automating xbar as ybar in pgfplots

Question

I am endeavoring to make a demonstration of the conceptual difference between the Lebesgue V Riemann integrals.  It's easy to automate the ybar command to give left right or midpoint sums as seen at Placing coloured rectangles on a plot, using points from the plot - Riemann Sums.

I could make the boxes by hand a couple of different ways, but I would like to avoid brute force if possible. The main trouble is that xbar always wants to go back to xmin (which again is an obvious means of one-at-a-timing it) Any ideas on how to automate the xbar command as with the ybar above to make sideways "Lebesgue Sums"?

user194703 · Answer

Like in this answer, you can use intersections here.

documentclass[tikz,border=3mm]{standalone}
usetikzlibrary{backgrounds,calc,intersections}
begin{document}
begin{tikzpicture}
 draw[thick,name path=A] plot[smooth] coordinates {(0,0) (1,0.8) (2,1) (3.5,0.6) (4,0)};
 begin{scope}[on background layer]
  foreach X [count=Z] in {0,0.5,...,4}
   {path[name path global=v-Z,overlay] (X,0) --  (X,0|-current bounding box.north);
   draw[name intersections={of=A and v-Z,by=i-Z},blue,fill=blue!20] 
   ifnumZ>1
    let p1=(i-thenumexprZ-1),p2=(i-Z) in
    (X-0.5,0) rectangle (X,{max(y1,y2)})
    fi;
   draw[name intersections={of=A and v-Z,by=i-Z},blue,fill=blue!80] 
   ifnumZ>1
    let p1=(i-thenumexprZ-1),p2=(i-Z) in
    (X-0.5,0) rectangle (X,{min(y1,y2)})
    fi;}
 end{scope}
 %
 begin{scope}[yshift=-2cm]
  draw[thick,name path=B] plot[smooth] coordinates {(0,0) (1,0.8) (2,1) (3.5,0.6) (4,0)};
  begin{scope}[on background layer]
   foreach Y [count=Z] in {0,0.2,0.4,0.6,0.8,1}
    {path[name path global=h-Z,overlay] (0,Y) --  (4,Y);
    draw[name intersections={of=B and h-Z,by={i-Z-1,i-Z-2}},red,fill=red!20] 
     ifnumZ>1
      let p1=(i-thenumexprZ-1relax-1),p2=(i-thenumexprZ-1relax-2),
       p3=(i-Z-1),p4=(i-Z-2)
       in
    ({min(x1,x3)},Y-0.2) rectangle ({max(x2,x4)},Y)
    fi;
    draw[name intersections={of=B and h-Z,by={i-Z-1,i-Z-2}},red,fill=red!60] 
     ifnumZ>1
      let p1=(i-thenumexprZ-1relax-1),p2=(i-thenumexprZ-1relax-2),
       p3=(i-Z-1),p4=(i-Z-2)
       in
    ({max(x1,x3)},Y-0.2) rectangle ({min(x2,x4)},Y)
    fi;
    }
  end{scope}
 end{scope}
end{tikzpicture}
end{document}

automating xbar as ybar in pgfplots

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