9 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 \(m > 0\). Find the \(y\)-coordinate of the centroid of \(R\).