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 $$A= \lim_{n\rightarrow\infty} R_n=\lim_{n\rightarrow\infty}[f(x_1)\Delta x+f(x_2)\Delta x+...+f(x_n)\Delta x]$$

(a) Use the above definition to determine which of the following expressions represents the area under the graph of $f(x)=x^5$ from $x$ = 0 to $x$ = 2.

A. $\displaystyle \underset{n \rightarrow \infty}{\lim}\, \frac{1}{n^6}\overset{n}{\underset{i=1}{\sum}} \,i^5$
B. $\displaystyle \underset{n \rightarrow \infty}{\lim}\, \frac{64}{n}\overset{n}{\underset{i=1}{\sum}} \,i^5$
C. $\displaystyle \underset{n \rightarrow \infty}{\lim}\, \frac{64}{n^6}\overset{n}{\underset{i=1}{\sum}} \,i$
D. $\displaystyle \underset{n \rightarrow \infty}{\lim}\, \frac{64}{n^6}\overset{n}{\underset{i=1}{\sum}} \,i^5$

(b) Evaluate the limit that is the correct answer to part (a). You may find the following formula helpful: $$1^5+2^5+3^5+...+n^5 = \overset{n}{\underset{i=1}{\sum}} \,i^5 = \frac{n^2(n+1)^2(2n^2+2n-1)}{12}.$$

Value of limit =