anti-derivative gives function = area under curve, help illustate with tikz

Question

I'm trying to reproduce this illustration I've made in Inkscape as a tikzpictures for use in LaTex. I've looked a lot at this, and around, but I'm stuck and it's late. Would one of you kind experts out there help me figure out how to make this illustration?

I also looked at this and this, I'll update if I work out some solution that works. IN the meantime please come in here with any suggestions of ideas. Thanks a lot!

Here's a first jab at a compilable MWE,

\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}[
    ticks=none,
    domain=0:10,
    samples=100,
    axis lines=middle
]
\addplot [thick] {f} node[pos=1] (f) {};
\addplot [
    red,
    integral=2:6
] {f};
\end{axis}
\end{tikzpicture}

\end{document}

@salim-bou and @zarko, I'm amazed, and truly grateful you took the time, and that it didn't take you more then some minutes. Would it be too much to ask how I overlay some text on my graph (obviously not the LaTeX code, but I only have some paint like software to show this here where I am).

Answer 1

Well, @salim bou bit me for 6 minutes, but anyway (slightly different):

I reproduce your image with the following pure TikZ code:

\documentclass[tikz,
              border=3mm]{standalone}
\usetikzlibrary{intersections,quotes}

\begin{document}
    \begin{tikzpicture}
% axes
\draw[->]  (-0.2,0) -- (8,0) node [below left] {$x$};
\draw[->]  (0,-0.2) -- (0,6) node [below left] {$y(x)$};
% grid
\draw[gray!30,thin]  (-1,-1) grid +(9,7);
% curve
\draw[red,very thick, name path=A]
    (1,2) .. controls +(1,0) and + (-1.5,-0.1) .. (4,3)
          .. controls +(1.5,0.1) and + (-1,0) .. (7,4);
% start, end of curve
\draw[dashed]  (1,-0.1) node[below] {$a$} -- (1,2);
\draw[dashed]  (7,-0.1) node[below] {$b$} -- (7,4);
% for determining intersections a and b
\path[name path=B]  (3,-0.1) node[below] {$x_0$} -- + (0,6);
\path[name path=C]  (5,-0.1) node[below] {$x_0 + \Delta x$} -- + (0,6);
% rectangles
\draw [name intersections={of=A and B, by=a},very thick,dashed, blue] 
    (5,0) -| (a) -| (5,0);
\draw [name intersections={of=A and C, by=b},very  thick,dotted,green] 
    (3,0) -| (b) -| (3,0);
% measure of \Delta x
\draw[<->]  (3,0.5) to ["$\Delta x$"] (5,0.5);
% comments:
\draw[<-]   (3.4,3.0) -- + (120:1) node[above] {$f(x_0+\Delta)\cdot\Delta x$};
\draw[<-]   (4.4,2.9) -- + (0:0.8) node[right] {$f(x_0+\Delta)\cdot\Delta x$};
\draw[<-]   (4.4,1.5) -- + (0:0.8) node[right] {$f(x_0)\cdot\Delta x$};
    \end{tikzpicture}
\end{document}

which gives:

Edit: with consideration with recent edit of question -- to MWE is added:

% comments:
\draw[<-]   (3.4,3.0) -- + (120:1) node[above] {$f(x_0+\Delta)\cdot\Delta x$};
\draw[<-]   (4.4,2.9) -- + (0:0.8) node[right] {$f(x_0+\Delta)\cdot\Delta x$};
\draw[<-]   (4.4,1.5) -- + (0:0.8) node[right] {$f(x_0)\cdot\Delta x$};

Answer 2

Code

\documentclass[margin=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}

\def\a{0.5}
\def\b{7}
\def\xA{2}
\def\xB{4}

\begin{document}

\begin{tikzpicture}[>=stealth,thick]
\draw [->](-0.5,0)--(8,0);
\draw [->](0,-0.5)--(0,6);
\draw [name path=graph](\a,1.5)coordinate(a)..controls (\xA,4) and (\b,2).. (\b,5)coordinate(b);
\path [name path=lineA](\xA,0)--+(0,6);
\path [name path=lineB](\xB,0)--+(0,6);
\draw [name intersections={of=lineA and graph},dashed,green]
(\xB,0)|-(intersection-1)--(intersection-1 |- 0,0)node [below,black]{\strut  $x_0$};
\draw [name intersections={of=lineB and graph},dashed,blue]
(\xA,0)|-(intersection-1)--(intersection-1 |- 0,0)node [below,black]{\strut $x_0+\Delta x$};
\draw [<->] ([yshift=3mm]\xA,0)--node[fill=white]{$\Delta x$}([yshift=3mm]\xB,0);
\draw [dashed,thin] (a)--(a|- 0,0)node [below]{\strut $a$};
\draw [dashed,thin] (b)--(b|- 0,0)node [below]{\strut $b$};
\end{tikzpicture}

\end{document}

Result