% Thanks to http://tex.stackexchange.com/a/30782/5645 for this 
% example!
\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}
\usepackage{pgfplots}
\pgfplotsset{
    integral segments/.code={\pgfmathsetmacro\integralsegments{#1}},
    integral segments=3,
    integral/.style args={#1:#2}{
        ybar interval,
        domain=#1+((#2-#1)/\integralsegments)/2:#2+((#2-#1)/\integralsegments)/2,
        samples=\integralsegments+1,
        x filter/.code=\pgfmathparse{\pgfmathresult-((#2-#1)/\integralsegments)/2}
    }
}

\begin{document}
\begin{preview}
  \begin{tikzpicture}[/pgf/declare function={f=-15*(x-5)+(x-5)^3+50;}]
    \begin{axis}[
        domain=0:10,
        samples=100,
        axis lines=middle
    ]
    \addplot [
        red,
        fill=red!50,
        integral=2:6
    ] {f};
    \addplot [thick] {f};
    \end{axis}
  \end{tikzpicture}
\end{preview}
\end{document}