Is it possible to draw a normal domain with Latex?

This is certainly not the most elegant solution, but can perhaps serve as a starting point.

I used pgfplots as a basis to draw the coordinate system as well as the "normal domain". Everything is drawn within a single axis block. The boundary lines at the top and bottom are created using the curve through interpolation functionality from the hobby tikzlibrary. Therefore I defined the coordinates (A) to (F) (lower boundary, from left to right) and (G) to (L) (upper boundary, from right to left). Of course, you can adjust the coordinates to your personal preference. For filling the domain I used pattern=north east lines from the tikzlibrary patterns.

Here is the complete code:

documentclass[tikz, border=5pt]{standalone}
usepackage{pgfplots}
usetikzlibrary{hobby, patterns}
pgfplotsset{compat=newest}
begin{document}
begin{tikzpicture}
 begin{axis}[axis lines=center, 
              xmin=-1, xmax=8, ymin=-1, ymax=8, 
              line width=1pt, 
              xtick={2, 6}, xticklabels={$a$, $b$}, 
              ytick={2, 6}, yticklabels={},
              xlabel={normalsize $x$}, xlabel style={yshift=-.5*pgfkeysvalueof{/pgfplots/major tick length},anchor=north east,inner xsep=0pt}, 
              ylabel={normalsize $y$}, ylabel style={xshift=-.5*pgfkeysvalueof{/pgfplots/major tick length}, anchor=north east, inner ysep=0pt}
              ]    
        
        % define the coordinates (interpolation points) for the lower boundary
        coordinate (A) at (2,2);
        coordinate (B) at (2.2,2.3);
        coordinate (C) at (3,1.9);
        coordinate (D) at (4,2.3);
        coordinate (E) at (5,1.8);
        coordinate (F) at (6,2);
        
        % define the coordinates (interpolation points) for the upper boundary
        coordinate (G) at (6,6); 
        coordinate (H) at (5,6.2);
        coordinate (I) at (4,5.8);
        coordinate (J) at (3,6.1);
        coordinate (K) at (2.2,5.7);
        coordinate (L) at (2,6);

        % draw the domain as a single closed curved that interpolates through A, B, C, D, E and F runs straight to G interpolates through H, I, J, K and L and runs again straight to A
        draw[pattern=north east lines, pattern color=red] (A) to [curve through = {(B) (C) (D) (E)}] (F) to (G) to [curve through = {(H) (I) (J) (K)}] (L) to (A);

        % draw the dashed red lines
        draw[red, dashed] (2,0) -- (A);
        draw[red, dashed] (6,0) -- (F);
        
        % add the function name to the upper and lower boundary
        node at (4,6.5) {$beta (x)$};
        node at (4,1.5) {$alpha (x)$};

    end{axis}
end{tikzpicture}
end{document}

And this is the result: