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.

Question

Show that the volume generated is 1/2π(4-ln5)

Answer 1

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)