$r = a(1 + 2\cos \theta)$

$A = {\displaystyle \frac{1}{2}{\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta$

$A = 2 \left[ {\displaystyle \frac{1}{2} \int_{2\pi/3}^\pi} a^2 (1 + 2\cos \theta)^2 \, d\theta \right]$

$A = {\displaystyle a^2 \int_{2\pi/3}^\pi} (1 + 4\cos \theta + 4\cos^2 \theta) \, d\theta$

$A = {\displaystyle a^2 \int_{2\pi/3}^\pi} \left[ 1 + 4\cos \theta + 4 \left( \dfrac{1 + \cos 2\theta}{2} \right) \right] \, d\theta$

$A = {\displaystyle a^2 \int_{2\pi/3}^\pi} [ \, 1 + 4\cos \theta + (2 + 2\cos 2\theta) \, ] \, d\theta$

$A = {\displaystyle a^2 \int_{2\pi/3}^\pi} [ \, 3 + 4\cos \theta + 2\cos 2\theta \, ] \, d\theta$

$A = a^2 \Big[ 3\theta + 4\sin \theta + \sin 2\theta \Big]_{2\pi/3}^\pi$

$A = a^2 \Big[ 3\pi + 4\sin \pi + \sin 2\pi \Big] - a^2 \Big[ 3(\frac{2}{3}\pi + 4\sin \frac{2}{3}\pi + \sin \frac{4}{3}\pi \Big]$

$A = a^2 [ \, 3\pi + 0 + 0 \, ] - a^2 [ \, 2\pi + 3.4641 - 0.8660 \,]$

$A = 0.5435a^2 \, \text{ unit}^2$ *answer*