Answer
$A = \lim\limits_{n \to \infty} \sum_{i=1}^{n}\sqrt{sin(\frac{\pi~i}{n}})\cdot \frac{\pi}{n}$
Work Step by Step
For each $x_i$, such that $1 \leq i \leq n$, note that $x_i = \frac{\pi~i}{n}$
$\Delta x = \frac{\pi}{n}$
We can express the area under the curve as a limit:
$A = \lim\limits_{n \to \infty} [f(x_1)\Delta x+f(x_2)\Delta x+...+f(x_n)\Delta x]$
$A = \lim\limits_{n \to \infty} [f(\frac{\pi}{n})(\frac{\pi}{n})+f(\frac{2\pi}{n})(\frac{\pi}{n})+...+f(\frac{n\pi}{n})(\frac{\pi}{n})]$
$A = \lim\limits_{n \to \infty} [\sqrt{sin(\frac{\pi}{n}})(\frac{\pi}{n}) + \sqrt{sin(\frac{2\pi}{n}})(\frac{\pi}{n}) +...+\sqrt{sin(\frac{n\pi}{n}})(\frac{\pi}{n}) ]$
$A = \lim\limits_{n \to \infty} \sum_{i=1}^{n}\sqrt{sin(\frac{\pi~i}{n}})\cdot \frac{\pi}{n}$
