If $S\subseteq\mathbb R^3$ is a surface (2-submanifold) described implicitly as $F(x,y,z)=0$ for $(x,y)\in D$, then $\int_S |F_z|d\sigma=\int_D\sqrt{F_x^2+F_y^2+F_z^2}dxdy$.
I can prove this in the case that $F(x,y,z)=0\iff z=G(x,y)$ for some function $G$. But this can't work in the case that $S$ is the unit sphere. Is there an easy way to find a proof? Apologize for possible mistakes.