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March 27, 2017

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Let V be the volume of the 3-dimensional structure bounded by the paraboloid z=1−x^2−y^2, planes x=0, y=0 and z=0 and by the cylinder x^2+y^2−x=0. If V=aπ/b, where a and b are coprime positive integers, what is the value of a+b?

  • calculus - ,

    The cylinder is centered at (1/2,0) and has radius 1/2

    Seems the best way to handle this one is with cylindrical coordinates (like polar coordinates, plus a z-axis):

    As with polar coordinates, the alement of area is

    dA = r dr dθ

    so, the volume element is

    dV = r dr dθ dz

    We have z = 1-(x^2+y^2) = 1-r^2
    x^2+y^2-x = 0 becomes
    r^2 - rcosθ = 0, or
    r = 1-cosθ

    So,

    V = ∫[0,pi/2] ∫[0,cosθ] z r dr dθ
    = ∫[0,pi/2] ∫[0,cosθ] r(1-r^2) dr dθ
    = ∫[0,pi/2] (1/2 cos^2(θ) - 1/3 cos^3(θ)) dθ
    = 1/72 (18θ - 18sinθ + 9sin2θ - 2sin3θ) [0,pi/2]
    = pi/8 - 2/9

    Hmm. I don't get a*pi/b
    Must have messed up. Check it out.

  • calculus - ,

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