Let
\(I_{n}=\int_{0}^{\frac{1}{2} \pi} \sin ^{n} \theta \mathrm{~d} \theta\)
where \(n\) is a non-negative integer. Show that \(I_{n+2}=\frac{n+1}{n+2} I_{n}\).

The region \(R\) of the \(x-y\) plane is bounded by the \(x\)-axis, the line \(x=\frac{\pi}{2 m}\) and the curve whose equation is \(y=\sin ^{4} m x\), where \(m\gt 0\). Find the \(y\)-coordinate of the centroid of \(R\).