A differential dz = f (x, y) dx + g(x, y) dy is exact if the integral ∫ f (x, y) dx + ∫ g(x, y) dy is independent of the path. Demonstrate that the differential dz = 2xy dx + x² dy is exact by integrating dz along the paths (1, 1) → (1, 8) → (6, 8) and (1, 1) → (1, 3) → (4, 3) → (4, 8) → (6, 8). The first number in each set of parentheses is the x coordinate, and the second number is the y coordinate.