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Math Help - Volume formula for cap of sphere

  1. #1
    HelpPlease
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    Volume formula for cap of sphere

    A hemispherical bowl of radius r contains water to a depth height h. Give a formula that you can use to measure the volume of the water in the bowl.
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  2. #2
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    I would use integration consider a semi-circle.
    y=\sqrt{r^2-x^2}.
    Let h be the height i.e. distance from the endpoint on semicircle. Then what distance from the origon is r-h.
    Thus, the volume by revolving about the x-axis is given by,
    \pi \int_{r-h}^r r^2-x^2dx

    Note: LaTeX (equation editor) is currectly disabled. In a few days these strange symbols I wrote down should turn into equations. If you know LaTeX you can follow this still.
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  3. #3
    Eater of Worlds
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    There are various ways, but you could come up with a formula by rotating about the y-axis.

    {\pi}\int_{-r}^{h-r}(r^{2}-y^{2})dy=\frac{h^{2}{\pi}(3r-h)}{3}


    Try it with a sphere of radius 4, completely filled:

    {\pi}\int_{-4}^{4}(16-y^{2})dy=\frac{256{\pi}}{3}

    Check this against the formula for sphere volume.
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  4. #4
    MHF Contributor Quick's Avatar
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    This should help you understand what they're saying:
    I realize it's of poor quality, but the things that were typed in latex are to the left, and the images they give are to the right
    Attached Thumbnails Attached Thumbnails Volume formula for cap of sphere-mathtype.gif  
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