2. Planar Region
$$
\left\{(x, y) \mid \sqrt{1-x^{2}} \sqrt{1-y^{2}} \geqslant x y\right\}
$$

The area of the region is $\qquad$ .
2. $2+\frac{\pi}{2}$.

From the problem, we know that $x \in[-1,1], y \in[-1,1]$. If $x y \leqslant 0$, then the inequality holds; if $x y>0$, then $1-x^{2}-y^{2} \geqslant 0$, i.e., $x^{2}+y^{2} \leqslant 1$. By drawing the graph, we can see that the area is $2+\frac{\pi}{2}$.