Integration to sum

The definite integral of a continuous function $f$ over the interval $[a,b]$, denoted by $\int_a^b f(x) dx$, is the limit of a Riemann sum as the number of subdivisions approach infinity.
\[ \int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n \delta x \cdot f(x_i) \]
where $\delta x = (b-a)/n$, and $x_i = a+\delta x \cdot i$.

Example
\[\int_{\pi}^{2\pi} cos(x)dx \]
\[\delta x = (b-a)/n = (2\pi-\pi)/n = pi/n\]
\[x_i = a + \delta x \cdot i = \pi + \pi/n \cdot i \]
Therefore
\[\int_{\pi}^{2\pi} \cos(x)dx = \lim_{n\to \infty}\sum_{i=1}^{n} \frac{\pi}{n}\cdot \cos(\pi+\frac{\pi i}{n}) \]

Reference:
https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum