Thread: Volume calculation - Trippel integrate

Volume T limited by : z=sqrt(x^2+y^2-4), z=0, z=sqrt(5)

Find volume T

Originally Posted by kjell

Volume T limited by : z=sqrt(x^2+y^2-4), z=0, z=sqrt(5)

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.

Thank you Very Much!!

Kjell