I have a problem that seems to be a cylindrical conversion problem, but I could not find bounds for r.
The problem asked me to find the volume bounded by $z = (2x)^2 + y^2$ and $z+y^2 = 2$.
I first drew the graph
The oval has a shape like this, disappears if z = 0
The surface curved like this along z-y continuous inf for all x
So I choose cylindrical conversion $x = r\cos\theta, y =r\sin\theta, z = z$ bounds are $0<z<2, 0<\theta<2\pi$, but what is r? I think it is changing from $\sqrt{z}/2$ to $\sqrt{z}$ governed by $\theta$.
I know I want to setup something like $\iiint dzrdrd\theta$ but I am not sure what do I put in there, just an empty integral?