Timing and integration with TikZ PGFlots

Question

How do I set intervals or integration with TikZ PGFlots? Something similar to the figure below, but I want to define the function, the intervals of integration and the partitions.

\documentclass{scrartcl}
\usepackage[utf8]{inputenc} 
\usepackage[upright]{fourier}
\usepackage[usenames,dvipsnames,svgnames]{xcolor}
\usepackage{amsmath,tkz-fct}
\usepackage[active,pdftex,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{5pt}   

\begin{document} 

\begin{tikzpicture}[scale=1]
\tkzInit[xmin=-3,xmax=6,ymin=-2,ymax=14,ystep=2]
\tkzDrawX \tkzDrawY
\tkzFct[line width=2pt,color = red, domain =-3:6]{(-\x-2)*(\x-5)}
\tkzDrawRiemannSumSup[fill=blue!40,opacity=.5,interval=-1:5,number=10] 
\tkzDrawRiemannSumInf[fill=green!40,opacity=.5,interval=-1:5,number=10] 
\end{tikzpicture} 

\end{document}   

Answer 1

If you're happy with using the midpoint of the partitions for the approximation (instead of the maximum or minimum), you can relatively easily do this in PGFplots.

I've defined a new style integral=<lower>:<upper> that draws the rectangles of the approximate integral. You can set the number of partitions by setting integral segments=<number> (default is 3).

The style could easily be adjusted to use the left or right point of the partition for the approximation.

For example

\pgfset{declare function={f=-15*(x-5)+(x-5)^3+50;}}
\addplot [thick] {f};
\addplot [
    red,
    integral=2:6
] {f}

would yield

Setting the number of partitions:

\addplot [
    fill=yellow,
    integral segments=10,
    integral=2:8
] {f};
\addplot [thick] {f};

Here's the complete code:

\documentclass{article}

\usepackage{pgfplots}

\begin{document}

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

\end{document}

Answer 2

Did you take a look at the tkz-fct package? It does whole bunch of stuff, including Riemann sums.