Surface Integrals — Question 7

Question 7 – Solution

Strategy. Pair opposite faces. Each pair produces the same contribution, avoiding six unrelated computations.

Step 1: Pair the $x$-faces On $x=0$, $dS=dy\,dz$ and $$\int_0^1\int_0^1(y+z)\,dy\,dz=1.$$ On $x=1$, $$\int_0^1\int_0^1(1+y+z)\,dy\,dz=2.$$ Thus the pair perpendicular to the $x$-axis contributes $3$.

Step 2: Use coordinate symmetry The cube and integrand are unchanged by permuting $x,y,z$. Consequently, the two $y$-faces and two $z$-faces also contribute $3$ apiece. Hence $$\boxed{\iint_S(x+y+z)\,dS=3+3+3=9}.$$

Step 3: Average The six unit-square faces give $\operatorname{area}(S)=6$, so $$\boxed{\operatorname{Avg}_S(x+y+z)=\frac 96=\frac 32}.$$

Verification The map $(x,y,z)\mapsto(1-x,1-y,1-z)$ preserves the boundary and pairs values whose sum is $3$. Therefore their average is $3/2$, confirming the computation.