Trapezoidal Rule

Question

I want to give a lecture note to my students explaining the trapezoidal rule. Can anyone help how to draw these shapes together using latex.

Answer 1

This is a solution with a macro for the axis and the function that also creates nodes which help to draw the rectangles or trapeziums:

The code:

\documentclass[tikz,border=2mm]{standalone}
\tikzset{declare function={f(\x)=0.25(\x-2.5)^3-0.75\x+5);}}
\newcommand{\myfunction}[2]
{
  \begin{scope}[shift={(#1,0)}]
  \draw[-latex] (-0.5,0) -- (5,0) node[below] {$x$};
  \draw[-latex] (0,-0.5) -- (0,5) node[left]  {$y$};
  \draw[thick,cyan] plot[domain=0.5:4.5,samples=41] (\x,{f(\x)});
  \node at (3.25,3.5) {$y=f(x)$};
  \foreach\i in {0,1,2,3,...,8}
  {
    \pgfmathsetmacro\j{0.5*\i+0.5}
    \coordinate (x\i) at (\j,0);
    \coordinate (y\i) at (\j,{f(\j)});
  }
  \foreach\i in {0,1,2}
    \node[below]        at (x\i) {$x_\i$};
  \node[below]          at (x7) {$x_{n-1}$};
  \node[below]          at (x8) {$x_n$};
  \node[yshift=-0.75cm] at (x0) {\strut$a$};
  \node[yshift=-0.75cm] at (x8) {\strut$b$};
  \node[yshift=-1cm]    at (x4) {(#2)};
  \end{scope}
}
\begin{document}
\begin{tikzpicture}[line join=round,line cap=round,scale=1.25]
% left
\myfunction{0}{A}
\foreach\i in {0,1,2,3,7}
{
  \pgfmathtruncatemacro\j{\i+1}
  \draw[gray] (x\j) rectangle (y\i);
  \fill (y\i) circle (1pt);
}
% center
\myfunction{6}{B}
\foreach\i in {0,1,2,3,7}
{
  \pgfmathtruncatemacro\j{\i+1}
  \draw[gray] (x\i) rectangle (y\j);
  \fill (y\j) circle (1pt);
}
% right
\myfunction{12}{C}
\foreach\i in {0,1,2,3,7,8}
{
  \pgfmathtruncatemacro\j{\i+1}
  \ifnum\i<8
    \draw[gray] (x\i) -- (y\i) -- (y\j) -- (x\j);
  \fi
  \fill (y\i) circle (1pt);
}
\end{tikzpicture}
\end{document}