If the Infinite Curve $y = e ^ { - x } , x \geq 0$

Multiple Choice

If the infinite curve $y = e ^ { - x } , x \geq 0$ , is rotated about the $x$ -axis, find the area of the resulting surface. Select the correct answer.

A) $\pi [ \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } ) ]$
B) $\frac { \pi } { 6 } [ \sqrt { 3 } + \ln ( 1 + \sqrt { 3 } ) ]$
C) $\frac { \pi } { 3 } [ \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } ) ]$
D) $\frac { \pi } { 4 } [ \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } ) ]$
E) $\frac { \pi } { 6 } [ \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } ) ]$