Given a surface defined by z=g(x,y) for (x,y) in a region R of the xy plane, then the surface area is given by:

.

For your problem, R is the rectangle bound by

,

,

, and

; while

and

, and so you have

. Like you said, substitute

and

. Then we get:

.

Now, a table of integrals will tell you that

(see #8

here, for example). Replacing x with v and

with

, the above applies to our inner integral, and we have:

.

Now this is a bit more complicated, but the integrand may be broken up into four terms, each of which may be integrated. For the terms

and

, we can use the form

, this time with x replaced by u, and with either

or

in place of

.

For the terms

and

, well, those may take more creative substitutions, but I'm pretty sure they can be done analytically as well. Some messy algebra later, you'll have your analytic answer.

--Kevin C.