In this section we will discuss a Maple program for calculating approximate values of for monotonic functions on the interval . The programs will be based on formulas discussed in theorem 5.40.
Let be a decreasing function from the interval to
,
and let
be a partition of .
We know that
In the programs below, leftsum(f,a,b,n)
calculates
rightsum(f,a,b,n)
calculates
average(f,a,b,n)
calculates the average of
leftsum(f,a,b,n)
and rightsum(f,a,b,n)
.
The
equation of the unit circle is , so the upper unit semicircle is the
graph of where
. The area of the unit circle is 4 times
the area of the portion of the circle in the first quadrant, so
My routines and calculations are given below. Here leftsum
,
rightsum
and average
are all
procedures with four arguments, f,a,b,
and n
.
f
is a function.
a
and b
are the endpoints of an interval.
n
is the number of subintervals in a partition of [a,b]
.
The functions F
and G
are defined by
F
and G
.
The command
average(F,1.,2.,10000);estimates by considering the regular partition of into equal subintervals. and the command
4*average(G,0.,1.,2000);estimates by considering the regular partition of into equal subintervals.
> leftsum := > (f,a,b,n) -> (b-a)/n*sum(f( a +((j-1)*(b-a))/n),j=1..n);
> rightsum := > (f,a,b,n) -> (b-a)/n*sum(f( a +(j*(b-a))/n),j=1..n);
> average := > (f,a,b,n) -> (leftsum(f,a,b,n) + rightsum(f,a,b,n))/2;
> F := t -> 1/t;
> leftsum(F,1.,2.,10000);
> rightsum(F,1.,2.,10000);
> average(F,1.,2.,10000);
> ln(2.);
> G := t -> sqrt(1-t^2);
> 4*leftsum(G,0.,1.,2000);
> 4*rightsum(G,0.,1.,2000);
> 4*average(G,0.,1.,2000);
> evalf(Pi);
Observe that in these examples, average
yields
much more accurate approximations than either leftsum
or rightsum
.