First slide

Introduction to definite integration

Question

$\underset{n \to \infty}{L t} \sum_{r = 1}^{n} \frac{1}{\sqrt{4 n^{2} - r^{2}}}$

Moderate

A

$π$
B

$\frac{π}{2}$
C

$\frac{π}{3}$
D

$π - \frac{π}{6}$

Solution

$\begin{array}{l} {Lt}_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{n} \frac{1}{\sqrt{4 - {(\frac{r}{n})}^{2}}} = \int_{0}^{1} \frac{1}{\sqrt{4 - x^{2}}} d x \\ = {(\sin^{- 1} (\frac{x}{2}))}_{0}^{1} = \sin^{- 1} (\frac{1}{2}) - \sin^{- 1} (0) \\ = \frac{π}{6} - 0 = \frac{π}{6} \end{array}$

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