what should the bounds of integration be? 
I'm confused if the region on xy-plane is a rectangle $R = [-4,4]*[-2,2]$ or something like $-2\le y\le2$ and $-4+y^2 \le x\le4-y^2$. If someone could provide me with an explanation that would be great.

Your region on the $xy$-plane is bounded by the horizontal parabolae
$$4-y^2-x=0, \quad 4-y^2+x=0.$$
This is because you know that the three-dimensional region $R$ whose volume you are after is defined as
$$R=\{(x,y,z)\in\mathbb R^3\ |\ 0 \leq z \leq f(x,y)\}$$
and the intersection of $R$ with the $xy$-plane happens where $z=0$, so that
$$0 \leq 4-y^2-|x|,$$
or $$|x| \leq 4-y^2.$$
This means that both $x\leq 4-y^2$ ($x$ is bounded on the right by a horizontal parabola) and $-x \leq 4-y^2$, which means $y^2-4\leq x$ ($x$ is bounded on the left by a horizontal parabola that is symmetric to the other one).

