<s>[INST] Triple integral : volume I need to calculate the volume between $x^2+y^2\le z^2-1$ and $2x^2+y^2+z^2\le 2$.
So It's a hyperboloid of two-sheets intersected with an ellipsoid.
their intersection leads to: $3x^2+2y^2=1$ which is an ellipse.
Using Cylindrical coordinates : $(x,y,z)=(\frac{1}{\sqrt{3}}r\cos\theta,\frac{1}{\sqrt{2}}r\sin\theta,z)$
Jacobian $|J|=\frac{\sqrt{6}}{6}r$
$\sqrt{1+x^2+y^2} \le z\le \sqrt{2-2x^2-y^2}$
$$
Volume =2\int_{0}^{2\pi}\int_{0}^{1}\int_{\sqrt{1+\frac{1}{3}r^2\cos^2\theta+\frac{1}{2}r^2\sin^2\theta}}^{\sqrt{2-\frac{2}{3}r^2\cos^2\theta-\frac{1}{2}r^2\sin^2\theta}}\frac{\sqrt{6}}{6}rdzdrd\theta?
$$ 
If it is correct Is there a simpler way to get the volume ?
 [/INST] The thought process was correct. 
Hint: the integrand should be $\frac{\sqrt{6}}{6}r^2$.
Not $\frac{\sqrt{6}}{6}r$, since the Jacobian is $\frac{\sqrt{6}}{6}r$ for the change of variables.
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