// A program for approximating the area under a function y = f(x)
// between a and b using the Trapezoidal Rule.
//    The Trapezoidal Rule divides [a, b] into n evenly spaced intervals
//    of width W = (b-a)/n.  The approximate area is
//    W*(f(a)/2 + f(a+W) + f(a+2W) + ... + f(a+(n-2)W) + f(a+(n-1)W) + f(b)/2)
// Version 1:  Approximating area under sqrt(4-x^2) on [0, 2].
//    As this is 1/4 of circle of radius 2, result should be Pi.

#include <iostream.h>
#include <math.h>

const double a = 0.0;      /* limits for the area under y = f(x) */ 
const double b = 2.0;      /* area will be computed under f(x) on [a,b] */

double f(double x) 
/* function to be used in the area approximation */
{  return (sqrt(4.0 - x*x));
}

int main (void)
{  int  n;
   double width, sum, xvalue, area;

   cout << "Program approximates the area under a function using the "
        << "Trapezoidal Rule." << endl;
   cout << "Enter number of subintervals to be used: ";
   cin  >> n;

   width = (b - a) / n; 

   /* compute the sum in the area approximation */
   sum = (f(a) + f(b)) / 2.0;            /* first and last terms in sum */
   for (xvalue = a + width; xvalue < b; xvalue += width)
      sum += f(xvalue);

   area = sum * width;

   cout << "The approximate area is " << area << endl;

   return 0;
}