| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 4 |
| Topic | Polar coordinates |
| Type | Area enclosed by polar curve |
| Difficulty | Standard +0.3 This is a straightforward application of the polar area formula A = ½∫r²dθ with clearly defined limits. The integrand (sin θ + cos θ)² expands to standard trigonometric terms that integrate routinely using double-angle identities. While polar coordinates is a Further Maths topic, this particular question requires only direct formula application with no geometric insight or problem-solving, making it easier than average overall. |
| Spec | 4.09c Area enclosed: by polar curve |
$A = k\int(\sin\theta + \cos\theta)^2\,\mathrm{d}\theta$ including squaring attempt; ignore limits and $k \neq \frac{1}{2}$ **M1**
$= \frac{1}{2}\int(1 + \sin 2\theta)\,\mathrm{d}\theta$ for use of the double-angle formula **B1**
**OR** integration of $\sin\theta\cos\theta$ as $k\sin^2\theta$ or $k\cos^2\theta$
$= \frac{1}{2}\left[\theta - \frac{1}{2}\cos 2\theta\right]_0^{\pi/2}$ **ft** (constants only) in the integration; MUST be 2 separate terms **A1**
$= \frac{1}{4}\pi + \frac{1}{2}$ **A1**
**[4]**2 Find the area enclosed by the curve with polar equation $r = \sin \theta + \cos \theta , 0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2012 Q2 [4]}}