Find the volume under the surface with several constraints I need to find the volume under the surface
$$z = a^2 - x^2$$
on domain restricted by these surfaces and planes:
$$y = 2x$$
$$x+y=a$$
$$z = 0$$
$$y=0$$
I made a drawing but I cannot even infer what the bounds of integration are.
When I drew all these lines (assuming $a=2$) I got this:

What are the bounds of integration in this case?

To find the limit we need to make a sketch of the surface and bounds in the planes $z-x$ and $y-x$.

From here it follows that the volume is given by the sum of the following integrals
$$\int_{-a}^{0}\,dx\int_{0}^{-x+a} (a^2-x^2) \,dy $$
$$\int_{0}^{a/3}\,dx\int_{2x}^{-x+a} (a^2-x^2) \,dy $$
$$\int_{a/3}^{a}\,dx\int_{-x+a}^{2x} (a^2-x^2) \,dy $$
