The volume of the solid obtained by rotating the region bounded by y=e^x, y=Inx, x=1, and x=2 about the line y-axis can be computed using the method of cylindrical shells. Using the method of cylindrical shells find the volume.

Answer 1

Draw the curves.

v = ∫[1,2] 2πrh dx
where r=x and h=e^x-ln(x)
v = ∫[1,2] 2πx(e^x-lnx) dx = 2π(e^2-ln4)

using discs requires breaking up the volume at boundary changes, and that's too much bother for now. It would be

∫[0,ln2] π((e^y)^2-1^2) dy
+ ∫[ln2,e] π(2^2-1^2) dy
+ ∫[e,e^2] π(2^2-(lny)^2) dy

check my setup