\documentclass{article}

\usepackage[latin1]{inputenc}
\usepackage{tikz}

% GNUPLOT required
\begin{document}
\pagestyle{empty}


\begin{tikzpicture}
    \draw[very thin,color=gray] (-0.1,-0.1) grid (3.9,3.9);
    \draw[->] (-0.2,0) -- (4.2,0) node[right] {$x$};
    \draw[->] (0,-1.2) -- (0,4.2) node[above] {$f(x)$};

    \foreach \x/\xtext in {1/a,2/b}
       \draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\xtext$};
%       \draw [color=red, thin] (1,0) -- (1,1);
%       \draw [color=red, thin] (2,0) -- (2,1.2599210498948731647672106072782);
%       \draw [color=red, thin] (1,0) -- (2,0);
%       \draw  [color=red, thin] plot[id=w3_ausschnitt, smooth, domain=1:2, samples=200] function{x**(1./3.)};
       \draw[color=orange] plot[id=w3, smooth, domain=0:3.5, samples=200] function{x**(1./3.)} node[right] {$f(x) = \sqrt[3]{x}$};
      %\fill[red!30] (1,0)  -- plot[id=f3,domain=0.658:3,samples=50] function {exp(-x*x*0.5/0.16)} -- (3,0) -- cycle;

    \fill[red!50] (1,0)  -- plot[id=w3_ausschnitt,domain=1:2,samples=200] function {x**(1./3.)} -- (2,0) -- cycle;
  
\end{tikzpicture}


\end{document}