Volume calculation

**kjell** · Jul 13th 2012, 02:41 PM

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

Find volume T

**Reckoner** · Jul 13th 2012, 04:57 PM

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 $z$ might make things difficult. Instead, choose the order $dr\,d\theta\,dz.$ The hyperboloid, in cylindrical form, is

$z = \sqrt{r^2-4}$

and, solving for $r,$

$r = \pm\sqrt{z^2+4}.$

This suggests the following limits:

$0\leq r\leq\sqrt{z^2+4}$

$0\leq\theta\leq\2\pi$

$0\leq z\leq\sqrt5.$

So, the volume of the region $T$ is

$V = \iiint\limits_TdV$

$=\iiint\limits_Tr\,dr\,d\theta\,dz$

$=\int_0^{\sqrt5}\int_0^{2\pi}\int_0^{\sqrt{z^2+4}} r\,dr\,d\theta\,dz.$

I leave the integration to you.

**kjell** · Jul 14th 2012, 12:23 AM

Thank you Very Much!!

Kjell

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