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Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y=1/x, y=0, x=1 and x=4

about the line y=−1

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  1. v = ∫[1,4] π(R^2-r^2) dx
    where R=(1/x + 1) and r=1
    v = ∫[1,4] π((1/x + 1)^2-1^2) dx = π(3/4 + ln16)

    check, using shells of thickness dy:
    v = ∫[0,1/4] 2πrh dy
    where r=y+1 and h=3
    + ∫[1/4,1] 2πrh dy
    where r=y+1 and h=(1/y - 1)

    Crank it out, and you get the same answer.

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