I tried to solve the following integral and I get either $\frac\pi4$ or $\frac\pi8$. Can't figure out which is right.
$$\iint_Sxz\,\mathrm dy\,\mathrm dz+x^2y\,\mathrm dx\,\mathrm dz+y^2z\,\mathrm dx\,\mathrm dy$$
where $S$ is the surface $x^2+y^2\le1$, $x\ge0$, $y\ge0$, and $0\le z\le x^2+y^2$.
I used Gauss's theorem:
