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I'd like to draw two wave-shaped contours, as in the figure below, and fill the area between. I tried the following code, but it produces an undesired black vertical edge on the right. How do I remove that?

documentclass[tikz]standalone
begindocument
begintikzpicture
 useasboundingbox (-1,-1) rectangle (6,2);
 draw[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) -- ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
endtikzpicture
enddocument

I also tried to remove the -- between ++(5,-0.6) and ++(0,0.6) in the 5th line, it doesn't draw the undesired edge, but it messes up the filling.

documentclass[tikz]standalone
begindocument
begintikzpicture
 useasboundingbox (-1,-1) rectangle (6,2);
 draw[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
endtikzpicture
enddocument

Quick and dirty: fill the area first and then draw what you want to draw. It is not easily possible to switch off the drawing of a part of a path (but it is possible yet considerably more effort).

documentclass[tikz]standalone
begindocument
begintikzpicture
 useasboundingbox (-1,-1) rectangle (6,2);
 path[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) -- ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
 draw (0,1) to[out=0,in=180] ++(5,-0.6)++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
endtikzpicture
enddocument

You can switch the path on and off e.g. using this answer.

documentclass[tikz]standalone
pgfkeystikz/.cd,
edge options/.code=tikzsetedge style/.style=#1,

begindocument
begintikzpicture[every edge/.append code = % https://tex.stackexchange.com/a/396092/121799 
 globalletcurrenttargettikztotarget % save tikztotarget in a global variable
 pgfkeysalsoappend after command=to[edge style] (currenttarget),
every edge/.append style=edge style ] 
 useasboundingbox (-1,-1) rectangle (6,2);
 path[fill=black!10] 
 (0,1) [edge options=out=0,in=180,draw=black] edge ++(5,-0.6) 
 -- ++(0,0.6) 
 [edge options=out=180,in=0,draw=black] edge ++(-5,-0.6);
endtikzpicture
enddocument

With use TikZ library pgfplots.fillbetween

documentclass[margin=3mm]standalone
usepackagepgfplots
pgfplotssetcompat=1.16
usetikzlibrarypgfplots.fillbetween

begindocument
 begintikzpicture
draw[name path=A] (0, 0.3) to [out=0,in=180] ++(5,-0.6);
draw[name path=B] (0,-0.3) to [out=0,in=180] ++(5, 0.6);
tikzfillbetween[of=A and B] fill=gray!30;
 endtikzpicture
enddocument

For comparison, with plain Metapost, where filling and drawing are kept separate, and where you can concatenate path variables directly.

documentclass[border=5mm]standalone
usepackageluatex85
usepackageluamplib
begindocument
beginmplibcode
beginfig(1);
 z0 = (-72, 8.5);

 path A, B;
 A = z0 right .. right z0 rotated 180;
 B = A reflectedabout(up, down);

 fill A -- B -- cycle withcolor 7/8[blue, white];
 draw A;
 draw B;
endfig;
endmplibcode
enddocument

(compile with lualatex...)