Let a Denote the Area Enclosed by the Graph $f ( x ) = \frac { 1 } { 2 } x ^ { 3 },$

Multiple Choice

Let A denote the area enclosed by the graph $f ( x ) = \frac { 1 } { 2 } x ^ { 3 },$ the x-axis, and the lines x = 0 and x = 2. Graphing the region and using plane geometry, we can find that A is

A) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$
B) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k - 1 } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$
C) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k + 2 } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$
D) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 2 } { n } \right)$
E) $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left( \frac { 2 k } { n } \right) ^ { 3 } \left( \frac { 1 } { n } \right)$