Definite Integration

Let \({g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2\), and \(f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R\) be functions such that \({g_1}(x) = 1,{g_2}(x) = |4x - \pi |\) and \(f(x) = {\sin ^2}x\), for all \(x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]\). Define \({S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx}\), i = 1, 2

The value of \({{48{S_2}} \over {{\pi ^2}}}\) is ___________.