Put $z=x+i y$
$ \begin{aligned} & \quad|z-1| \leq 2 \Rightarrow(x-1)^2+y^2 \leq 4 \\ & \quad(z+\bar{z})+i(z-\bar{z}) \leq 2 \Rightarrow 2 x+i(2 i y) \leq 2 \\ & \operatorname{Im}(\mathrm{z}) \geq 0 \Rightarrow \mathrm{y} \geq 0 \end{aligned} $

Required area
$\begin{aligned} & =\text { Area of semi-circle - area of sector } \mathrm{A} \\ & \frac{1}{2} \pi(2)^2-\frac{\pi}{2} \\ & =\frac{3 \pi}{2}\end{aligned}$