Which of These Integrals Can Be Used to Calculate the Surface

Multiple Choice

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \sqrt [ 3 ] { x },$ the -axis, x = 1, and x = 5 about the x-axis?

A) $\int _ { 1 } ^ { 5 } x ^ { 1 / 3 } \sqrt { 1 + \left( \frac { 1 } { 3 } x ^ { - 2 / 3 } \right) ^ { 2 } } d x$
B) $2 \pi \int _ { 1 } ^ { 5 } \sqrt { 1 + \left( \frac { 1 } { 3 } x ^ { - 2 / 3 } \right) ^ { 2 } } d x$
C) $2 \pi \int _ { 1 } ^ { 5 } x ^ { 1 / 3 } \sqrt { 1 + \left( \frac { 1 } { 3 } x ^ { - 2 / 3 } \right) ^ { 2 } } d x$
D) $2 \pi \int _ { 1 } ^ { 5 } x ^ { 1 / 3 } \sqrt { 1 + \frac { 1 } { 3 } x ^ { - 2 / 3 } } d x$
E) $2 \pi \int _ { 1 } ^ { 5 } x ^ { 1 / 3 } \sqrt { 1 + \left( x ^ { 1 / 3 } \right) ^ { 2 } } d x$