Standard +0.8 This is a Further Maths polar coordinates question requiring sketching and area calculation. While the integral setup is standard (½∫r²dθ), students must correctly handle the algebraic form r=(π/2-θ)², expand it carefully, and integrate a polynomial in θ. The algebra is moderately involved and errors are easy to make, placing this above average difficulty but not exceptionally hard for FM students.
3 The curve \(C\) has polar equation
$$r = \left( \frac { 1 } { 2 } \pi - \theta \right) ^ { 2 } ,$$
where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\).
Find the area of the region bounded by \(C\) and the initial line, leaving your answer in terms of \(\pi\).
3 The curve $C$ has polar equation
$$r = \left( \frac { 1 } { 2 } \pi - \theta \right) ^ { 2 } ,$$
where $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. Draw a sketch of $C$.
Find the area of the region bounded by $C$ and the initial line, leaving your answer in terms of $\pi$.
\hfill \mbox{\textit{CAIE FP1 2008 Q3 [6]}}