calculus help

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
x = 4(square root of (3y)), x = 0, y = 3;
about the y-axis

V=?????

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  1. x=4√(3y)
    x^2 = 48y

    To use discs, note that the radius of each disc is x, and the thickness is dy, so

    v = ∫[0,3] πx^2 dy
    = 48π∫[0,3] y dy
    = 216π

    To use shells, each shell has radius x, and height 3-y, so

    v = ∫[0,12] 2πx(3-y) dx
    = 2π∫[0,12] x(3-x^2/48) dx
    = 216π

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