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 $$A = \lim_{n\to\infty} R_n = \lim_{n\to\infty} \bigl[ f(x_1)\Delta x + f(x_2) \Delta x + \dotsb + f(x_n)\Delta x \bigr].$$ (a) Use the above definition to determine which of the following expressions represents the area under the graph of $f(x)=x^3$ from $x = 0$ to $x = 1$.
A. $\displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \left(\frac{i}{n}\right)\frac{5}{n}$
B. $\displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \left(\frac{i}{n}\right)^3\frac{5}{n}$
C. $\displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \left(\frac{i}{n}\right)\frac{1}{n}$
D. $\displaystyle \lim_{n \to \infty} \sum_{i = 1} ^ n \left(\frac{i}{n}\right)^3\frac{1}{n}$


(b) Evaluate the limit that is the correct answer to part (a). You may find the following formula for the sum of cubes helpful: $$1^3 + 2^3 + 3^3 + \dotsb + n^3 = \sum_{i = 1} ^ n i^3 = \left(\frac{n(n+1)}{2}\right)^2.$$

The value of the limit is .