Mathematics

Problem

A region is enclosed by the graph of

y=\ln (x+1)

, the

x

-axis, and the vertical line

x=e^{2}-1

. When this region is revolved about the

y

-axis, which of the following definite integrals equals the volume of the resulting solid? a)

\displaystyle 2\pi\int_{0}^{2}\bigl(e^{y}-1\bigr)\,dy

\displaystyle \pi\int_{0}^{2}\bigl(e^{y}\bigr)^{2}\,dy

\displaystyle \pi\int_{0}^{2}\bigl(e^{y}-1\bigr)^{2}\,dy

\displaystyle \pi\int_{0}^{e^{2}-1}\bigl(\ln (x+1)\bigr)^{2}\,dx