Question

Let $g _{ i }:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R , i =1,2$ and $f :\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R$ be functions such that $g_{1}(x)=1, g_{2}(x)=|4 x-\pi|$ and $f(x)=\sin ^{2} x$, for all $x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] .$ Define $S_{i}=\int\limits_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2$
The value of $\frac{16 S _{1}}{\pi}$ is_____

Answer 1

$g _{1}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R , i =1,2, f :\left[\frac{\pi}{ r }, \frac{3 \pi}{ r }\right] \rightarrow R$
$g _{1}=1, g _{2}=|4 x -\pi|, f ( x )=\sin ^{2} x$
$S _{1}=\int\limits_{\pi / 8}^{3 / 8} f ( x ) \cdot g _{1}( x ) d x$
$S _{1}=\int\limits_{\pi / 8}^{3 \pi / 8} \sin ^{2} x dx =\int\limits_{\pi / 8}^{3 \pi / 8} \sin ^{2}\left(\frac{\pi}{2}- x \right) dx $
$\Rightarrow 2 S _{1}=\int\limits_{\pi / 8}^{3 \pi / 8} 1 dx$
$\Rightarrow S _{1}=\frac{1}{2}\left(\frac{3 \pi}{8}-\frac{\pi}{8}\right)=\frac{\pi}{8} $
$\Rightarrow \frac{16 S _{1}}{\pi}=2$