## Volume of Cylindrical Surface Bounded By Hyperboloid

Suppose the infinite cylindrical surface
$x^2+y^2=4, \;$
is bounded above and below by the hyperboloid
$x^2+y^2-z^2=1$
. What is the volume of the solid enclosed?
In cylindrical polar coordinates
$x^2+y^2=r%2, \; z=z, \; \theta = \theta$
and we can write
$r^2-z^2=1 \rightarrow 2z= \sqrt{r^2-1} \sqrt{r^2-1}$
. The volume element is
$dV=2z r dr d \theta = 2r \sqrt{r^2-1} dr d \theta$
. The volume is
\begin{aligned} V &= \int^{2 \pi}_0 \int^2_1 r \sqrt{r^2-1} dr d \theta \\ &= \int^{2 \pi}_0 [ \frac{2}{3} (\sqrt{r^2-1})^{\frac{3}{2}} ]^2_1 d \theta \\ &= \int^{2 \pi}_0 2 \sqrt{3} d \theta = 4 \pi \sqrt{3}\end{aligned}

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