Volume T limited by : z=sqrt(x^2+y^2-4), z=0, z=sqrt(5)
Find volume T
Cylindrical coordinates would be easiest. The first surface is the upper half of a hyperboloid of one sheet (see the attached graph). Because of the "hole" in the center of the graph, integrating first with respect to $\displaystyle z$ might make things difficult. Instead, choose the order $\displaystyle dr\,d\theta\,dz.$ The hyperboloid, in cylindrical form, is
$\displaystyle z = \sqrt{r^2-4}$
and, solving for $\displaystyle r,$
$\displaystyle r = \pm\sqrt{z^2+4}.$
This suggests the following limits:
$\displaystyle 0\leq r\leq\sqrt{z^2+4}$
$\displaystyle 0\leq\theta\leq\2\pi$
$\displaystyle 0\leq z\leq\sqrt5.$
So, the volume of the region $\displaystyle T$ is
$\displaystyle V = \iiint\limits_TdV$
$\displaystyle =\iiint\limits_Tr\,dr\,d\theta\,dz$
$\displaystyle =\int_0^{\sqrt5}\int_0^{2\pi}\int_0^{\sqrt{z^2+4}} r\,dr\,d\theta\,dz.$
I leave the integration to you.