$\iint_{S} z+y+\sqrt{(a^2-x^2)} \,ds$ , surface integral

Question

$$\iint_{S} z+y+\sqrt{(a^2-x^2)} \,ds$$

$$ S: x^2+y^2=a^2,0\leqslant z
\leqslant c $$

$$ a,c>0 $$

Evaluate surface integral.

I wanted to express x(y),then with it evaluate $ dS $, then project this on yOz. The limits of y would be from – a to a, z from 0 to c.
I have seen a solution in which $ dS $ was calculated from y. And the integral was divide into 2 parts for positive and negative y.

Is my idea okay?

Answer 1

Sounds right to me. We can then express the integral as following$$I=\iint_Sz+y+|y|ds=\iint_{y>0}z+2yds+\iint_{y<0}zds$$where$$ds=ad\phi dz$$therefore $$I=\int_{0}^{a}\int_{0}^{\pi}z+2a\sin\phi ad\phi dz+2a\pi\int_{0}^{a}zdz$$