Consider the unit square, A, with opposite vertices at (a, b) and (a+1, b+1). Now consider the elemental strip within A, made up of all points (x, y), for which

a + b + s < x + y < a + b + s + ds

For 0 <= s <= 1(i.e. in the triangular region, A1, below the diagonal)

Integrand = [x + y]{x + y}(x + y) = (a + b)s(a + b + s) = cs(c + s)where c = a + b

Area of elemental strip = s ds

So the integral over A1 = Int(s=0 to 1) of cs(c + s) s ds

= c(cs^{3}/3 + s^{4}/4)between 0 and 1

= c(4c + 3)/12(1)

For 1 <= s <= 2(i.e. in the triangular region, A2, above the diagonal)

Integrand = [x + y]{x + y}(x + y) = (a + b + 1)(s - 1)(a + b + s)

= (c + 1)(s - 1)(c + s)

Area of elemental strip = (2 - s) ds

Integral over A2 = Int(s=1 to 2) of (c + 1)(s - 1)(c + s)(2 - s) ds