Contributor: GLENN GROTZINGER
program integration; uses crt;
{ program below demonstrates Pascal code used to compute a definite }
{ integral. Useful for many calculus-related functions such as }
{ finding areas of irregular shapes when a functional relation is }
{ known. You may freely use this code, but do please give me the }
{ credits. }
{ A negative area as an answer, is the result of incorrectly defining
the lower and upper bounds for a function. For example, using the
function
6 - 6x^5, a perfectly justifiable lower bound would be 0, while - 5
would not be. a perfectly justifiable upper bound would be 1, while
6 would not be. The non-justifiable bounds used as examples, are not
defined in the function used, so a negative area would result in this
case
{ Tutorial: this program uses Simpson's rule as a method of finding }
{ the area under a graphed curve. A lower and an upper limit is set }
{ where the area is calculated. The area is cut up into a number of }
{ rectangles dictated by the 'number of divisions'. The more you }
{ divide up this area, the more accurate an approximation becomes. }
var
lower, upper, divisions, sum, width, counter, x, left, right, middle,
c: real;
procedure formula;
{ procedure set apart from rest of program for ease of changing the }
{ function if need be. The function is defined as: f(x) = }
{ , expression being set in a Pascal-type statement }
begin
c := 6 - ( 6 * x * x * x * x * x ); { current function set: 6 - 6x^5 }
end;
begin
clrscr;
{ read in lower bound }
writeln('Input lower limit.');
readln(lower);
{ read in upper bound }
writeln('Input upper limit.');
readln(upper);
{ read in the number of divisions.. The higher you make this number, }
{ the more accurate the results, but the longer the calculation... }
Writeln('number of divisions?');
readln(divisions);
{ set the total sum of the rectangles to zero }
sum := 0;
{ determine width of each rectangle }
width := (upper - lower) / (2 * divisions);
{ initalize counter for divisions loop }
counter := 1;
clrscr;
writeln('Working...');
{ start computations }
repeat
{ define left, right, and middle points along each rectangle }
left := lower + 2 * (counter - 1) * width;
right := lower + 2 * counter * width;
middle := (left + right) / 2;
{ compute functional values at each point }
x := left;
formula;
left := c;
x := middle;
formula;
middle := c;
x := right;
formula;
right := c;
{ calculate particular rectangle area and increment the area to the }
{ sum of the areas. }
sum := (width * (left + 4 * middle + right)) / 3 + sum;
{ write sum to screen as a "working" status }
writeln;
write(sum:0:9);
gotoxy(1,2);
{ increment counter }
counter := counter + 1;
{ stop loop when all areas of rectangles are computed }
until counter = divisions;
{ output results }
clrscr;
writeln('The area under the curve is ', sum:0:9, '.');
{ ^^^^^^^^ }
end. { format code used to eliminate }
{ scientific notation in answer }