Draw Riemann Sum in two dimensions using Tikz

Question

Anyone have ay idea how can I draw this figure

using Tikz or PgfPlots? For one dimension I can bur, for two I have no idea. I would like to provide a function $z=f(x,y)$, the ranges for $x; (a\le x\le b)$ and $y; (c\le y\le d)$ and the number of divisions $m$ and $n$ in each interval. Many thanks.

Answer 1

Probably it is better to use the cube* plot mark from pgfplots, but for simple enough functions the following may do. You can define a function, the number of steps, and the plot range.

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\tikzset{pics/3d bar/.style={code={%
 \tikzset{3d bar/.cd,#1}
 \path[3d bar/x face] (\mydx/2,\mydy/2,0) -- (\mydx/2,\mydy/2,\myh)
    -- (-\mydx/2,\mydy/2,\myh) -- (-\mydx/2,\mydy/2,0) -- cycle;
 \path[3d bar/y face] (\mydx/2,\mydy/2,0) -- (\mydx/2,\mydy/2,\myh)
    -- (\mydx/2,-\mydy/2,\myh) -- (\mydx/2,-\mydy/2,0) -- cycle;
 \path[3d bar/z face] (\mydx/2,\mydy/2,\myh) -- (-\mydx/2,\mydy/2,\myh)
    -- (-\mydx/2,-\mydy/2,\myh) -- (\mydx/2,-\mydy/2,\myh) -- cycle;
    }},3d bar/.cd,dx/.store in=\mydx,dx=1,dy/.store in=\mydy,dy=1,
        h/.store in=\myh,h=1,x face/.style={draw=blue!50,fill=cyan!20},
        y face/.style={draw=blue!50,fill=cyan!50},
        z face/.style={draw=blue!50,fill=cyan!30}}
\begin{document}
\tdplotsetmaincoords{70}{110}%
\begin{tikzpicture}[tdplot_main_coords]
 \begin{scope}[declare function={f(\x,\y)=1+3*exp(-\x/5-\y/4);% function
    n=5;% steps
    xmin=0;xmax=5;ymin=0;ymax=5;}]
  \pgfmathtruncatemacro{\myn}{n}
  \tikzset{3d bar/dx/.evaluated={(xmax-xmin)/n},
     3d bar/dy/.evaluated={(ymax-ymin)/n}}
  \foreach \i in {1,...,\myn}
   {\foreach \j in {1,...,\myn}
   {\pgfmathsetmacro{\myx}{xmin+(\i-0.5)*(xmax-xmin)/\myn}
    \pgfmathsetmacro{\myy}{ymin+(\j-0.5)*(ymax-ymin)/\myn}
    \pgfmathsetmacro{\myf}{f(\myx,\myy)}
    \path (\myx,\myy,0) pic{3d bar={h=\myf}};}}  
 \end{scope}
 %
 \begin{scope}[xshift=7cm,
     declare function={f(\x,\y)=0.5+3*exp(-\x/5-\y/4);% function
     n=20;% steps
     xmin=0;xmax=5;ymin=0;ymax=5;}]
  \pgfmathtruncatemacro{\myn}{n}
  \tikzset{3d bar/dx/.evaluated={(xmax-xmin)/n},
     3d bar/dy/.evaluated={(ymax-ymin)/n}}
  \foreach \i in {1,...,\myn}
   {\foreach \j in {1,...,\myn}
   {\pgfmathsetmacro{\myx}{xmin+(\i-0.5)*(xmax-xmin)/\myn}
    \pgfmathsetmacro{\myy}{ymin+(\j-0.5)*(ymax-ymin)/\myn}
    \pgfmathsetmacro{\myf}{f(\myx,\myy)}
    \path (\myx,\myy,0) pic{3d bar={h=\myf}};}}  
 \end{scope}
\end{tikzpicture}
\end{document}

Answer 2

If I may add a tiny cherry (a cat guru taught me once) on the beautiful answer by user240002...

To avoid using (annoying IMHO, \pgfmathsetmacro), include the variables to evaluate directly in the loop.

\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\tikzset{pics/3d bar/.style={code={%
 \tikzset{3d bar/.cd,#1}
 \path[3d bar/x face] (\mydx/2,\mydy/2,0) -- (\mydx/2,\mydy/2,\myh)
    -- (-\mydx/2,\mydy/2,\myh) -- (-\mydx/2,\mydy/2,0) -- cycle;
 \path[3d bar/y face] (\mydx/2,\mydy/2,0) -- (\mydx/2,\mydy/2,\myh)
    -- (\mydx/2,-\mydy/2,\myh) -- (\mydx/2,-\mydy/2,0) -- cycle;
 \path[3d bar/z face] (\mydx/2,\mydy/2,\myh) -- (-\mydx/2,\mydy/2,\myh)
    -- (-\mydx/2,-\mydy/2,\myh) -- (\mydx/2,-\mydy/2,\myh) -- cycle;
    }},3d bar/.cd,dx/.store in=\mydx,dx=1,dy/.store in=\mydy,dy=1,
        h/.store in=\myh,h=1,x face/.style={draw=blue!50,fill=cyan!20},
        y face/.style={draw=blue!50,fill=cyan!50},
        z face/.style={draw=blue!50,fill=cyan!30}}
\begin{document}
\tdplotsetmaincoords{70}{110}%
\begin{tikzpicture}[tdplot_main_coords]
    \begin{scope}[declare function={f(\x,\y)=1+3*exp(-\x/5-\y/4);% function
                n=5;% steps
            xmin=0;xmax=5;ymin=0;ymax=5;}]
        \pgfmathtruncatemacro{\myn}{n}
        \tikzset{3d bar/dx/.evaluated={(xmax-xmin)/n},
            3d bar/dy/.evaluated={(ymax-ymin)/n}}
          
        \foreach \i in {1,...,\myn}
        {% here
            \foreach [evaluate ={
                    \myx = xmin+(\i-0.5)*(xmax-xmin)/\myn ;
                    \myy = ymin+(\j-0.5)*(ymax-ymin)/\myn ;
                    \myf = f(\myx,\myy) ;
                }] \j in {0,...,\myn}
            {\path (\myx,\myy,0) pic{3d bar={h=\myf}};}
        }  
    \end{scope}
    %
    \begin{scope}[xshift=7cm,
            declare function={f(\x,\y)=0.5+3*exp(-\x/5-\y/4);% function
                n=20;% steps
                xmin=0;xmax=5;ymin=0;ymax=5;}]
            \pgfmathtruncatemacro{\myn}{n}
            \tikzset{3d bar/dx/.evaluated={(xmax-xmin)/n},
                3d bar/dy/.evaluated={(ymax-ymin)/n}}
            \foreach \i in {1,...,\myn}
            {% here
                \foreach [evaluate ={
                        \myx = xmin+(\i-0.5)*(xmax-xmin)/\myn ;
                        \myy = ymin+(\j-0.5)*(ymax-ymin)/\myn ;
                        \myf = f(\myx,\myy) ;
                    }] \j in {0,...,\myn}
                {\path (\myx,\myy,0) pic{3d bar={h=\myf}};}
            }  
        \end{scope}
    \end{tikzpicture}
\end{document}
           

More details in pfg manual v3.1.5.b p1003, section 89 Repeating Things: The Foreach Statement.