Answer
$\frac{6}{5}{\pi}a^{2}$
Work Step by Step
$x$ = $a\cos^3\theta$
$y$ = $a\sin^3\theta$, $0$ $\leq\theta\leq\frac{\pi}{2}$
$\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2 = (-3a\cos^2\theta\sin\theta)^2+(3a\sin^2\theta\cos\theta)^2$
$=9a^2\cos^4\theta\sin^2\theta+9a^2\sin^4\theta\cos^2\theta$
$=9a^2\sin^2\theta\cos^2\theta(\cos^2\theta+\sin^2\theta)$
$=9a^2\sin^2\theta\cos^2\theta$
$S$ = $\int_0^{\frac{\pi}{2}}2{\pi}(a\sin^3\theta)(3a\sin\theta\cos\theta)d\theta$
$=6\pi a^2\int_0^{\frac{\pi}{2}}2{\pi}\sin^4\theta\cos\theta d\theta$
$=6\pi a^2\left(\frac{\sin^5\theta}{5}\right)_0^{\pi/2}$
$=\frac{6}{5}{\pi}a^{2}$
