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The purpose of this lab is to use Maple to study planar domains and
the solids of
revolution. Solids of revolution are created by rotating curves in
the *x*-*y* plane about an axis, generating a three dimensional object.
They are discussed in Chapter 6 of *Calculus* by Varberg and Pursell (sections 2 and 3). The specific properties of them that we
wish to study are their volume, surface area, and graph.

If we have a curve lying entirely on or above the *x*-axis, which is the
graph of a function *y* = *f*(*x*), and *x* is restricted to being between *a*
and *b*, then we may rotate it about the *x*-axis, generating a three
dimensional object, a *solid of revolution*.

For example, if the curve is simply *f*(*x*) = *ax*/*h* (a straight line)
and *x* is between and *h*, we obtain a *cone* of height *h* and radius *a*.

Consider *f*(*x*) = *x ^{2}* + 1 for which appears in Figure 1.

If we take the region between the graph of *f*(*x*) and the *x*-axis and
revolve it about the *x*-axis then it becomes the surface of the solid
pictured in Figure 2.

If *f*(*x*) = *x ^{1/2}* and

If

The

(see p. 263). The region

(see p. 321).

The

(see Example 4a on page 316 of the textbook for illustration)

The volume is obtained by adding up volumes of cylindrical disks,
while its **surface area** (p. 324) is given by

Note that both of these formulas are for rotating the graph of *f*(*x*)
about the **x-axis**.

We will consider planar domains defined as a union of the
region *R* introduced above and the region produced by
reflection of *R* in the *x*-axis. Clearly, the area of equals 2*A*, and the perimeter of equals 2[*P* + *f*(*a*) +
*f*(*b*)]. Particularly, if the region *R* is defined as the interior
of a half of the ellipse

belonging to the upper half plane *y* > 0 then becomes the
interior of a half of the reflected ellipse

belonging to the lower half plane *y* < 0. So, - a complete ellipse, and its area is equal to 2*A*, whereas
its perimeter equals 2*P* since *f*(*a*) = *f*(*b*) = 0 in this case.

The *area radius* of the planar domain is defined as
the radius of the circle possessing the same area as , i.e.

The *perimeter radius* of the planar domain is defined
as the radius of the circle having the same perimeter as , i.e.

The *volume radius* of the body is defined as the
radius of the sphere possessing the volume *V*, i.e.

The *surface radius* of the body is defined as the radius of the sphere having the surface area *S*, i.e.

To plot the graph of the function *y* = *f*(*x*) given by the list of
points we use the command `plot`
in the relevant format, e.g.

> plot([[5,5],[6,7],[7,10],[8,12]]);Check it!

To do graphics in three dimensions, we utilize the `tubeplot`
function from the `plots` library of Maple, along with suitable
options.

For sake of demonstration, suppose we wish to plot the surface of
revolution one gets from rotating the curve about the *x*-axis. Maple commands to accomplish this would be:

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

> with(plots): PltStyle:= axes= NORMAL, style= PATCHNOGRID, tubepoints=40;

> tubeplot([x,0,0], x=0..9, radius=f(x),PltStyle);

The last command uses the function `tubeplot` from the
`plots` library, rotating about the *x*-axis (as controlled by the first
argument, [*x*,0,0], letting *x* range from to 9, using the function
*f*(*x*) to determine the radius at each point, and setting the options
as set up by the line assigned PltStyle). The only especially
noteworthy thing set by PltStyle is the last option, `
tubepoints`. This effectively determines the resolution of the graph;
the higher the value, the smoother it is (but the longer it takes). A
value as low as 10 gives a fairly crude surface, by comparison. (Try
it!)

Since this involves straightforward integration, one may simply issue commands such as:

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

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

> evalf(vol);

Assuming the function has been defined, one might issue:

> 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

For the function,

the graph in the (*x*,*y*) plane is a half of ellipse;
revolving this graph about the x-axis, we generate an ellipsoid of
revolution.

- 1.
- (a)
- Plot the graph of
*y*=*f*(*x*). - (b)
- Find the area
*A*of the planar domain*R*. - (c)
- Find the length
*P*of a half of the ellipse. - (d)
- Calculate the area radius and the perimeter radius
for the ellipse and make sure that

- (e)
- Consider the family of half-ellipses

depending on parameter*k*. This parameter will be assumed taking the values from the interval . - (f)
- Calculate and for the ellipses related to the following set of values of parameter
*k*:

and plot the graph of the ratio

versus*k*.

- 2.
- (a)
- Plot the surface of the ellipsoid defined above.
- (b)
- Find the volume
*V*of this ellipsoid. - (c)
- Find the area
*S*of the surface of this ellipsoid. - (d)
- Calculate the volume radius and the surface radius
and make sure that

- (e)
- Consider the family of ellipsoids generated by rotating the
half-ellipses

about the*x*-axis; here*k*is the parameter taking the values from the interval . - (f)
- Calculate
and for the ellipsoids related to the following
set of values of parameter
*k*:

and plot the graph of the ratio

versus*k*.

- 3.
- Given the graph of
*n*=*n*(*k*), what can you say about the ellipsoid of revolution for which*n*(*k*) attains its maximum value? What is this maximum value? Make a plausible hypothesis related to the maximum value of*n*(*k*) calculated for all convex bodies, not necessarily the solids of revolution. (We call convex the bodies that contain along with any two points the whole segment bounded by these points.) Develop similar consideration for the graph of*m*=*m*(*k*) and the relevant elliptical domains in the plane.

9/14/1998