I am having some difficulty identifying the bounds of a three-dimensional region.
I am asked to evaluate $\iiint_R (xz+3z)dV$, where $R$ is the region bounded by the cylinder $x^2 + z^2 = 9$ and the planes $x+y=3$, $z=0$, and $y=0$, above the $xy$-plane.
So now that I've a sketch of the region $R$, I am trying to find the bounds of $x$, $y$, and $z$ but I'm always confused when it comes to identifying the correct bounds.
For example, I don't know which of the following for $x$ are correct:
$-3 \le x \le 3$
$-3 \le x \le 3-y$
(are both of them wrong?)
Could someone please give me some tips on how I should go about constructing the inequalities for this 3D region?