Question

Set up, but do not evaluate, the integral which gives the volume when the region bounded by the curves y = Ln(x), y = 1, and x = 1 is revolved around the line y = -1. (10 points)

Answer 1

Using the washer method, the volume is given by

$\pi\displaystyle\int_1^e((1+1)^2-(\ln x+1)^2)\,\mathrm dx=\pi\int_1^e(4-(\ln x+1)^2)\,\mathrm dx$

or by the shell method,

$2\pi\displaystyle\int_0^1(e^y-1)(y+1)\,\mathrm dy$