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

The curve intersects x=2 at (2,1/5)
The yintercept is at (0,1)
Using discs,
v = ∫π(2^2)(1/5) dy [0,1/5] + ∫π((1y/2y)^2 (y1/5) dy [1/5,1]
4π/5 + π/4 (y^212ylny)/4y [1/5,1]
= π/2 (4ln5)
Using shells,
v = ∫2πx(1/(1+2x)) dx [0,2]
= π/2 ((2x+1)ln(2x+1)) [0,2]
= π/2 (4ln5)