/* 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 <stdio.h>
#include <math.h>

const float a = 0.0;         /* limits for the area under y = f(x) */ 
const float b = 2.0;         /* area will be computed under f(x) on [a,b] */
const int n = 100;           /* number of subintervals to be used */

/* function to be used in the area approximation
 *  Preconditions: x*x does not underflow or overflow
 *  Postconditions: f(x) = sqrt(4-x^2) 
 */
double f(double x) 
{  
  return sqrt(4.0 - x*x);
}

/* main: runs trapezoid approximation for semicircle of radius 2 centered at
 * origin over [0,2] 
 * 
 * Preconditions: f(x) is defined
 * Postconditions: 
 *  Prints the area to terminal
 *  returns 0
 */
int main (void)
{  
   double width, sum, xvalue, area;

   printf ("Program approximates the area under a function using the ");
   printf ("Trapezoidal Rule, based on %d intervals.\n", 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;

   printf ("The approximate area is %7.4f\n", area);

   return 0;
}