Answer
$${\text{Area}} = 15$$
Work Step by Step
$$\eqalign{
& y = \frac{1}{4}{x^3},{\text{ }}\left[ {2,4} \right] \cr
& f\left( x \right) = \frac{1}{4}{x^3} \cr
& f\left( 2 \right) = 2{\text{ and }}f\left( 4 \right) = 16,{\text{ }} \cr
& f\left( x \right){\text{ is continuous and not negative on the interval }}\left[ {2,4} \right]. \cr
& {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {{c_i}} \right)} \Delta x,{\text{ }}\Delta x = \frac{{4 - 2}}{n} = \frac{2}{n} \cr
& {c_i} = a + i\Delta x \to 2 + \frac{2}{n} \cr
& {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left[ {\frac{1}{4}{{\left( {2 + \frac{{2i}}{n}} \right)}^3}\left( {\frac{2}{n}} \right)} \right]} \cr
& {\text{Area}} = \frac{1}{2}\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{i = 1}^n {\left( {\frac{{8{i^3}}}{{{n^3}}} + \frac{{24{i^2}}}{{{n^2}}} + \frac{{24i}}{n} + 8} \right)} \cr
& {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left( {\frac{{4{i^3}}}{{{n^4}}} + \frac{{12{i^2}}}{{{n^3}}} + \frac{{12i}}{{{n^2}}} + \frac{4}{n}} \right)} \cr
& {\text{Using the Summation Formulas from THEOREM 4}}{\text{.2}} \cr
& {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \frac{4}{{{n^4}}}\left( {\frac{{{n^2}{{\left( {n + 1} \right)}^2}}}{4}} \right) + \mathop {\lim }\limits_{n \to \infty } \left( {\frac{{12}}{{{n^3}}}\left( {\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}} \right)} \right) \cr
& + \mathop {\lim }\limits_{n \to \infty } \frac{{12}}{{{n^2}}}\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right) + 4\left( {\mathop {\lim }\limits_{n \to \infty } \frac{n}{n}} \right) \cr
& {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{{n^2}}} + \frac{2}{n} + 1} \right) + \mathop {\lim }\limits_{n \to \infty } \left( {\frac{2}{{{n^2}}} + \frac{6}{n} + 4} \right) \cr
& + \mathop {\lim }\limits_{n \to \infty } \left( {\frac{6}{n} + 6} \right) + 4\mathop {\lim }\limits_{n \to \infty } \left( 1 \right) \cr
& {\text{Evaluate the limit when }}n \to \infty \cr
& {\text{Area}} = \left( {0 + 0 + 1} \right) + \left( {0 + 0 + 4} \right) + \left( {0 + 6} \right) + 4 \cr
& {\text{Area}} = 1 + 4 + 6 + 4 \cr
& {\text{Area}} = 15 \cr} $$
