1. Let the plane point sets be
$$
\begin{array}{l}
A=\left\{(x, y) \left\lvert\,(y-x)\left(y-\frac{1}{x}\right) \geqslant 0\right.\right\}, \\
B=\left\{(x, y) \mid(x-1)^{2}+(y-1)^{2} \leqslant 1\right\} .
\end{array}
$$

Then the area of the plane figure represented by $A \cap B$ is ).
(A) $\frac{4 \pi}{3}$
(B) $\frac{3}{5}$
(C) $\frac{4 \pi}{7}$
(D) $\frac{\pi}{2}$
-,1.D.
$A \cap B$ represents the shaded area in Figure 5.

Given that the curve $y=\frac{1}{x}$ and the circle $(x-1)^{2}+(y-1)^{2}=1$ are both symmetric about the line $y=x$, the shaded area is half of the circle's area. Therefore, the required area is $\frac{\pi}{2}$.