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The region bounded by y=-x^2+14-45 and y=0 is rotated about the y-axis, find the volume.

  • Calculus -

    The region goes from x=5 to x=9, So,

    using shells,

    v = ∫[5,9] 2πrh dx
    where r = x and h=y
    v = 2π∫[5,9] x(-x^2+14x-45) dx
    = 448/3 π

    Using discs (washers), things get a bit more complicated, because there are two branches to the parabola.

    y = 4-(x-7)^2
    x = 7±√(4-y)

    v = ∫[0,4] π(R^2-r^2) dy
    where R = 7+√(4-y) and r = 7-√(4-y)
    v = π∫[0,4] (7+√(4-y))^2 - (7-√(4-y))^2) dy
    = 448/3 π

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