Now that this has been satisfactorily solved, I'll post a
youtube video of this problem that I literally saw the week before Ether posted. I didn't feel right in just posting it, or claiming it as my own solution.
The problem gets to the 4ʃʃ(1-x)(1-y)√(x^2+y^2)dxdy GeeTwo derived, but then it does a polar coordinate substitution to make the integration "easier"
It's another approach which gives a closed form solution, demonstrating that there can be multiple ways to validly solve a problem.