6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \cos 2 t \quad y = 4 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. 1. Show, making your working clear, that the area of \(R = \int _ { 0 } ^ { \frac { \pi } { 4 } } 32 \sin ^ { 2 } t \cos t d t\)
   2. Hence find, by algebraic integration, the exact value of the area of \(R\).
2. Show that all points on \(C\) satisfy \(y = \sqrt { a x + b }\), where \(a\) and \(b\) are constants to be found. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where f is the function $$f ( x ) = \sqrt { a x + b } \quad - 2 \leqslant x \leqslant 2$$ and \(a\) and \(b\) are the constants found in part (b).
3. State the range of f.