Challenging +1.2 This is a standard polar area calculation requiring the formula A = ½∫r²dθ. While the integrand (2+cos θ)²sin θ requires expansion and basic trigonometric identities, it's a routine Further Maths exercise with straightforward integration techniques. The bounds are given, and no geometric insight or novel problem-solving is needed beyond applying the standard formula.
3 The diagram shows a sketch of a loop, the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_305_553_383_740}
The polar equation of the loop is
$$r = ( 2 + \cos \theta ) \sqrt { \sin \theta } , \quad 0 \leqslant \theta \leqslant \pi$$
Find the area enclosed by the loop.
Valid method to reach \(k(2+\cos\theta)^3\) or \(a\cos\theta + b\cos 2\theta + c\cos^3\theta\) OE; {SC: M1 if expands then integrates to get either \(a\cos\theta + b\cos 2\theta\) OE or \(c\cos^3\theta\) OE in a valid way}
A1
OE e.g. \(-4\cos\theta - \cos 2\theta - \frac{1}{3}\cos^3\theta\)
3 The diagram shows a sketch of a loop, the pole $O$ and the initial line.\\
\includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_305_553_383_740}
The polar equation of the loop is
$$r = ( 2 + \cos \theta ) \sqrt { \sin \theta } , \quad 0 \leqslant \theta \leqslant \pi$$
Find the area enclosed by the loop.
\hfill \mbox{\textit{AQA FP3 2009 Q3 [6]}}