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calculus

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The region bounded by the curve y=1÷(1+2x) , the line X=2 , the x-axis and the y axis is rotated completely about the y-axis.
Show that the volume generated is 1/2π(4-ln5)

  • calculus - ,

    The curve intersects x=2 at (2,1/5)
    The y-intercept is at (0,1)

    Using discs,

    v = ∫π(2^2)(1/5) dy [0,1/5] + ∫π((1-y/2y)^2 (y-1/5) dy [1/5,1]
    4π/5 + π/4 (y^2-1-2ylny)/4y [1/5,1]
    = π/2 (4-ln5)

    Using shells,

    v = ∫2πx(1/(1+2x)) dx [0,2]
    = π/2 ((2x+1)-ln(2x+1)) [0,2]
    = π/2 (4-ln5)

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