6. \begin{figure}[h] \end{figure} Figure 2 shows the curve with equation $$y = x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right) , \quad 0 < x \leq 2 \pi .$$ The finite region \(R\) bounded by the line \(x = \pi\), the \(x\)-axis, and the curve is shown shaded in Fig 2.

1. Find the exact value of the area of \(R\), by integration. Give your answer in terms of \(\pi\). The table shows corresponding values of \(x\) and \(y\).

   \(x\)	\(\pi\)	\(\frac { 5 \pi } { 4 }\)	\(\frac { 3 \pi } { 2 }\)	\(\frac { 7 \pi } { 4 }\)	\(2 \pi\)
   \(y\)	9.8696	14.247	15.702	\(G\)	0

2. Find the value of \(G\).
3. Use the trapezium rule with values of \(x ^ { 2 } \sin \left( \frac { 1 } { 2 } x \right)\)
   1. at \(x = \pi , x = \frac { 3 \pi } { 2 }\) and \(x = 2 \pi\) to find an approximate value for the area \(R\), giving your answer to 4 significant figures,
   2. at \(x = \pi , x = \frac { 5 \pi } { 4 } , x = \frac { 3 \pi } { 2 } , x = \frac { 7 \pi } { 4 }\) and \(x = 2 \pi\) to find an improved approximation for the area \(R\), giving your answer to 4 significant figures.
      6. continued