CAIE A-Level Further Math AS 1.5.3 Polar Area Questions

Practise using polar integration to find exact areas bounded by polar curves and lines.

Exam points

set up polar area integrals
calculate exact areas bounded by polar curves
apply polar area methods to composite regions

[Maximum number: 4]

Find the exact value of the area of the region enclosed by C.

$\frac{1}{2} \int_{0}^{\frac{1}{4} \pi} \sin ^{2} 4 \theta \mathrm{~d} \theta$
Uses $\frac{1}{2} \int r^{2} \mathrm{~d} \theta$ with correct limits.

$=\frac{1}{4} \int_{0}^{\frac{1}{4} \pi} 1-\cos 8 \theta \mathrm{~d} \theta=\frac{1}{4}\left[\theta-\frac{1}{8} \sin 8 \theta\right]_{0}^{\frac{1}{4} \pi}$
Applies double angle formula and integrates.

$=\frac{1}{16} \pi$
4

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CAIE A-Level Further Math AS 1.5.3 Polar Area Questions question 1