<p>It is given that</p>
<p style="text-align: center">$\mathrm{f}\left( x \right)=\left\{ \begin{matrix}
&amp; \sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}}\,\,\,\,\mathrm{for}\,\,-a\le x\le a,\\&amp;0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,a&lt;x\le2a, \\\end{matrix} \right.$</p>
<p>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.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-4a\le x\le 6a$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use the substitution $x=a\sin \theta $ to find the exact area of $\displaystyle\int\nolimits_{\frac{1}{2}a}^{\frac{\sqrt{3}}{2}a}{\mathrm{f}\left( x \right)\,\mathrm{d}x}$ in terms of $a$ and $\pi $.</span></div><p style="text-align: right;">[5]</p></li></ol>

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

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.

Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-4a\le x\le 6a$.

Use the substitution $x=a\sin \theta $ to find the exact area of $\displaystyle\displaystyle \displaystyle\int\nolimits_{\frac{1}{2}a}^{\frac{\sqrt{3}}{2}a}{\mathrm{f}\left( x \right)\,\mathrm{d}x}$ in terms of $a$ and $\pi $.

N2013 P1 Q5

N2013 P1 Q5

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