Answer
$2 \pi a^2$
Work Step by Step
Applying Stoke's Theorem, we have
$\oint F \cdot dr=\iint _S (\nabla \times F) \cdot n d\sigma$
Here, $F \cdot \dfrac{dr}{dt}=ay \sin t+ax \cos t=a^2 \sin^2 t+a^2 \cos^2 t=a^2$
Then, we have
Flux of $(\nabla \times F)$=$\iint _S (\nabla \times F) \cdot n d\sigma=\int _{0}^{2\pi} a^2 dt$
This implies that
$\int _{0}^{2\pi} a^2 dt=2 \pi a^2$
