Mathematics - Definite Integration Question with Solution | TestHub

${\mathrm{S}}_{\mathrm{n}}={\sum }_{\mathrm{r}=0}^{\mathrm{n}-1}\frac{1}{\sqrt{4{\mathrm{n}}^2-{\mathrm{r}}^2}}\Rightarrow {\mathrm{S}}_{\mathrm{n}}=\frac{1}{\mathrm{n}}{\sum }_{\mathrm{r}=0}^{\mathrm{n}-1}\frac{1}{\sqrt{4-{\left(\frac{\mathrm{r}}{\mathrm{n}}\right)}^2}}$

$$\begin{array}{rlr}& S_n={\int }_0^1\frac{dx}{\sqrt{4-x^2}}={\left[\frac{1}{2}{\sin }^{-1}\left(\frac{x}{2}\right)\right]}_0^1& \end{array}$$

$$S_n={\sin }^{-1}\left(\frac{1}{2}\right)\Rightarrow S_n=\frac{\pi }{6}\Rightarrow S_n=\frac{\pi }{6}$$