Surface integral of a scalar over a unit cube.

Question

Evaluate the following integral $$\iint_S (x+y+z) \, dS$$

where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$

Honestly, I don't know what to do. All I know is that you have to evaluate the integral for each of the 6 surfaces of the cube. The normals for them I can do and the equations of the planes, but other than that I couldn't do much.

Can someone break it down for me, please?

PS: No divergence theorem yet for me.

Answer 1

you have 6 surfaces. surface 1. $x = 0, y\times z = [0,1]\times[0,1]$

$\int_0^1\int_0^1 (0+y+z) dy dz = 1$ you get to verify if this is true.

surface 2. $x = 1, y\times z = [0,1]\times[0,1]$

$\int_0^1\int_0^1 (1+y+z) dy dz$

Now do to the symmetry of the problem each of the remaining 4 surfaces will have an identical integration to one of the above.

Answer 2

Due to symmetry, your integral is just $$ 3 \iint_{S} x\,d\mu =3\left(1+4\int_{0}^{1}\int_{0}^{1}x\,dx\,dy\right)=\color{red}{9}.$$