Definition: The AREA A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles $$A= \lim_{n\rightarrow\infty} R_n=\lim_{n\rightarrow\infty}[f(x_1)\Delta x+f(x_2)\Delta x+...+f(x_n)\Delta x]$$

Consider the function $f(x)=\sqrt[4]{x},\quad 1 \le x \le 16.$ Using the above definition, determine which of the following expressions represents the area under the graph of $f$ as a limit.

A. $\underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \sqrt[4]{\frac{16i}{n}}$
B. $\underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{15}{n}\sqrt[4]{\frac{15i}{n}}$
C. $\underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{16}{n}\sqrt[4]{1+\frac{16i}{n}}$
D. $\underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{15}{n}\sqrt[4]{1+\frac{15i}{n}}$