Textbook Problem

Find an iterated integral for the surface integral ${\displaystyle \underset{S}{\iint }f(x,y,z)dS}$ where f(x, y, z) = x + y and the surface S is given by r(u, v) = (u + v)i + (u − v)j + (uv)k, 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.

1. a) ${\displaystyle {\int }_0^1{\displaystyle {\int }_0^{1}2u\sqrt{2u^2+2v^2}dudv}}$
2. b) ${\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\sqrt{2u^2+2v^2}dudv}}$
3. c) ${\displaystyle {\int }_0^1{\displaystyle {\int }_0^{1}2u\sqrt{2u^2+2v^2+4}dudv}}$
4. d) ${\displaystyle {\int }_0^1{\displaystyle {\int }_0^1\sqrt{2u^2+2v^2+4}dudv}}$