Simpson’s 1/3rd rule is an extension of the trapezoidal rule in which the integrand is approximated by a second-order polynomial. Simpson rule can be derived from the various way using Newton’s divided difference polynomial,  Lagrange polynomial and the method of coefficients. Simpson’s 1/3rd rule is defined by:

This rule is known as Simpson’s One-third rule.

Simpson’s 1/3rd Rule Example

Evaluate ∫₀¹e^xdx, by Simpson’s ⅓rd rule.

Solution:

Let us divide the range [0, 1] into six equal parts by taking h = 1/6.

If x₀ = 0 then y₀ = e⁰ = 1.

If x₁ = x₀ + h = ⅙, then y₁ = e^1/6 = 1.1813

If x₂ = x₀ + 2h = 2/6 = 1/3 then, y₂ = e^1/3 = 1.3956

If x₃ = x₀ + 3h = 3/6 = ½ then y₃ = e^1/2= 1.6487

If x₄ = x₀ + 4h = 4/6 ⅔ then y₄ = e^2/3 = 1.9477

If x₅ = x₀ + 5h = ⅚ then y₅ = e^5/6 = 2.3009

If x₆ = x₀ + 6h = 6/6 = 1 then y₆ = e¹ = 2.7182

We know by Simpson’s ⅓ rule;

∫_a^b f(x) dx = h/3 [(y₀ + y_n) + 4(y₁ + y₃ + y₅ + …. + y_n-1) + 2(y₂ + y₄ + y₆ + ….. + y_n-2)]

Therefore,

∫₀¹e^xdx = (1/18) [(1 + 2.7182) + 4(1.1813 + 1.6487 + 2.3009) + 2(1.39561 + 1.9477)]

=  (1/18)[3.7182 + 20.5236 + 6.68662]

= 1.7182 (approx.)

C Program

//Simpson's one-third rule
//This code is written by Souvik Ghosh
#include<stdio.h>
#include<conio.h>
#include<math.h>
/* Define function here */
float f(float x){
return exp(x);
}
// main function
void main()
{
 float lower, upper, integration=0.0, h, k,simIntegration,error;
 int i, subInterval;
 /* Input */
 printf("Enter lower limit of integration: ");
 scanf("%f", &lower);
 printf("Enter upper limit of integration: ");
 scanf("%f", &upper);
 printf("Enter number of sub intervals: ");
 scanf("%d", &subInterval);

 /* Calculation */
 /* Finding h */
 h = (upper - lower)/subInterval;

 /* Finding Integration Value */
 integration = f(lower) + f(upper);
 for(i=1; i<= subInterval-1; i++)
 {
  k = lower + i*h;
  if(i%2==0)
  {
   integration +=2 * f(k);
  }
  else
  {
   integration +=4 * f(k);
  }
 }
 integration = integration * h/3;
 printf("\nRequired value of integration is: %.6f\n", integration);
}

Output Terminal