Needham Multivariable Calculus: Surface Integral of a Scalar Field

Well, well, well! You've gotten this far. Congratulations! This is where the problems get ABSOLUTELY HORRIBLE, because there are so many steps.
In this problem, we go over many different concepts. There's how to parametrize a surface, how to calculate a cross product, blah blah blah. At least the answer doesn't end up being zero!!!