Change of Variables & the Jacobian · Superchallenge problem

Let $\Phi$ be the time-$1$ map of the system $x' = 2x + \sin y$, $y' = y + \cos x$: $\Phi(p)$ is the position at time $1$ of the solution that starts at $p$ at time $0$. Find the area of $\Phi(D)$, where $D$ is the unit disk $x^2 + y^2 \le 1$.