/* ---------------------------------- */
/* save as x_dydx.h                   */
/* ---------------------------------- */
double int_dy(
double (*P_f)(double x, double y),
double x,
double (*P_u)(double x),
double (*P_v)(double x),
   int ny
)
{
   int i = 0;
double m = 0.;
double M = 0.;

 for(i = 0; i <= ny; i++)
 {
    if(i ==0 || i== ny){m = 1.;}
  else                {m = 2.;}

  M += m * (*P_f)( x, ((*P_u)( x)) + i*(((*P_v)( x))-((*P_u)( x))) /ny);
 }                   /*    y                                    */

  return( ((((*P_v)( x)) -((*P_u)( x)))*M) / (2*ny) );
}
/* ---------------------------------- */
double trapezoid_dydx(
double (*P_f)(double x, double y),
double a,
double b,
   int n,
double (*P_u)(double x),
double (*P_v)(double x),
   int ny
)
{
   int i = 0;
double m = 0.;
double M = 0.;

 for(i = 0; i <= n; i++)
 {
    if(i ==0 || i== n){m = 1.;}
  else                {m = 2.;}

  M += m * int_dy((*P_f),(a+i*(b-a)/n), (*P_u),(*P_v),ny);
 }                       /*    x     */

  return( ((b-a)*M) / (2*n) );
}
/* ---------------------------------- */