<s>[INST] evaluate this elliptic hyperbloid volume? How can I calculate the volume of this region in cylindrical coordinates?
$D=\{2x^2+y^2=z^2+4,|z| \le 2\}$
I think I got this wrong :
$$\operatorname{Volume} = 2\int_{0}^{2\pi}\int_{0}^{2} \int_{0}^{\sqrt{2r^2\cos^2\theta+r^2\sin^2\theta-4}}rdzdrd\theta$$
The problem is that the region is an ellipse and not a circle. I can integrate for $x^2+y^2=z^2+4$. Could you explain to me how to do it?
I think, what's wrong is also the Jacobian  :
$x=2r\cos\theta$
$y=\frac{1}{\sqrt{2}}r\cos\theta$
$|J|=2\sqrt{2}$ ?
so maybe z range like this : $z=\{0,\sqrt{8r^2\cos^2\theta+\frac{1}{2}r^2\sin^2\theta-4}\}$
 [/INST] It's a single sheet hyperboloid and its central axis of symmetry is the $z$-axis.
Easiest way to find its volume is to consider that at every $z$ we have an ellipse, and integrate over the areas of those ellipses.
For a constant $z$ we have the ellipse:
$$\frac{x^2}{\frac{z^2+4}{2}}+\frac{y^2}{z^2+4}=1$$
It has semi axes $a=\sqrt{\frac{z^2+4}{2}}$ and $b=\sqrt{z^2+4}$.
Since the area of an ellipse is $\pi a b$, the volume is:
$$\text{Volume}=\int_{-2}^2 \pi a b\,dz = \int_{-2}^2 \pi\cdot\frac{z^2+4}{\sqrt 2}\,dz$$
</s>