I need to calculate the surface integral of $xy$ over the surface $y^2+z^2=36$ in the first quadrant contained within $x^2+y^2=25$.
I started off trying to take a shortcut by solving $y^2+z^2=36$ for $z$: $z=\sqrt{36-y^2}$. While I think this is not incorrect, it gave an integral that was too difficult to solve. If it was incorrect to do this in the first place, let me know why.
So I took the standard approach and parameterized the equation using cylindrical coordinates, as the equation is that of a cylinder. The parameterization I chose was $r=
Therefore, I need to integrate $6\int\int_D{xy}\ dA$. Now to integrate this, do I parameterize x and y in the same way as before? So $x=x$ and $y=6\cos(\theta)$? Or can I choose a new parameterization like $x=5\cos(\theta)$ or $y=5\sin(\theta)$? I'm assuming I can't use a different parameterization, and if not, then what would the domain of integration be? $\theta$ would range from $0$ to $\pi/2$ since it's in the first quadrant, but what would I choose for $x$? If I can choose a different parameterization, why can I?