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.