ChapterZero

[illust mp]
pair xaxisend, yaxisend,y,x;
path xaxis;
height := 5cm;
overshoot = height/15;
res := 400;
stretch := 1.4;
radstodeg := 180/3.14159/cm;
x := (1,0);
y := (0,1);

% Draw axes
yaxisend = origin + height*y;
xaxisend = origin + height*x;
drawarrow origin-overshoot*y — yaxisend;
drawarrow origin-overshoot*x — xaxisend+overshoot*x;

% Draw nice function
path graph;
graph = (0, height/2)
for i = 0 upto res: .. (i/res*height, height/2 + sind(stretch*i/res*height*radstodeg)*
2cm ) endfor;
draw graph;

% Draw horiz. lines
path topline, bottomline;
yi = 13*height/50 + height/2;
yim = 3*height/50 + height/2;

label.lft(btex $y_i$ etex, (0, yi));
label.lft(btex $y_{i-1}$ etex, (0, yim));

topline = (origin — xaxisend+overshoot*x) shifted (0, yi);
bottomline = (origin — xaxisend+overshoot*x) shifted (0, yim);
draw topline dashed evenly;
draw bottomline dashed evenly;

% Get intersection points between horiz lines and func
% and corresponding points on the xaxis

(t[1], crap[1]) = graph intersectiontimes bottomline;
(t[2], crap[2]) = graph intersectiontimes topline;
(t[3], crap[3]) = subpath (t[2]+epsilon, length graph) of graph intersectiontimes topline + (t[2],
0); % the epsilon necessary to prevent it from finding t3=t2 incorrectly
(t[4], crap[4]) = subpath (t[1]+epsilon, length graph) of graph intersectiontimes
bottomline + (t[1],0);
(t[5], crap[5]) = subpath (t[4]+epsilon, length graph) of graph intersectiontimes
bottomline + (t[4], 0);
(t[6], crap[6]) = subpath (t[3]+epsilon, length graph) of graph intersectiontimes
topline + (t[3], 0);
for i = 1 upto 6:
z[i] = point t[i] of graph;
endfor

% Check location of intersection points
% fill fullcircle scaled 3bp shifted point t1 of graph;
% fill fullcircle scaled 3bp shifted point t2 of graph;
% fill fullcircle scaled 3bp shifted point t3 of graph;
% fill fullcircle scaled 3bp shifted point t4 of graph;
% fill fullcircle scaled 3bp shifted point t5 of graph;
% fill fullcircle scaled 3bp shifted point t6 of graph;

% Draw the differential areas
path a[];
for i = 0 upto 2:
a[1+i] = (x[1+2*i],0) — subpath (t[1+2*i], t[2*(i+1)]) of graph –
(x[2*(i+1)],0) — cycle;
fill a[1+i] withcolor .9white; % Change to hatching
endfor;
draw graph;

% Draw and label the subsets of the xaxis associated with the differential areas
pair setlabelpos, startlabelarrowpsn;
setlabelpos = (height, -height/4);
startlabelarrowpsn = (height/1.5, -height/5);

pickup pencircle scaled 1.25;
for i = 0 upto 2:
draw (x[1+2*i],0) — (x[2*(1+i)],0);
endfor;
pickup defaultpen;

label.lft(btex $E_i= \{ x : y_{i-1} \leq f(x) \leq y_i \}$ etex, setlabelpos);
for i = 0 upto 2:
drawarrow startlabelarrowpsn — ((x[1+2*i]+x[2*(1+i)])/2, -overshoot) — ((x[1+2*i]+x[2*
(1+i)])/2, -overshoot/5);
endfor;

% Label the main idea of Lebesgue integration
pair idealabelpos;
idealabelpos = (height, height);
label.lft(btex $y_{i-1} l(E_i) \leq A_i \leq y_i l(E_i)$ etex, idealabelpos);

[/illust]