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Math Help - volumes by slicing

  1. #1
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    volumes by slicing

    A spherical cap of radius p and height h is cut from a sphere of radius r. Show that the volume of the spherical cap can be expressed as:
    a) V = (1/3)πh^2(3r - h)
    b) V = (1/6)πh(3p^2 + h^2)

    i tried setting up a coordinate system with the origin at the center of the sphere. so i am taking he integral fro (r - h) to r. i used similar triangles and established the fact that a = p(h-y) / h where a is the radius at any point along the y axis of the spherical cap and y is the height at any point along the y axis of the spherical cap. Then i found the area of a cross section by πa^2 = πp^2(h-y)^2 / h^2 and integrated that with respect to y from (r-h) to r. i treated πp^2/h^2 as a constant and found the integral of (h-y)^2. i ended up with something like π(p^2)(h)/3 - πp^2 + π(p^2)(r^2)/h. i am not sure how to get my answer to look like a or b. is my method correct or did i do something wrong?
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  2. #2
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    Just rotate the circle x^{2}+y^{2}=r^{2} through 360 degrees around the x-axis.
    The required volume will be
    \pi\int^{r}_{r-h}{(r^{2}-x^{2})} dx.
    That evaluates to your answer (a).
    Now use a property of intersecting chords in a circle (or some other method) to get the relationship
    (2r-h)h=p^{2},
    and use this to substitute for r.
    That will get you to answer (b).
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