Consider the part of the surface $z = xy$, which lies within the cylinder $x^2+y^2 = 9$ and call it $S$. Compute the upward flux of $F = (y,x,3)$ through $S$.
Clearly, the normal to the given surface is: $\mathbb{n} = <y,x,-1>$, so we have the integral: $$ \int \int_D x^2+y^2 -3$$ to calculate, or at least that's what I got.
However, I'm a bit confused on the domain of integration here. Clearly, it cannot be the entire disc. Can someone give me pointers on how to find the domain of integration in such cases?
Yes, you can do it like that. But you want the upward flux. So $$ \iint_D(3-x^2-y^2)\,dA=\int_0^{2\pi}\int_0^3(3-r^2)r\,dr\,d\theta=-\frac{27\pi}2 $$