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Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

xy = 2, x = 0, y = 2, y = 4

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  1. Each shell has thickness dy and height x = 2/y

    So, the volume v is

    v = ∫[2,4] 2πrh = ∫[2,4] 2πy(2/y) dy
    = ∫[2,4] 4π dy
    = 8π

    As a check, you can use discs, getting

    v = π(4^2-2^2)(1/2) + ∫[1/2,1] π(R^2-r^2) dx
    where R=y and r=2
    v = 6π + ∫[1/2,1] π((2/x)^2-2^2) dx
    v = 6π+2π = 8π

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