This is a Mathematical Analysis problem that requires the proof. This proof is based on Integration in Several Variables - Integration over a rectangle. Problem: Let R = [0, 1] [0, 1] be the square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1) and let P be the partition of R consisting of the partition {0, 1/4, 1/2, 3/4, 1} in both factors of [0, 1] [0, 1]. Find U(f, P) and L(f, P) if f(x, y) = xy