# 
# 
# 
> restart;
> f:=x->x^2;
                                     2
                          f := x -> x 
> 
> d:=x->-abs(x)+sqrt(x^2+er);
                                      / 2     \
                 d := x -> -|x| + sqrt\x  + er/
> er:=.1;
                           er := 0.1
> with(plots):
Warning, the name changecoords has been redefined
> read "khtools.m":
> khbox(f,0,1,d);

The Riemann sum is   0.301548

> tagdiv;nops(tagdiv);
[[[0., 0.2377977880], 0.], [[0.2377977880, 0.3956612036], 0.2377977880], 

  [[0.3956612036, 0.5065054669], 0.3956612036], 

  [[0.5065054669, 0.6210772557], 0.5345436910], 

  [[0.6210772557, 0.6969483177], 0.6210772557], 

  [[0.6969483177, 0.7714019855], 0.7036057300], 

  [[0.7714019855, 0.8337031985], 0.7714019855], 

  [[0.8337031985, 0.9175139126], 0.8612965690], 

  [[0.9175139126, 0.951191152], 0.9175139126], [[0.951191152, 1.], 1.]]
                               10
> 
> dd:=x->min(.15,er/(2*x+.15));
                              /          er    \
                dd := x -> min|0.15, ----------|
                              \      2 x + 0.15/
> er:=.1;
                           er := 0.1
> khbox(f,0,1,dd);

The Riemann sum is   0.309189

> tagdiv;nops(tagdiv);
[[[0., 0.15], 0.], [[0.15, 0.3883720930], 0.2383720930], 

  [[0.3883720930, 0.4962767354], 0.3883720930], 

  [[0.4962767354, 0.6123434707], 0.5296511630], 

  [[0.6123434707, 0.6850873057], 0.6123434707], 

  [[0.6850873057, 0.7623437502], 0.6976296960], 

  [[0.7623437502, 0.8220563832], 0.7623437502], 

  [[0.8220563832, 0.9115114509], 0.8579160610], 

  [[0.9115114509, 0.9534883721], 0.9115114509], [[0.9534883721, 1.], 1.]]
                               10
> 
> f2:=x->1-x^4;
                                       4
                       f2 := x -> 1 - x 
> d2:=x->.025/(abs(x)+sqrt(sqrt(.025)))^3;
                                  0.025           
             d2 := x -> --------------------------
                                                 3
                        (|x| + sqrt(sqrt(0.025))) 
> 
> khbox(f2,0,1,d2);

The Riemann sum is   0.806949

> f3:=x->piecewise(x<0,x^2,x=0,-1,1-x^4);
                           /        2                  4\
       f3 := x -> piecewise\x < 0, x , x = 0, -1, 1 - x /
> 
> d3:=x->piecewise(x<0,.025/(abs(x)+sqrt(sqrt(.025)))^3,x=0,.1,min(.15,er/(2*x+.15)));
                    /                 0.025                         
d3 := x -> piecewise|x < 0, --------------------------, x = 0, 0.1, 
                    |                                3              
                    \       (|x| + sqrt(sqrt(0.025)))               

     /          er    \\
  min|0.15, ----------||
     \      2 x + 0.15/|
                       /
> khbox(f3,-1,1,d3);

The Riemann sum is   1.105795

> f4:=x->1/(1+x^2);
                                    1   
                       f4 := x -> ------
                                       2
                                  1 + x 
> d4:=x->-abs(x)+sqrt(x^2+.1*(1+x^2)*(1+max(0,abs(x)-1)));
                      / 2       /     2\                      \
d4 := x -> -|x| + sqrt\x  + 0.1 \1 + x / (1 + max(0, |x| - 1))/
> khbox(f4,-4,4,d4);

The Riemann sum is   2.616072

> 
> 
                          1.539600718
>