Answer
(a) $A = \lim\limits_{n \to \infty} \sum_{i=1}^{n} (\frac{i}{n})^3(\frac{1}{n})$
(b) $A = \frac{1}{4}$
Work Step by Step
(a) For each $x_i$, such that $1 \leq i \leq n$, note that $x_i = \frac{i}{n}$
$\Delta x = \frac{1}{n}$
We can express the area under the graph as a limit:
$A = \lim\limits_{n \to \infty} [f(x_1)\Delta x+f(x_2)\Delta x+...+f(x_n)\Delta x]$
$A = \lim\limits_{n \to \infty} [f(\frac{1}{n})(\frac{1}{n})+f(\frac{2}{n})(\frac{1}{n})+...+f(\frac{n}{n})(\frac{1}{n})]$
$A = \lim\limits_{n \to \infty} [ (\frac{1}{n})^3(\frac{1}{n}) + (\frac{2}{n})^3(\frac{1}{n})+...+ (\frac{n}{n})^3(\frac{1}{n})]$
$A = \lim\limits_{n \to \infty} \sum_{i=1}^{n} (\frac{i}{n})^3(\frac{1}{n})$
(b) We can evaluate the limit in part a:
$A = \lim\limits_{n \to \infty} \sum_{i=1}^{n} (\frac{i}{n})^3(\frac{1}{n})$
$A = \lim\limits_{n \to \infty} [ (\frac{1}{n})^3(\frac{1}{n}) + (\frac{2}{n})^3(\frac{1}{n})+...+ (\frac{n}{n})^3(\frac{1}{n})]$
$A = \lim\limits_{n \to \infty} [ \frac{1^3}{n^4} + \frac{2^3}{n^4} +...+ \frac{n^3}{n^4} ]$
$A = \lim\limits_{n \to \infty} [ \frac{1^3+2^3+...+n^3}{n^4}]$
$A = \lim\limits_{n \to \infty} \frac{n^2(n+1)^2}{4n^4}$
$A = \lim\limits_{n \to \infty} \frac{n^2(n^2+2n+1)}{4n^4}$
$A = \lim\limits_{n \to \infty} \frac{n^4+2n^3+n^2}{4n^4}$
$A = \lim\limits_{n \to \infty} \frac{1+\frac{2}{n}+\frac{1}{n^2}}{4}$
$A = \frac{1}{4}$
