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
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.
Warning: new definition for norm
Warning: new definition for trace
> sphere := (s,t) -> [cos(s)*sin(t),sin(s)*sin(t),cos(t)] ;
> t1 := VDiff(sphere(s,t),t);
> t2 := VDiff(sphere(s,t),s);
Note that VMag puts an extra absolute value around the result and that Maple doesn't automatically replace with . An alternative is the command
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.
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.
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.
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.