Answer
$A$ = $\frac{\sqrt 3}{2}+\frac{\pi}{3}$
Work Step by Step
$A$ = $\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\int_1^{2\cosθ}rdrdθ$
$A$ = $2\int_{0}^{\frac{\pi}{3}}\int_1^{2\cosθ}rdrdθ$
$A$ = $\int_{0}^{\frac{\pi}{3}}[r^2]_1^{2\cosθ}dθ$
$A$ = $\int_{0}^{\frac{\pi}{3}}[4\cos^2{θ}-1]dθ$
$A$ = $\int_{0}^{\frac{\pi}{3}}[2\cos2θ+1]dθ$
$A$ = $[\sin2θ+θ]_{0}^{\frac{\pi}{3}}$
$A$ = $\sin{\frac{2\pi}{3}}+\frac{\pi}{3}$
$A$ = $\frac{\sqrt 3}{2}+\frac{\pi}{3}$
