1. Let the plane point set $A=\left\{(x, y) \left\lvert\,(y-x)\left(y-\frac{1}{x}\right) \geqslant 0\right.\right\}, B=\left\{(x, y) \mid(x-1)^{2}+(y-1)^{2} \leqslant 1\right\}$, then the area of the plane figure represented by $A \cap B$ is
A. $\frac{4}{3} \pi$
B. $\frac{3}{5}$
C. $\frac{4}{7} \pi$
D. $\frac{\pi}{2}$
Since $y=x$ and $y=\frac{1}{x}$ both pass through the center of the circle $(x-1)^{2}+(y-1)^{2}=1$, by symmetry, the area of the plane figure represented by $A \cap B$ is $\frac{\pi}{2}$, so the answer is $D$.