First slide

Introduction to definite integration

Question

The value of $lim_{n \to \infty} \frac{1}{n} (\sin \frac{π}{n} + \sin \frac{2 π}{n} + \dots + \sin \frac{n π}{n})$ is

Easy

A

$π$
B

$- \frac{π}{2}$
C

$\frac{1}{π}$
D

$\frac{2}{π}$

Solution

Read. $limit = lim_{n \to \infty} \frac{1}{n} \sum_{n = 1}^{n} \sin \frac{r π}{n}$
$\begin{array}{l} = \int_{0}^{1} \sin π x d x = - {\frac{\cos π x}{π}|}_{0}^{1} \\ = \frac{- 1}{π} [- 1 - 1] = \frac{2}{π} \end{array}$

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