Previous:** Labs and Projects for

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 *f*(*x*) = *x ^{2}*+1
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 *f*(*x*) about the line
*y*=-2 instead of the default *y*=0.

> 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);

It turns out that the volume of the solid obtained by revolving the
region in Figure 1 between the graph and the *x*-axis
about the *x*-axis can
be determined from the integral

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));

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

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

> evalf(vol);

> S:= int(2*Pi*f(x)*sqrt(1+D(f)(x)^2), x= 0..9);

> evalf(S);

to find the surface area of the solid obtained by rotating

- 1.
- For the following function find the volume of the solid generated by the equation
Once you have generated the volume for this equation, please plot the graph of the solid, using a radius =2
*ft*and a height = 5*ft*. What geometric object is this? - 2.
- For the following function find the volume of the solid generated by the equation
Once you have generated the volume for this equation, please plot the graph of the solid, centered around the origin, with
*r*=12*ft*. - 3.
- The equation for an
*ellipsoid*is as follows Find the volume of the solid generated by this equation. Plot the solid formed when*a*=3 and*b*=7. What geometric object is this? - 4.
- A torus is formed by revolving the equation
*x*+ (^{2}*y*-*a*)^{2}=*r*^{2}*x*-*axis*. Find the volume of the*torus*generated by this equation. After finding the general form of the volume of a torus, plot this solid by substituting numerical values for*a*and*r*. Note: In order to create a*torus*,*r*<*a*. Why must this be true?

3/27/1998