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# 5.6 Computer Calculation of Area

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

where
 (5.83) (5.84)

Let be the average of and , so

Now represents the area of the trapezoid with vertices , , and , so represents the area under the polygonal line obtained by joining the points and for .

In the programs below, leftsum(f,a,b,n) calculates

which corresponds to (5.84) when is the regular partition of into equal subintervals, and rightsum(f,a,b,n) calculates

which similarly corresponds to (5.83). The command 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

Also

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.

Next: 6. Limits of Sequences Up: 5. Area Previous: 5.5 Brouncker's Formula For   Index
Ray Mayer 2007-09-07