N2013 P1 Q5

It is given that

\(\mathrm{f}\left( x \right)=\left\{ \begin{matrix} & \sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}}\,\,\,\,\mathrm{for}\,\,-a\le x\le a,\\&0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,a<x\le2a, \end{matrix} \right.\)

and that \(\mathrm{f}\left( x+3a \right)=\mathrm{f}\left( x \right)\) for all real values of \(x\), where \(a\) is a real constant.

1. Sketch the graph of \(y=\mathrm{f}\left( x \right)\) for \(-4a\le x\le 6a\).[3]
2. Use the substitution \(x=a\sin \theta \) to find the exact area of \(\displaystyle\displaystyle \displaystyle\int_{\frac{1}{2}a}^{\frac{\sqrt{3}}{2}a}{\mathrm{f}\left( x \right)\,\mathrm{d}x}\) in terms of \(a\) and \(\pi \).[5]