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Math Help - I have no idea how to calculate volume of cap (=top) of a sphere (Image attached)

  1. #1
    s3a
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    I have no idea how to calculate volume of cap (=top) of a sphere (Image attached)

    I'm not sure if this isn't part of Calculus but it's in my Calculus text book so sorry if I am posting in the wrong place.

    How do I calculate the volume of the cap (=top) of a sphere such as the part indicated in blue of my drawing where the black colour refers to the rest of the sphere? I don't even know how to get started. All I know is the volume of the whole sphere is 4/3 * pi*r^3 and I'm thinking that the 4/3 needs to be 4/6 = 2/3 or something after because it's like a "mini-semisphere."

    The question as written from the text book is:
    "Find the volume of the described solid S; A cap of a sphere with radius r and height h."

    The answer is:
    V = pi * h^2 * (r - 1/3 * h)

    Any help would be greatly appreciated!
    Thanks in advance!
    Attached Thumbnails Attached Thumbnails I have no idea how to calculate volume of cap (=top) of a sphere (Image attached)-cap-sphere.jpg  
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  2. #2
    Member Nacho's Avatar
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    wiht integrals: <br />
V_{cap}  = \pi \int_h^r {\left( {\sqrt {r^2  - x^2 } } \right)^2 dx} , but if you solve this, you donīt get the same solution of your book

    Whit a little of euclidean geometry and trigonometry you can follow tha same way how we calculate the circular segment (althoug I donīt know if is correct to make the proportion <br />
\frac{{2\pi }}<br />
{\alpha } = \frac{{\frac{4}<br />
{3}\pi r^3 }}<br />
{{V_x }}<br />
), but youīll find with <br />
\arccos \left( {\frac{h}<br />
{r}} \right)<br />
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