```/***************************************************************
/
/ 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);
}```