calculus

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The area bounded by the curve 2y^2=x and the line 4y=x is rotated around the y-axis. The volume of the resulting structure can be expressed as V=a/bπ, where a and b are coprime positive integers. What is the value of a+b?

  • calculus -

    something's wrong. The region is not closed.

  • calculus -

    i believe its bounded from y=0 to y=2

  • calculus -

    Ah. In that case, using discs (washers),

    v = ∫[0,2] π(R^2-r^2) dy
    where R = 4y and r = 2y^2, so

    v = ∫[0,2] π((4y)^2-(2y^2)^2) dy
    = 4π∫[0,2] 4y^2 - y^4 dy
    = 1024/15 π

  • calculus -

    no, i think that
    ∫[0,2] π((4y)^2-(2y^2)^2) dy is 256/15π

  • calculus -

    you may be right. maybe I mixed in an unneeded factor of 4 somewhere.

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