[Tex/LaTex] Riemann Sum for Double Integral

Question

Is there any way to produce a figure like this:

or this

Using TikZ?

Answer 1

With stippling

Since Mark Wibrow has shown us how to get stippling, let's use it:

The code:

\documentclass[border=5pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations,calc,intersections}

%Mark Wibrow's code for stippling
% https://tex.stackexchange.com/a/267037/3954
\pgfkeys{/pgf/decoration/.cd,
  stipple density/.store in=\pgfstippledensity,
  stipple density=.1,
  stipple scaling function/.store in=\pgfstipplescalingfunction,
  stipple scaling function=sin(\pgfstipplex*180)*0.875+0.125,
  stipple radius/.store in=\pgfstippleradius,
  stipple radius=0.25pt
}
\pgfdeclaredecoration{stipple}{draw}{
\state{draw}[width=\pgfdecorationsegmentlength]{
  \pgfmathparse{\pgfdecoratedcompleteddistance/\pgfdecoratedpathlength}%
  \let\pgfstipplex=\pgfmathresult%
  \pgfmathparse{int(\pgfstippledensity*100)}%
  \let\pgfstipplen=\pgfmathresult
  \pgfmathloop
  \ifnum\pgfmathcounter<\pgfmathresult\relax
    \pgfpathcircle{%
      \pgfpoint{(rnd)*\pgfdecorationsegmentlength}%
        {(\pgfstipplescalingfunction)*(rnd^4)*\pgfdecorationsegmentamplitude+\pgfstippleradius}}% 
    {\pgfstippleradius}%
  \repeatpgfmathloop
}
}

\tikzset{stipple/.style={
  decoration={stipple, segment length=2pt, #1},
  decorate,
  fill
}}
% end of stippling code

\newcommand\DrawBlock[3]{
\ifx#1b\relax
  \path[draw]
    (lm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklf)
    (bm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklb)
    (lm#2) -- ++(0,0,#3) coordinate (blockrf)
    (bm#2) -- ++(0,0,#3) coordinate (blockrb);
  \filldraw[fill=white,draw=black]
    (lm\the\numexpr#2-1\relax) -- (blocklf) -- (blocklb) -- (blockrb) -- (blockrf) -- (lm#2);
\else  
  \ifx#1f\relax
    \path[draw]
      (fm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklf)
      (lm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklb)
      (fm#2) -- ++(0,0,#3) coordinate (blockrf)
      (lm#2) -- ++(0,0,#3) coordinate (blockrb);
    \filldraw[fill=white,draw=black]
      (fm\the\numexpr#2-1\relax) -- (blocklf) -- (blocklb) -- (blockrb) -- (blockrf) -- (fm#2);
  \fi
\fi
\draw (blocklf) -- (blockrf);
}

\begin{document}
\begin{tikzpicture}[y={(0:1cm)},x={(225:0.86cm)}, z={(90:1cm)}]

% coordinates for the lower grid
\path
  (1,3,0) coordinate (bm0) -- 
  (4,3,0) coordinate (fm0) coordinate[midway] (lm0) --
  (4,8,0) coordinate[pos=0.25] (fm1) coordinate[midway] (fm2) coordinate[pos=0.75] (fm3) coordinate (fm4) --
  (1,8,0) coordinate (bm4) coordinate[midway] (lm4)--
  (bm0) coordinate[pos=0.25] (bm3) coordinate[midway] (bm2) coordinate[pos=0.75] (bm1);
\draw[dashed]
  (lm0) -- 
  (lm4) coordinate[pos=0.25] (lm1) coordinate[midway] (lm2) coordinate[pos=0.75] (lm3);

% the blocks
\DrawBlock{b}{1}{4}
\DrawBlock{b}{2}{3.7}
\DrawBlock{b}{3}{4.3}
\DrawBlock{b}{4}{5}
\DrawBlock{f}{1}{3.3}
\DrawBlock{f}{2}{3.5}
\DrawBlock{f}{3}{4}
\DrawBlock{f}{4}{4.7}

\foreach \Point/\Height in {lm1/3.7,lm2/4.3,lm3/5}
  \draw[ultra thin,dashed,opacity=0.2] (\Point) -- ++(0,0,\Height);

% the lower grid
\foreach \x in {1,2,3}
  \draw[dashed] (fm\x) -- (bm\x);
\draw[dashed] (fm0) -- (bm0) -- (bm4);
\draw (fm0) -- (fm4) -- (bm4);
\draw[dashed] (lm0) -- (lm4);

% coordinates for the surface
\coordinate (curvefm0) at ( $ (fm0) + (0,0,4) $ );
\coordinate (curvebm0) at ( $ (bm0) + (0,0,4) $ );
\coordinate (curvebm4) at ( $ (bm4) + (0,0,6) $ );
\coordinate (curvefm4) at ( $ (fm4) + (0,0,5.7) $ );

% the surface
\filldraw[ultra thick,fill=gray!25,fill opacity=0.2]
  (curvefm0) to[out=-30,in=210] 
  (curvefm4) to[out=-4,in=260]
  (curvebm4) to[out=215,in=330]
  (curvebm0) to[out=240,in=-20]
  (curvefm0);

% lines from grid to surface
\draw[very thick,name path=leftline] (curvefm0) -- (fm0);
\draw[very thick] (curvefm4) -- (fm4);
\draw[very thick,name path=rightline] (curvebm4) -- (bm4);
\draw[very thick,dashed] (curvebm0) -- (bm0);

% coordinate system
\coordinate (O) at (0,0,0);
\draw[-latex] (O) -- +(5,0,0) node[above left] {$x$};
\path[name path=yaxis] (O) -- +(0,10,0) coordinate (yaxisfinal) node[above] {$y$};
\draw[-latex] (O) -- +(0,0,5) node[left] {$z$};
\path[name intersections={of=yaxis and leftline,by={yaxis1}}];
\path[name intersections={of=yaxis and rightline,by={yaxis2}}];
\draw (O) -- (yaxis1);
\draw[densely dashed,opacity=0.1] (yaxis1) -- (yaxis2);
\draw[-latex] (yaxis2) -- (yaxisfinal);

% the stippling
\path[postaction={stipple={amplitude=1cm,stipple density=0.15}}]
  ( $ (fm4) + (0,0,4.7) $ ) -- (fm4);
\path[postaction={stipple={amplitude=1cm,stipple density=0.05}}]
  ( $ (lm4) + (0,0,4.7) $ ) -- (lm4);

% for debugging
%\foreach \Name in {bm0,fm0,lm0,fm1,fm2,fm3,fm4,bm4,lm4,bm1,bm2,bm3,lm1,lm2,lm3,%
%curvefm0,curvebm0,curvebm4,curvefm4}
%  \node at (\Name) {\Name};  
\end{tikzpicture}

\end{document}

Without stippling

The code:

\documentclass[border=5pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,intersections}

\newcommand\DrawBlock[3]{
\ifx#1b\relax
  \path[draw]
    (lm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklf)
    (bm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklb)
    (lm#2) -- ++(0,0,#3) coordinate (blockrf)
    (bm#2) -- ++(0,0,#3) coordinate (blockrb);
  \filldraw[fill=white,draw=black]
    (lm\the\numexpr#2-1\relax) -- (blocklf) -- (blocklb) -- (blockrb) -- (blockrf) -- (lm#2);
\else  
  \ifx#1f\relax
    \path[draw]
      (fm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklf)
      (lm\the\numexpr#2-1\relax) -- ++(0,0,#3) coordinate (blocklb)
      (fm#2) -- ++(0,0,#3) coordinate (blockrf)
      (lm#2) -- ++(0,0,#3) coordinate (blockrb);
    \filldraw[fill=white,draw=black]
      (fm\the\numexpr#2-1\relax) -- (blocklf) -- (blocklb) -- (blockrb) -- (blockrf) -- (fm#2);
  \fi
\fi
\draw (blocklf) -- (blockrf);
}

\begin{document}
\begin{tikzpicture}[y={(0:1cm)},x={(225:0.86cm)}, z={(90:1cm)}]

% coordinates for the lower grid
\path
  (1,3,0) coordinate (bm0) -- 
  (4,3,0) coordinate (fm0) coordinate[midway] (lm0) --
  (4,8,0) coordinate[pos=0.25] (fm1) coordinate[midway] (fm2) coordinate[pos=0.75] (fm3) coordinate (fm4) --
  (1,8,0) coordinate (bm4) coordinate[midway] (lm4)--
  (bm0) coordinate[pos=0.25] (bm3) coordinate[midway] (bm2) coordinate[pos=0.75] (bm1);
\draw[dashed]
  (lm0) -- 
  (lm4) coordinate[pos=0.25] (lm1) coordinate[midway] (lm2) coordinate[pos=0.75] (lm3);

% the blocks
\DrawBlock{b}{1}{4}
\DrawBlock{b}{2}{3.7}
\DrawBlock{b}{3}{4.3}
\DrawBlock{b}{4}{5}
\DrawBlock{f}{1}{3.3}
\DrawBlock{f}{2}{3.5}
\DrawBlock{f}{3}{4}
\DrawBlock{f}{4}{4.7}

\foreach \Point/\Height in {lm1/3.7,lm2/4.3,lm3/5}
  \draw[ultra thin,dashed,opacity=0.2] (\Point) -- ++(0,0,\Height);

% the lower grid
\foreach \x in {1,2,3}
  \draw[dashed] (fm\x) -- (bm\x);
\draw[dashed] (fm0) -- (bm0) -- (bm4);
\draw (fm0) -- (fm4) -- (bm4);
\draw[dashed] (lm0) -- (lm4);

% coordinates for the surface
\coordinate (curvefm0) at ( $ (fm0) + (0,0,4) $ );
\coordinate (curvebm0) at ( $ (bm0) + (0,0,4) $ );
\coordinate (curvebm4) at ( $ (bm4) + (0,0,6) $ );
\coordinate (curvefm4) at ( $ (fm4) + (0,0,5.7) $ );

% the surface
\filldraw[ultra thick,fill=gray!25,fill opacity=0.2]
  (curvefm0) to[out=-30,in=210] 
  (curvefm4) to[out=-4,in=260]
  (curvebm4) to[out=215,in=330]
  (curvebm0) to[out=240,in=-20]
  (curvefm0);

% lines from grid to surface
\draw[very thick,name path=leftline] (curvefm0) -- (fm0);
\draw[very thick] (curvefm4) -- (fm4);
\draw[very thick,name path=rightline] (curvebm4) -- (bm4);
\draw[very thick,dashed] (curvebm0) -- (bm0);

% coordinate system
\coordinate (O) at (0,0,0);
\draw[-latex] (O) -- +(5,0,0) node[above left] {$x$};
\path[name path=yaxis] (O) -- +(0,10,0) coordinate (yaxisfinal) node[above] {$y$};
\draw[-latex] (O) -- +(0,0,5) node[left] {$z$};
\path[name intersections={of=yaxis and leftline,by={yaxis1}}];
\path[name intersections={of=yaxis and rightline,by={yaxis2}}];
\draw (O) -- (yaxis1);
\draw[densely dashed,opacity=0.1] (yaxis1) -- (yaxis2);
\draw[-latex] (yaxis2) -- (yaxisfinal);

% for debugging
%\foreach \Name in {bm0,fm0,lm0,fm1,fm2,fm3,fm4,bm4,lm4,bm1,bm2,bm3,lm1,lm2,lm3,%
%curvefm0,curvebm0,curvebm4,curvefm4}
%  \node at (\Name) {\Name};  
\end{tikzpicture}

\end{document}