Question

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

Answer 1

Answer: 2

$g_1:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to R,i=1,2,f:\left[\frac{\pi }{r},\frac{3\pi }{r}\right]\to R$
$g_1=1,g_2=|4x-\pi |,f(x)={\sin }^{2}x$
$S_1=\underset{\pi /8}{\overset{3/8}{\int }}f(x)\cdot g_1(x)dx$
$S_1=\underset{\pi /8}{\overset{3\pi /8}{\int }}{\sin }^{2}xdx=\underset{\pi /8}{\overset{3\pi /8}{\int }}{\sin }^2\left(\frac{\pi }{2}-x\right)dx$
$\Rightarrow 2S_1=\underset{\pi /8}{\overset{3\pi /8}{\int }}1dx$
$\Rightarrow S_1=\frac{1}{2}\left(\frac{3\pi }{8}-\frac{\pi }{8}\right)=\frac{\pi }{8}$
$\Rightarrow \frac{16S_1}{\pi }=2$