Originally Posted by

**james121515** Hi,

I am trying to calculate:

$\displaystyle \int_0^{\pi}\sin x \,dx $

using the definition only, that is, that $\displaystyle \int_0^{\pi}\sin x\,dx = \lim_{n \to \infty}\sum_{i = 1}^{\infty}\sin(x_{i}^{*})\Delta x$, where $\displaystyle \Delta x = \frac{\pi}{n}$. If we take take $\displaystyle x_i^* = i\Delta x = \frac{\pi}{n}$ we obtain:

$\displaystyle \int_0^{\pi}\sin x\,dx = \lim_{n \to \infty}\left(\frac{\pi}{n}\right)\sum_{i=1}^n \sin\left(\frac{i\pi}{n}\right)$.

I am stuck at this point. Is there any kind of identity, perhaps trigonometric or power series related, to help me evaluate this limit by hand only?

Thank in advanced or any help,

James