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Thread: Volume calculation - Trippel integrate

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    Volume calculation - Trippel integrate

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

    Find volume T
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    Re: Volume calculation - Trippel integrate

    Quote Originally Posted by kjell View Post
    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.
    Attached Thumbnails Attached Thumbnails Volume calculation - Trippel integrate-mhf_20120713a.png  
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    Re: Volume calculation - Trippel integrate

    Thank you Very Much!!

    Kjell
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