TIKZ: Evaluate declared functions inside `foreach`

I'm trying to write a loop to plot a rectangle showing the Midpoint Rule using pgfplots, but cannot evaluate the function to draw it. (around the end of code). Actual code isn't compiling.

Midpoint Rule

documentclassarticle
usepackagepgfplots
begindocument
pgfplotsset
integral axis/.style=
axis lines=middle,
enlarge y limits=upper,
axis equal image, width=12cm,
xlabel=$x$, ylabel=$y$,
ytick=empty,
xticklabel style=font=small, text height=1.5ex, anchor=north,
samples=100
,
integral/.style=
domain=2:8,
samples=7
,
integral fill/.style=
integral,
draw=none, fill=#1,
%on layer=axis background
,
integral fill/.default=cyan!10,
integral line/.style=
integral,
very thick,
draw=#1
,
integral line/.default=black



begintikzpicture[
% The function that is used for all the plots
declare function=f=x/5-cos(deg(x*1.85))/2+2;
]
beginaxis[
integral axis,
ymin=0,
xmin=0.75, xmax=9.25,
domain=1.5:8.5,
xtick=2,2.5,3,3.5,...,8,
xticklabels=$x_0$, $x_1/2$, $x_1$,$x_3/2$,$x_2$,$x_5/2$,$x_3$,,,,$x_N-1$,$x_N-frac12$,$x_N$,
axis on top
]
% The filled area under the approximate integral
addplot [integral fill=cyan!15] f closedcycle;

% The highlighted segment
%addplot [integral fill=cyan!35, domain=6:7, samples=2] f closedcycle;

% The function
addplot [very thick, cyan!75!blue] f node [anchor=south] $y=f(x)$;

% The approximate integral
addplot [integral line=black] f;

% The vertical lines between the segments
addplot [integral, ycomb] f;

% Issue is here.
foreach i in 2.5,3.5,...,7.5
pgfmathsetmacroyf i;
draw (i-.5,y) rectangle (i+.5,0);


endaxis
endtikzpicture

enddocument


1 Answer
1

Pgfplots comes with its own foreach version, which you may want to use here. Notice that you did not specify a version. In older versions of pgfplots you need to add axis cs: to the coordinates in the loop. I adjusted your function such that it is really a function of a variable, otherwise you cannot use it in the draw commands

axis cs:

draw

documentclassarticle
usepackagepgfplots
begindocument
pgfplotssetcompat=1.16,
integral axis/.style=
axis lines=middle,
enlarge y limits=upper,
axis equal image, width=12cm,
xlabel=$x$, ylabel=$y$,
ytick=empty,
xticklabel style=font=small, text height=1.5ex, anchor=north,
samples=100
,
integral/.style=
domain=2:8,
samples=7
,
integral fill/.style=
integral,
draw=none, fill=#1,
%on layer=axis background
,
integral fill/.default=cyan!10,
integral line/.style=
integral,
very thick,
draw=#1
,
integral line/.default=black



begintikzpicture[
% The function that is used for all the plots
declare function=f(x)=x/5-cos(deg(x*1.85))/2+2;
]
beginaxis[
integral axis,
ymin=0,
xmin=0.75, xmax=9.25,
domain=1.5:8.5,
xtick=2,2.5,3,3.5,...,8,
xticklabels=$x_0$, $x_1/2$, $x_1$,$x_3/2$,$x_2$,$x_5/2$,$x_3$,,,,$x_N-1$,$x_N-frac12$,$x_N$,
axis on top
]
% The filled area under the approximate integral
addplot [integral fill=cyan!15] f(x) closedcycle;

% The highlighted segment
%addplot [integral fill=cyan!35, domain=6:7, samples=2] f closedcycle;

% The function
addplot [very thick, cyan!75!blue] f(x) node [anchor=south] $y=f(x)$;

% The approximate integral
addplot [integral line=black] f(x);

% The vertical lines between the segments
addplot [integral, ycomb] f(x);
% Issue is here.

pgfplotsinvokeforeach2.5,3.5,...,7.5
draw[thick,blue] (#1-0.5,f(#1)) rectangle(#1+.5,0);

endaxis
endtikzpicture
enddocument


Of course, you could plot these bars also with a bar plot. That is, you could replace the loop by

bar

addplot [blue,thick,ybar,bar width=1,samples at=2.5,3.5,...,7.5] f(x);


f(x)

only marks

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