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Math Help - 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 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.
    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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