Question

Let $g_i:[\frac{\pi }{8},\frac{3\pi }{8}]\to R,i=1,2$ and $f:[\frac{\pi }{8},\frac{3\pi }{8}]\to R$ be functions such that $g_1\left(x\right)=1,g_2\left(x\right)=|4x-\pi |$ and $f\left(x\right)={\sin }^{2}x,$ for all $x\in [\frac{\pi }{8},\frac{3\pi }{8}].$
If $S_i=\underset{\pi /8}{\overset{3\pi /8}{\int }}f\left(x\right)\cdot g_i\left(x\right)dx,i=1,2,$ then the value of $\frac{16S_1}{\pi }$ is

• A. $3$
• B. $\frac{1}{2}$
• C. $2$
• D. $1$

Answer 1

The correct option is C $2$

$S_1=\underset{\pi /8}{\overset{3\pi /8}{\int }}f\left(x\right)\cdot g_1\left(x\right)dx$
Given : $g_1\left(x\right)=1$
$\Rightarrow S_1=\underset{\pi /8}{\overset{3\pi /8}{\int }}{\sin }^{2}xdx$
$\Rightarrow S_1=\frac{1}{2}\underset{\pi /8}{\overset{3\pi /8}{\int }}(1-\cos 2x)dx$
$S_1=\frac{1}{2}[\frac{\pi }{4}-{\left(\frac{\sin 2x}{2}\right)}_{\pi /8}^{3\pi /8}]$
$S_1=\frac{\pi }{8}$
$\therefore \frac{16S_1}{\pi }=2$