/***************************************************************
/
/ trapezoid.c
/
/ A program in C to calculate the area under a curve by using the
/ Trapezoid Rule.  This is also called the First Degree Closed
/ Newton-Cotes Formula.
/
/ Paul Soper
/
/ March 2, 2018
/
/***************************************************************/

#include "stdio.h"

/***************************************************************
/ double f(double x)
/ 
/ We need to define the curve by an algebraic function.  We will / use a polynomial here, but the function could be any continuous / function
/
/***************************************************************/
double f(double x)
{
  /* use 5x^3 - 12x^2 + 7x - 3 as the polynomial function */
  return (5.0*x*x*x - 12.0*x*x + 7.0*x - 3.0);
}

int main()
{
  int n = 10;        /* n = the number of segments */
  int i = 0;         /* a counter */
  double area = 0;   /* hold the area under the curve */  
                     /* as we sum over the areas of */
                     /* each segment */

  /* We will be calculating the area under the curve y = f(x)
     from x = a to x = b, using n segments */

  /* x1 and x2 are the endpoints for the whole calculation */
  double x1 = 2.0;   
  double x2 = 6.0;

  double width = (x2 - x1)/(float)n;

  /* a and b are the endpoints for each segment */
  double a;          
  double b;

  for (i = 0; i < n; ++i)
  {
    a = x1 + i * width;
    b = x1 + (i + 1) * width;
    /* this is obviously not optimized, but is clear */
    area = area + ((b-a)/2.0) * (f(a + 0.0 * (b - a)/1.0) + 
                  f(a + 1.0 * (b - a)/1.0));
  }

  printf ("Area = %16.10f\n", area);

  /* We can easily check our answer, because f is just a 
     polynomial and is easily integrated, so we can solve the   
     problem analytically rather than numerically.
     The integral is 
         g(x) = (5/4)x^4 - (12/3)x^3 + (7/2)x^2 -3x + C
     The definite integral (which is the area under the curve, is
         g(b) - g(a)
     In our case that is  864 - (-4) = 868.0 */

  printf ("Analytic solution:  area =  868.0000000000\n");

  return (0);
}