[Tex/LaTex] drawing arrows on curves and shading regions in TikZ

Question

I need to draw the following figure in my latex using TikZ. I did the following:

\begin{tikzpicture}
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw (0,0) circle (1cm);
\draw (1,-.3) node[right] {(2,0)}; 
\draw (-1,-.3) node[left] {(-2,0)}; 

\end{tikzpicture}

But how can I shade the upper semi-circle and how can I draw the arrows in between?

Additional Question
Thank you so much for your answer. But I have one more doubt: If I use these command in my question I get the diagram right aligned and above the question. But I need it to be beside the question. And sometimes I need to define exactly where I want it to be. What should I do for this?

\question Evaluate $\displaystyle\iint_R dxdy,$ where $R$ is the shaded region in the figure.
\begin{tikzpicture}
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw (0,0) circle (1cm);
\draw (1,-.3) node[right] {(2,0)}; 
\draw (-1,-.3) node[left] {(-2,0)}; 
\draw[pattern=north west lines] (-1,0) --(1,0) arc[start angle=0, end angle=180, radius=1cm];
\end{tikzpicture}

Answer 1

Perhaps there is a better way, but here is one brute force way. You need to load the tikzlibraries patterns and arrows.meta in your preamble.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{patterns, arrows.meta}
\begin{document}
\begin{tikzpicture}
\draw (-2,0) -- (2,0);
\draw (0,-2) -- (0,2);
\draw (0,0) circle (1cm);
\draw (1,-.3) node[right] {(2,0)};
\draw (-1,-.3) node[left] {(-2,0)};
\draw[pattern=north west lines] (0,-1) --(0,1) arc[start angle=90, end angle=-90, radius=1cm];
\draw[-{To[length=5pt,width=10pt]}] (0,-1) arc[start angle=-90, end angle=-45, radius=1cm];
\draw[-{To[length=5pt,width=10pt]}] (0,1)  arc[start angle=90, end angle=135, radius=1cm];

\end{tikzpicture}
\end{document}