I have to calculate the volume of the solid bounded by: $$z=x^2+y^2 \qquad z=8-x^2-y^2 $$ using double integrals.
My problem is that I don't know what solid is. When I drew it, I see a kind of "egg" between the two paraboloids, but I don't know if the solid in the question is this or the solid around the inferior part of the "egg" limited by the plane $z=0$.
Finally, in the case that the solid is this "egg" I purpose an integral and I want to know if this is correct: $$V=\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\left(8-x^2-y^2\right)-\left(x^{2}+y^{2}\right)dy\,dx$$
I know that is so much easily making this by polar coordinates, but I want to know if is this correct with cartesian coordinates. I hope get help soon.