We have seen how double and triple integrals can be used to compute areas of plane regions, volumes, and centers of mass. This lab deals with using double integrals to compute areas of surfaces. You might have that sinking feeling that surface areas are like arc length, but worse, and you would be right! Fortunately, Maple can help a lot with doing the actual integrals.

To begin with, consider a surface defined parametrically by a vector
function
, where **s** and **t** lie in some
domain **D**.
For example, the unit sphere, , has the parametric
representation in spherical coordinates

with and . Note that any surface defined by can be represented parametrically as

The problem of computing the surface area via integration is really to
find the differential element of surface area **dS**. This turns out to
be a little involved, but the development is a typical calculus
exercise: first approximate a small element of surface area, then take
a limit as the size of the element goes to zero.

For the first step in the development, we need to recall the tangent plane at a point on a surface. Remember that this is the plane that touches the surface at and contains all vectors tangent to the surface at . Fortunately, to generate the plane, we only need the two tangent vectors

and

both evaluated at the point . The tangent plane to our surface at some particular point , is just the linear approximation

In the derivation of **dS**, we approximate a small area with
the area of a small piece of the tangent plane that we get by using
the linear approximation to map
a small rectangle in the **s-t** plane with sides of length
and . The linear
approximation maps this rectangle into a parallelogram, whose sides
are the vectors and . From
earlier in calculus we know that the area of this parallelogram can be
computed by taking the cross product of these two vectors, giving

Taking limits, we obtain

which gives a formula for the differential element of surface area.
So, if our surface is defined on a region **D** in the **s-t** plane, the
area **A** is given by

In any given case, computing **dS** is clearly not an easy task. In the
special case that the surface is defined by for some region
**D** in the **x-y** plane, the formula for the surface area
reduces to

The similarity to the case of arc length should be evident, and even in the case of relatively simple surfaces, the area often cannot be computed analytically.

In the case of a parametrically defined surface, Maple and the
functions for vector calculus from the `CalcP` package can be
useful. The following example shows how to use `VDiff`, `
VMag`, and the `linalg` procedures `dotprod` and `
crossprod` to compute the area of a sphere. If you've forgotten what
any of these procedures do, look at the help screens.

> with(CalcP):

> with(linalg):

Warning: new definition for normWarning: new definition for trace

> sphere := (s,t) -> [cos(s)*sin(t),sin(s)*sin(t),cos(t)] ;

> plot3d(sphere(s,t),s=0..2*Pi,t=0..Pi);

> t1 := VDiff(sphere(s,t),t);

> t2 := VDiff(sphere(s,t),s);

> VMag(crossprod(t1,t2));

> int(int(sin(t),s=0..2*Pi),t=0..Pi);

Note that `VMag` puts an extra absolute value around the result
and that Maple doesn't automatically replace with .
An alternative is the command

> sqrt(simplify(dotprod(crossprod(t1,t2),crossprod(t1,t2))));

but Maple is not good about using trig identities and, furthermore,
often makes mistakes choosing branches of the square root function, as
shown the wrong answer below. You might want to experiment with
peeling off the `sqrt` and `simplify` commands above , to see
why they were added.

> int(int(sqrt(1-cos(t)^2),s=0..2*Pi),t=0..Pi);

It isn't hard at all to come up with problems that Maple has trouble with. For example, consider finding the surface area of above the rectangle , . Using the formula we get

for the area. If you try to do this integral in Maple you get the following result.

> int(int(sqrt(1+4*x^2+4*y^2),x=0..1),y=0..1);

Error, (in int/cook/IIntd1c) cannot evaluate boolean

This is clearly a bug of some kind in Maple's integration routines. To get an answer, we can evaluate the double integral numerically with the following command.

> evalf(Int(Int(sqrt(1+4*x^2+4*y^2),x=0..1),y=0..1));

Note the use of the `Int` command, the inert form of `
int`. This is crucial in getting
Maple to evaluate the integral numerically, because it tells Maple not
to try to do the integrals analytically first.

Mon Sep 25 09:30:08 EDT 1995