CAIE A-Level Further Math AS 1.5.3 Polar Area Questions

$\frac{1}{2} \int_{0}^{\frac{1}{6} \pi} \sin ^{2} 2 \theta \mathrm{~d} \theta+\frac{1}{2} \int_{\frac{1}{6} \pi}^{\frac{1}{2} \pi} \cos ^{2} \theta \mathrm{~d} \theta$
Uses $\frac{1}{2} \int r^{2} \mathrm{~d} \theta$ with correct limits.

$\frac{1}{2} \int_{0}^{\frac{1}{6} \pi} \sin ^{2} 2 \theta \mathrm{~d} \theta=\frac{1}{4} \int_{0}^{\frac{1}{6} \pi} 1-\cos 4 \theta \mathrm{~d} \theta$
Integrates $\sin ^{2} 2 \theta$ using identity.

$=\frac{1}{4}\left[\theta-\frac{1}{4} \sin 4 \theta\right]_{0}^{\frac{1}{6} \pi}$$\frac{1}{2} \int_{\frac{1}{6} \pi}^{\frac{1}{2} \pi} \cos ^{2} \theta \mathrm{~d} \theta=\frac{1}{4} \int_{\frac{1}{6} \pi}^{\frac{1}{2} \pi} 1+\cos 2 \theta \mathrm{~d} \theta$
Integrates $\cos ^{2} \theta$ using identity.

$=\frac{1}{4}\left[\theta+\frac{1}{2} \sin 2 \theta\right]_{\frac{1}{6} \pi}^{\frac{1}{2} \pi}$$\frac{1}{4}\left(\frac{1}{6} \pi-\frac{1}{8} \sqrt{3}\right)+\frac{1}{4}\left(\frac{1}{2} \pi-\frac{1}{6} \pi-\frac{1}{4} \sqrt{3}\right)=\frac{1}{8}\left(\pi-\frac{3}{4} \sqrt{3}\right)$
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