First slide

Introduction to definite integration

Question

$lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r}{\sqrt{n^{2} + r^{2}}}$ equals

Very Easy

A

$1 + \sqrt{5}$
B

$- 1 + \sqrt{5}$
C

$- 1 + \sqrt{2}$
D

$1 + \sqrt{2}$

Solution

Required limit = $lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \frac{r / n}{\sqrt{1 + r^{2} / n^{2}}}$
$= \int_{0}^{2} \frac{x}{\sqrt{1 + x^{2}}} d x = {\sqrt{1 + x^{2}}|}_{0}^{2} = \sqrt{5} - 1 .$

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