Challenging +1.2 This is a Further Maths polar coordinates question requiring a sketch and area calculation using the standard formula ½∫r²dθ. While it involves integration of trigonometric expressions and comparing two regions, the method is direct and the integration is routine (involving sin²θ). The conceptual demand is moderate for FM students who have learned polar areas, making it slightly above average difficulty overall but not requiring novel insight.
8 The curve \(C\) has polar equation \(r = 1 + \sin \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\).
The area of the region enclosed by the initial line, the half-line \(\theta = \frac { 1 } { 2 } \pi\), and the part of \(C\) for which \(\theta\) is positive, is denoted by \(A _ { 1 }\). The area of the region enclosed by the initial line, and the part of \(C\) for which \(\theta\) is negative, is denoted by \(A _ { 2 }\). Find the ratio \(A _ { 1 } : A _ { 2 }\), giving your answer correct to 1 decimal place.
8 The curve $C$ has polar equation $r = 1 + \sin \theta$ for $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. Draw a sketch of $C$.
The area of the region enclosed by the initial line, the half-line $\theta = \frac { 1 } { 2 } \pi$, and the part of $C$ for which $\theta$ is positive, is denoted by $A _ { 1 }$. The area of the region enclosed by the initial line, and the part of $C$ for which $\theta$ is negative, is denoted by $A _ { 2 }$. Find the ratio $A _ { 1 } : A _ { 2 }$, giving your answer correct to 1 decimal place.
\hfill \mbox{\textit{CAIE FP1 2011 Q8 [10]}}