If we have two paraboloids, the first being $z_1 = ax^2 + by^2$ and $z_2 = 6 − cx^2 − dy^2$, how does one show that the volume of the region enclosed by the two surfaces is $$\frac{18π}{\sqrt{(a + c)(b + d)}}$$
I tried finding the intersection first of all, where I would get $x^2(a+c) + y^2(b+d)$ = 6. If I switch my coordinates by using $ x' = x/\sqrt{a+c}$ and $ y' = y/\sqrt{b+d}$, then I simply get that the region will be $x'^2+y'^2=6$. And, if I evaluate the jacobian - I get $\sqrt{(a + c)(b + d)}$.
But now evaluating the actual volume integral becomes the tricky part, if I try to evaluate each section individually - it becomes way too complicated. However, if I try to calculate the volume integral of ($z_2-z_1$) in cylindrical coordinates (while remaining in my new coordinates), I get the answer by integrating between $0<r'<\sqrt{6}$ and $0<\theta<2\pi$ - but I'm not exactly sure as to why. Surely this doesn't give the volume enclosed - but the volume that's missing? Any thoughts?