CAIE A-Level Further Mathematics 1.5.3 Polar Area

Practise selecting angular limits, integrating one-half of the squared radius exactly and subtracting polar regions from triangles or sectors when required.

Exam points

identify the bounding angles and integrate one-half of the squared radius over those limits
use a trigonometric identity to obtain an exact integral rather than a decimal area
for a remaining region, subtract the polar integral from the relevant triangle or sector

[Maximum number: 4]

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

$\frac{1}{2} \int_{0}^{\frac{1}{4} \pi} \sin ^{2} 4 \theta \mathrm{~d} \theta$

M1

Uses $\frac{1}{2} \int r^{2} \mathrm{~d} \theta$ with correct limits.

$=\frac{1}{4} \int_{0}^{\frac{1}{4} \pi} 1-\cos 8 \theta \mathrm{~d} \theta=\frac{1}{4}\left[\theta-\frac{1}{8} \sin 8 \theta\right]_{0}^{\frac{1}{4} \pi}$

M1 A1

Applies double angle formula and integrates.

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

A1

4

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1.5.3—Polar area question 1