Compute $\iint_S xz^2dydz+yz^2dzdx+z^3dxdy$

Question

Problem

Compute $$\displaystyle \iint_S xz^2dydz+yz^2dzdx+z^3dxdy$$ where $S$ denotes the outside surface of the common part $\Omega$ of $x^2+y^2+z^2\leq R^2$ and $x^2+y^2+z^2 \leq 2Rx$.

Comment

It would be fair to say this could be solved by Gauss's Formula, but how to make the transformation? Anyone can do me a favour?

Answer 1

Note that $x^2+y^2+z^2 \leq 2Rx$.

is actually : $(x-R)^2+y^2+z^2 \leq R^2$.