\documentclass{article}

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


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

    %\pgfmathsetmacro\wert2{\pgfmathpow{1.5}{\pgfmathdivide{1.}{3.}}}
    %\pgfmathsetmacro\wert2{1.5**0.33333333333}


%\pgfmathpow{1.5}{\pgfmathdivide{1.}{3.}}

     \draw [fill=black!50] (1,0) rectangle +(0.5,1);
%     \draw [fill=black!50] (1.5,0) rectangle +(0.5,(\pgfmathparse{pow(2,7)} \pgfmathresult));

    \pgfmathparse{pow(1.5, (1/3))};
    \pgfmathsetmacro\wert{\pgfmathresult} %%Die Variable darf keine Zahl in Namen enthalten
    \draw [fill=black!50] (1.5,0) rectangle +(0.5,\wert);

    % \draw [fill=black!50] (1.5,0) rectangle +(0.5,\pgfmathparse{\pgfmathpow{1.5}{\pgfmathdivide{1.}{3.}}} \pgfmathresult);
%     \fill [color=black!50] (1.5,0) rectangle +(0.5, 1.1447142425533318678);

\end{tikzpicture}


\end{document}