We have the set $M=\{(x,y,z)\in \mathbb{R}^3 : x^2+y^2-z^2\leq 1, \ 0 \leq z\leq 3\}$. Draw $M$ and calculate the volume of $M$.
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I have done the following :
\begin{equation*}\int_M\, dV=\int\int\int\, dx\, dy\, dy\end{equation*} Which are the boundaries of the integrals? Do we have to use spherical coordinates?
Or do we set $x=r\cos\theta$ and $y=r\sin\theta$ and $z$ remains $z$ with $0\leq z\leq 3$ ?
Note that the solid is bounded by a hyperboloid of one sheet between z=0 and z=3.
The volume is found by the disc method as follows
$$ V = \int _0 ^3 \pi (1+z^2) dz = 12\pi $$