My Prof. probably didn't think this one through too much. I found a similar question to this one here, yet from it I discerned an issue with the question (which is listed below, word-for-word as usual): it doesn't state which surface the region $\displaystyle R$ is bounded above, and which surface the region $\displaystyle R$ is bounded below. Anyhow, here is the question.

Find the volume of the region $\displaystyle R$ between the surfaces $\displaystyle z=3x^2+y^2$ and $\displaystyle z=2+x^2-y^2$.

(I know, the wording is horrible; so open to loopholes)

All I really know is that I have to get to here somehow:

$\displaystyle V=\iint_R f(x,y) dx dy$?

So far, here is what I have, borrowing bits from the link above.

For the point of intersection between the two surfaces, setting $\displaystyle z=0$ (I think), I got the following:

$\displaystyle 3x^2+y^2=2+x^2-y^2\Rightarrow 0=2x^2+2y^2-2\Rightarrow 0=x^2+y^2-1$

From here, however, I'm not sure what to do. It's very little, I know, but I'm just not getting this material drilled in right because of a lack of legible examples.