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calculus

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Use integrals to prove that the volume of a sphere of radius
R is equal to (4/3)(pi)R^3

  • calculus -

    well the really easy way is to assume that you know the area of a sphere is 4 pi r^2
    integral dr (4 pi r^2)
    = 4 pi r^3 / 3 = (4/3) pi r^3

  • calculus -

    Otherwise radius of plane parallel to x,y axis as a function of height above the center plane is
    r = R cos theta
    area of circular plane at height z= R sin theta = pi r^2 = pi R^2 cos^2 theta
    so hemisphere above axis
    integral dz pi R^2 cos^2 theta
    but dz = d theta R cos theta
    so
    integral d theta R cos theta pi R^2 cos^2 theta
    from theta = 0 to theta = pi/2
    pi R^3 d theta cos^3 theta
    integral of cos^3 = sin - (1/3) sin^3
    so
    pi R^3 [sin theta - (1/3 sin^3 theta]
    at pi/2 - at 0
    pi R^3 [ 2/3]
    that is for the upper hemisphere, multiply by 2 to include the bottom half
    pi R^3 [4/3]

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