Let there be a surface bounded below by the cone $\displaystyle { z }^{ 2 }={ x }^{ 2 }+{ y }^{ 2 }$ and above by $\displaystyle 1={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }$. Calculate the flux through this surface due to a vector field F(x, y, z), whose divergence cleanly works out to be 2z.

This turns out to be a triple integral $\displaystyle \int _{ 0 }^{ 2\pi }{ \int _{ 0 }^{ \frac { \sqrt { 2 } }{ 2 } }{ \int _{ r }^{ \sqrt { 1-r^{ 2 } } }{ ...\quad dz\quad dr\quad d\theta } } } $ BUT I'm not sure if I should integrate 2z or 2z*r, with the r coming from changing coordinate systems from rectangular to polar. I don't know if I'm even supposed to be changing coordinate systems since z is a polar coordinate. Do I need to treat this as if I changed variables, or should I just plainly integrate 2z?

Thanks
Anthony