<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=3\cos \theta -2\cos 3\theta $,</p>
<p style="text-align: center">$y=9\sin \theta -\sin 3\theta $,</p>
<p>for $0\le \theta \le 2\pi $.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch $C$ and state the cartesian equations of its lines of symmetry.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Given that $\cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta $, find the exact values of $\theta $ at the points where $C$ meets the $y$-axis.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Show that the area enclosed by the axes and the part of $C$ in the first quadrant is given by</div><div class="list-item-paragraph" style="text-align: center">$\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( 27{{\cos }^{2}}\theta -27\cos \theta \cos 3\theta +6{{\cos }^{2}}3\theta \right)\,\mathrm{d}\theta }$,</div><div class="list-item-paragraph">where the values of ${{\theta }_{1}}$ and ${{\theta }_{2}}$ should be stated.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence find the exact total area enclosed by $C$.</div><p style="text-align: right;">[5]</p></li></ol>

Sketch $C$ and state the cartesian equations of its lines of symmetry.

Given that $\cos 3\theta =4{{\cos }^{3}}\theta -3\cos \theta $, find the exact values of $\theta $ at the points where $C$ meets the $y$-axis.

Show that the area enclosed by the axes and the part of $C$ in the first quadrant is given by

$\displaystyle\displaystyle \int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{\left( 27{{\cos }^{2}}\theta -27\cos \theta \cos 3\theta +6{{\cos }^{2}}3\theta \right)\,\mathrm{d}\theta }$,

where the values of ${{\theta }_{1}}$ and ${{\theta }_{2}}$ should be stated.

Hence find the exact total area enclosed by $C$.

Applications of Intergration - General

Applications of Intergration

H2 Math

International Baccalaureate Higher Level Mathematics

2020 YIJC P2 Q3

2020 YIJC P2 Q3

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