$A = 2\left\{ \left[ {\displaystyle \frac{1}{2}{\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta \right]_{cardioid} - \left[ {\displaystyle \frac{1}{2}{\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta \right]_{circle}\right\}$

$A = \left[ {\displaystyle {\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta \right]_{cardioid} - \left[ {\displaystyle {\int_{\theta_1}}^{\theta_2}} r^2 \, d\theta \right]_{circle}$

$A = {\displaystyle {\int_0}^{\pi/2}} a^2(1 + \cos \theta)^2 \, d\theta - {\displaystyle {\int_0}^{\pi/2}} a^2 \, d\theta$

$A = {\displaystyle a^2 {\int_0}^{\pi/2}} (1 + 2\cos \theta + \cos^2 \theta) \, d\theta - {\displaystyle a^2 {\int_0}^{\pi/2}} d\theta$

$A = {\displaystyle a^2 {\int_0}^{\pi/2}} d\theta + {\displaystyle 2a^2 {\int_0}^{\pi/2}} \cos \theta \, d\theta + {\displaystyle a^2 {\int_0}^{\pi/2}} \cos^2 \theta \, d\theta - {\displaystyle a^2 {\int_0}^{\pi/2}} d\theta$

$A = {\displaystyle 2a^2 {\int_0}^{\pi/2}} \cos \theta \, d\theta + {\displaystyle a^2 {\int_0}^{\pi/2}} \cos^2 \theta \, d\theta$

$A = {\displaystyle 2a^2 {\int_0}^{\pi/2}} \cos \theta \, d\theta + {\displaystyle a^2 {\int_0}^{\pi/2}} \frac{1}{2}(1 + \cos 2\theta) \, d\theta$

$A = 2a^2 \Big[ \sin \theta \Big]_0^{\pi/2} + \frac{1}{2}a^2 \Big[ \theta + \frac{1}{2} \sin 2\theta \Big]_0^{\pi/2}$

$A = 2a^2 \left[ \sin \frac{1}{2}\pi - \sin 0 \right] + \frac{1}{2}a^2 \left[ (\frac{1}{2}\pi + \frac{1}{2} \sin \pi) - (0 + \frac{1}{2} \sin 0) \right]$

$A = 2a^2 [ \, 1 \, ] + \frac{1}{2}a^2 [ \, \frac{1}{2}\pi \, ]$

$A = (2 + \frac{1}{4}\pi)a^2$

$A = 2.7854a^2 \, \text{ unit}^2$ *answer*