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 $f(x)=x\cos(x),\quad 0 \le x \le \pi/2.$ Using the above definition, determine which of the following expressions represents the area under the graph of $f$ as a limit.

A. $\displaystyle \underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \left(\frac{\pi i}{2n}\cos\left(\frac{\pi i}{2n}\right)\right)$
B. $\displaystyle \underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{\pi}{2n}\left(\cos\left(\frac{\pi i}{2n}\right)\right)$
C. $\displaystyle \underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{\pi i}{2n}\left(\frac{\pi i}{2n}\cos\left(\frac{\pi i}{2n}\right)\right)$
D. $\displaystyle \underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{\pi}{2n}\left(\frac{\pi}{2n}\cos\left(\frac{\pi}{2n}\right)\right)$
E. $\displaystyle \underset{n \rightarrow \infty}{\lim} \overset{n}{\underset{i=1}{\sum}} \frac{\pi}{2n}\left(\frac{\pi i}{2n}\cos\left(\frac{\pi i}{2n}\right)\right)$