Mathematics

Problem

Consider the region

P

bounded on the left by the

y

–axis, above by the curve

y=e^{x}

, and below by the

x

–axis for

0\le x\le 2

. When this region is revolved about the vertical line

x=2

, a solid is formed. (a) Express the horizontal radius of a typical circular cross–section of the solid as a function of

y

. (b) Write a definite integral with respect to

y

that represents the exact volume of the solid. (c) Evaluate the integral from part (b) to obtain the exact volume.