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Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = 3e^(-x), y = 3, x = 2;
about y = 6

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  1. Using discs (washers) of thickness dx,

    v = ∫[0,2] π(R^2-r^2) dx
    where R=6-y and r=3
    v = ∫[0,2] π((6-3e^(-x))^2-3^2) dx
    = 9π(5e^4+8e^2-1)/(2e^4)

    using washers of thickness dy,
    v = ∫[3/e^2,3] 2πrh dy
    where r=6-y and h=2-x
    v = ∫[3/e^2,3] 2π(6-y)(2+ln(y/3)) dy
    = 9π(5e^4+8e^2-1)/(2e^4)

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