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Text "Riemann sums.

Just a (very) quick rough-up.  

Version 0.21
To do:
(i) Sanity check a<<b or define b in terms of a and a variable width
(ii) Include and sanity check case where f(x)<<0
(iii) ...

A function:";
Color 6;
Expr function(f,x)=x^3-(3/2*x^2)+1;
Color 2;
Expr function(f,x);
Text "Left and right ends of integration interval.   Order of partition.   Numerical values of Riemann sums.  (N = 199 the maximum value for which rectangles still show.  N=-1 to disable display.)";
Color 7;
MathPaneSlider 62;
Expr a=slider([-1,1]);
Color 8;
MathPaneSlider 162;
Expr b=slider([-1,1.5]);
MathPaneSlider 33;
Expr M=slider([1,100,99]);
Expr N=-1;
Color 5;
Expr I_R;
Color 6;
Expr I_L;
Color 7;
Expr I_M;
Color 8;
Expr I_m;
Color 3;
Expr x=a,y<function(f,a);
Color 3;
Expr x=b,y<function(f,b);
Text "Width of partition interval.  Locations of partition boundaries.";
Color 3;
Expr D=(b-a)/N;
Color 5;
Expr function(X,s)=a+s*D;
Text "Right and left partitions and Riemann sums.";
Color 3;
Expr function(X,s-1)<x<function(X,s),min([function(f,function(X,s)),0])<y<max([function(f,function(X,s)),0]),s=set(1,ldots,N);
Text "DEBUG – NOTE THE FOLLOWING LOGICAL EQUIVALENTS DO NOT WORK, EVEN WITH M -> floor(M):";
Color 17;
Expr function(X,s-1)<x<function(X,s),0<y<function(f,function(X,s)),s=set(1,ldots,M);
Color 17;
Expr function(X,s-1)<x<function(X,s),0<y<function(f,function(X,s)),s=set(1,ldots,n);
Text "END DEBUG";
Color 17;
Expr function(X,s-1)<x<function(X,s),min([function(f,function(X,s-1)),0])<y<max([function(f,function(X,s-1)),0]),s=set(1,2,ldots,N);
Color 4;
Expr I_R=D*sum(function(f,function(X,k)),k=1,N);
Color 2;
Expr I_L=D*sum(function(f,function(X,k-1)),k=1,N);
Text "Upper and lower partitions and Riemann sums ... sort of.   Rather than finding the max or min on each interval, this simply chooses the max/min at the boundaries of the interval.   This will yield the proper result for any interval on which there are no turning points – therefore essentially not an issue for differentiable functions on small intervals.";
Color 17;
Expr function(X,s-1)<x<function(X,s),min([max([function(f,function(X,s)),function(f,function(X,s-1))]),0])<y<max([max([function(f,function(X,s)),function(f,function(X,s-1))]),0]),s=set(1,2,ldots,N);
Color 17;
Expr function(X,s-1)<x<function(X,s),min([min([function(f,function(X,s)),function(f,function(X,s-1))]),0])<y<max([min([function(f,function(X,s)),function(f,function(X,s-1))]),0]),s=set(1,2,ldots,N);
Expr I_M=D*sum(max([function(f,function(X,k)),function(f,function(X,k-1))]),k=1,N);
Color 4;
Expr I_m=D*sum(min([function(f,function(X,k)),function(f,function(X,k-1))]),k=1,N);
Text "

<size=15>Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";