Programming Assignment 5:
Numerical Integration
CS 2073, Computer Programming with Engineering Applications
Spring Semester, 1992


For this assignment, we want a program that will do numerical integration.  You don't really 
need to know any calculus, since for us the integral of a function will just be the area under its 
graph, or its average value.  This assignment will use three numerical integration methods:

·	the trapezoid method,
·	Simpson's method, and
·	a Monte-Carlo method.

We will be finding the value of the integral of a function f(x), for x from a to b.  You will also 
start with an integer n representing the number of intervals to divide the segment from a to b 
into.  The size of each interval is h = (b - a)/n.  Given these starting values, the trapezoid method 
uses the formula

	   Integral » (h/2)[f(a) + 2´f(a+h) + 2´f(a+2h) + 2´f(a+3h) + 2´f(a+4h) + . . . 
                                                                           + 2´f(a+(n-2)h) + 2´f(a+(n-1)h) + f(b)]

Similarly, Simpson's method uses the formula

	   Integral » (h/3)[f(a) + 4´f(a+h) + 2´f(a+2h) + 4´f(a+3h) + 2´f(a+4h) + . . . 
                                                                           + 2´f(a+(n-2)h) + 4´f(a+(n-1)h) + f(b)]

Finally, a Monte-Carlo method might use

	   Integral » h´[f(x1) + f(x2) + f(x3) + . . . + f(xn-1) +  f(xn)]

Here the numbers  x1, x2, . . . , xn are randomly chosen from the interval from a to b.

You should use the two specific functions

		f(x) = 1/(1 + x2), for x from 0 to 1, and 

		g(x) = e-x2, for x from -2 to 2.

Finally for each of the two functions, and for each of the three integration methods (6 cases), you 
should try n = 10, and n = 1000.  Thus you should have 12 answers altogether, and your answers 
should be clearly labeled with the function (f or g above), the values of a and b, the value of n, 
and the integration method.

You must use Pascal functions to calculate f and g as above.

You must use a procedure generate_f that will take as inputs the numbers a, b, and n, and 
will return (as a reference parameter) an array of function values funcval, with f(a), f(a+h), 
f(a+2h), . . . , f(a+(n-1)h), f(b) stored in array locations 0 through n.  Similarly for a procedure 
generate_g:

	const Maxval = 1000;
	type funcvaltype = array[0..Maxval] of real;
	procedure generate_f (var funcval: funcvaltype; a, b: real; n: integer);
	procedure generate_g (var funcval: funcvaltype; a, b: real; n: integer);

Another Pascal procedure generate_fm should take a, b, and n as inputs and generate n 
function values f(x1), f(x2), f(x3), . . . , f(xn-1),  f(xn) in the array funcval, and similarly for a 
procedure generate_gm:

	procedure generate_fm(var funcval: funcvaltype; a, b: real; n: integer);
	procedure generate_gm(var funcval: funcvaltype; a, b: real; n: integer);

Then you must have three Pascal functions trap, simp and monte that use the array 
funcval and the values a, b, and n, to calculate the integral according to the above formulas:

	function trap  (funcval: funcvaltype; a, b: real; n: integer): real;
	function simp  (funcval: funcvaltype; a, b: real; n: integer): real;
	function monte (funcval: funcvaltype; a, b: real; n: integer): real;

The random numbers can be generated using a random number generator that will be separately 
distributed and discussed in class.  Let's use type double for all real numbers, rather than 
real.  Print answers with 16 significant digits.

(Note: for efficiency sake above, one might want to make the funcval parameters to the three 
functions above reference parameters.)

Extras: 
    In addition to the above, find an approximate value for the integral of g(x) from -¥ to +¥.