I need to calculate $$\int\int_{S} x^2yz^4 dA$$ whereas $$ S: z=4-x^2-y^2 \\ x\geq0\\z\geq0$$ I suspect that I should use the fact that the function is odd "with respect to $y$", but what I'm having trouble is determining the boundaries. Since $S$ is a surface in $\mathbb{R}^3$, I don't understand how one should take the slope of the surface in $z$-direction into account when determining the boundaries.
My second question is how do you double integrate a function with three variables? Will the answer contain a $z$ ?