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Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 1−y^4, x = 0; about x = 1.

  • calculus -

    using shells of thickness dx, and taking advantage of symmetry,

    v = 2∫[0,1] 2πrh dx
    where r=1-x and h=y=∜(1-x)
    v = 2∫[0,1] 2π(1-x)∜(1-x) dx
    = 4π∫[0,1] (1-x)^(5/4) dx = 16π/9

    using discs (washers) of thickness dy, we have

    v = 2∫[0,1] π(R^2-r^2) dy
    where R=1 and r=1-x
    v = 2∫[0,1] π(1-(1-(1-y^4))^2) dy
    = 2∫[0,1] π(1-y^8) dy
    = 16π/9

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