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

$A = 4 \left[ {\displaystyle \frac{1}{2}{\int_0}^{\pi/2}} (2a \, \sin^2 \theta)^2 \, d\theta \right]$

$A = 2{\displaystyle {\int_0}^{\pi/2}} (4a^2)[ \, \frac{1}{2} (1 - \cos 2\theta) \, ]^2 \, d\theta$

$A = 8a^2{\displaystyle {\int_0}^{\pi/2}} \frac{1}{4} (1 - 2\cos 2\theta + \cos^2 2\theta) \, d\theta$

$A = 2a^2{\displaystyle {\int_0}^{\pi/2}} [ \, 1 - 2\cos 2\theta + \frac{1}{2}(1 + \cos 4\theta) \, ] \, d\theta$

$A = 2a^2{\displaystyle {\int_0}^{\pi/2}} [ \, \frac{3}{2} - 2\cos 2\theta + \frac{1}{2} \cos 4\theta \, ] \, d\theta$

$A = 2a^2 \left[ \, \frac{3}{2}\theta - \sin 2\theta + \frac{1}{8} \sin 4\theta \right]_0^{\pi/2}$

$A = \left[ \, 3a^2\theta - 2a^2\sin 2\theta + \frac{1}{4}a^2 \sin 4\theta \right]_0^{\pi/2}$

$A = \left[ \, 3a^2(\pi/2) - 2a^2\sin \pi + \frac{1}{4}a^2 \sin 2\pi \right] - \left[ \, 3a^2(0) - 2a^2\sin 0 + \frac{1}{4}a^2 \sin 0 \right]$

$A = \frac{3}{2}\pi a^2$ *answer*