> restart;
> f := x -> sqrt(1-x^2);
>  n := 10: h := 1.0/n:
>  x1 := h: x2 := 2*h:  u2 := 1 - 2*h:  u1 := 1 - h:
>  y1 := f(x1): y2 := f(x2):  v2 := f(u2):  v1 := f(u1):
> [x1,y1], [x2,y2], [u2,v2],[u1,v1];
>  lenNear0 := sqrt( (x2- x1)^2 + (y2-y1)^2 );
>  lenNear1 := sqrt( (u2-u1)^2 + (v2-v1)^2 );
#  Now suppose that we are a bit off in our computation of the
# endpoints.  What will the 
#  effect be?
> epsilon := 0.0001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
>  nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
> 
> epsilon := 0.000001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
> nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
> n := 100: h := 1.0/n:
>  x1 := h: x2 := 2*h:  u2 := 1 - 2*h:  u1 := 1 - h:
>  y1 := f(x1): y2 := f(x2):  v2 := f(u2):  v1 := f(u1):
> lenNear0 := sqrt( (x2- x1)^2 + (y2-y1)^2 );
>  lenNear1 := sqrt( (u2-u1)^2 + (v2-v1)^2 );epsilon := 0.000001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
> nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
> n := 1000: h := 1.0/n:
>  x1 := h: x2 := 2*h:  u2 := 1 - 2*h:  u1 := 1 - h:
>  y1 := f(x1): y2 := f(x2):  v2 := f(u2):  v1 := f(u1):
> lenNear0 := sqrt( (x2- x1)^2 + (y2-y1)^2 );
>  lenNear1 := sqrt( (u2-u1)^2 + (v2-v1)^2 );epsilon := 0.000001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
> nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
> epsilon := 0.00000001:
> n := 10: h := 1.0/n:
>  x1 := h: x2 := 2*h:  u2 := 1 - 2*h:  u1 := 1 - h:
>  y1 := f(x1): y2 := f(x2):  v2 := f(u2):  v1 := f(u1):
> lenNear0 := sqrt( (x2- x1)^2 + (y2-y1)^2 );
>  lenNear1 := sqrt( (u2-u1)^2 + (v2-v1)^2 );epsilon := 0.000001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
> nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
> n := 100: h := 1.0/n:
>  x1 := h: x2 := 2*h:  u2 := 1 - 2*h:  u1 := 1 - h:
>  y1 := f(x1): y2 := f(x2):  v2 := f(u2):  v1 := f(u1):
> lenNear0 := sqrt( (x2- x1)^2 + (y2-y1)^2 );
>  lenNear1 := sqrt( (u2-u1)^2 + (v2-v1)^2 );epsilon := 0.000001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
> nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
> n := 1000: h := 1.0/n:
>  x1 := h: x2 := 2*h:  u2 := 1 - 2*h:  u1 := 1 - h:
>  y1 := f(x1): y2 := f(x2):  v2 := f(u2):  v1 := f(u1):
> lenNear0 := sqrt( (x2- x1)^2 + (y2-y1)^2 );
>  lenNear1 := sqrt( (u2-u1)^2 + (v2-v1)^2 );epsilon := 0.000001:
> nx1 := x1 + epsilon:    ny1 := f(nx1):
> nx2 := x2 + epsilon:    ny2 := f(nx2):
> nu1 := u1 + epsilon:    nv1 := f(nu1):  
> nu2 := u2 + epsilon:    nv2 := f(nu2):
> [nx1,ny1], [nx2,ny2],[nu2,nv2],[nu1,nv1];
> nlenNear0 := sqrt( (nx2- nx1)^2 + (ny2-ny1)^2 );
>  errorNear0 := lenNear0 - nlenNear0;
>  nlenNear1 := sqrt( (nu2-nu1)^2 + (nv2-nv1)^2 );
>  errorNear1 := lenNear1 - nlenNear1;
>