Finding Integrand using Weedle’s Rule

Given a function and two integers a and b. The task is to find the integrand of the given function from lower limit(a) to the upper limit(b) using Weedle’s Rule.
The given function is:

Examples:

Input: a = 0, b = 6
Output: 1.373448

Input: a = 10, b = 13
Output: f(x) = 0.022897

Approach:
The integration of any function using Weedle’s Formula is given by:

=
where,
h =
and i = [0, 6]

Below are the steps:

Find the value of h from the above formula i.e., h = .
Find the value from to and calculate the value from to
Substitute the above values in Weedle’s Formula to find the integral value.

Below is the implementation of the above approach:

C++

// C++ program to Implement Weedle's Rule
#include <bits/stdc++.h>

using namespace std;
 
// A sample function f(x) = 1/(1+x^2)

float y(float x)
{

    float num = 1;

    float denom = 1.0 + x * x;
 
    return num / denom;
}
 
// Function to find the integral value
// of f(x) with step size h, with
// initial lower limit and upper limit
// a and b

float WeedleRule(float a, float b)
{

    // Find step size h

    double h = (b - a) / 6;
 
    // To store the final sum

    float sum = 0;
 
    // Find sum using Weedle's Formula

    sum = sum + (((3 * h) / 10) * (y(a)

                                + y(a + 2 * h)

                                + 5 * y(a + h)

                                + 6 * y(a + 3 * h)

                                + y(a + 4 * h)

                                + 5 * y(a + 5 * h)

                                + y(a + 6 * h)));
 
    // Return the final sum

    return sum;
}
 
// Driver Code

int main()
{

    // lower limit and upper limit

    float a = 0, b = 6;
 
    // Function Call

    cout<< "f(x) = "<< fixed <<  WeedleRule(a, b);

    return 0;
}
// This code is contributed by shivanisinghss2110

// C program to Implement Weedle's Rule
#include <math.h>
#include <stdio.h>
 
// A sample function f(x) =  1/(1+x^2)

float y(float x)
{

    float num = 1;

    float denom = 1.0 + x * x;
 
    return num / denom;
}
 
// Function to find the integral value
// of f(x) with step size h, with
// initial lower limit and upper limit
// a and b

float WeedleRule(float a, float b)
{

    // Find step size h

    double h = (b - a) / 6;
 
    // To store the final sum

    float sum = 0;
 
    // Find sum using Weedle's Formula

    sum = sum + (((3 * h) / 10) * (y(a)

                                   + y(a + 2 * h)

                                   + 5 * y(a + h)

                                   + 6 * y(a + 3 * h)

                                   + y(a + 4 * h)

                                   + 5 * y(a + 5 * h)

                                   + y(a + 6 * h)));
 
    // Return the final sum

    return sum;
}
 
// Driver Code

int main()
{

    // lower limit and upper limit

    float a = 0, b = 6;
 
    // Function Call

    printf("f(x) = %f", WeedleRule(a, b));

    return 0;
}

Java

// Java program to Implement Weedle's Rule

import java.util.*;

class GFG{
 
    // A sample function f(x) = 1/(1+x^2)

    static float y(float x)

    {

        float num = 1;

        float denom = (float)1.0 + x * x;

        return num / denom;

    }

    // Function to find the integral value

    // of f(x) with step size h, with

    // initial lower limit and upper limit

    // a and b

    static float WeedleRule(float a, float b)

    {

        // Find step size h

        float h = (b - a) / 6;

        // To store the final sum

        float sum = 0;

        // Find sum using Weedle's Formula

        sum = sum + (((3 * h) / 10) * (y(a)

                                    + y(a + 2 * h)

                                    + 5 * y(a + h)

                                    + 6 * y(a + 3 * h)

                                    + y(a + 4 * h)

                                    + 5 * y(a + 5 * h)

                                    + y(a + 6 * h)));

        // Return the final sum

        return sum;

    }

    // Driver Code

    public static void main(String args[])

    {

        // lower limit and upper limit

        float a = 0, b = 6;

        // Function Call

        float num=WeedleRule(a, b);

        System.out.format("f(x) = "+"%.6f", num);

    }
}
 
// This code is contributed by AbhiThakur

Python3

# Python3 program to Implement Weedle's Rule
 
# A sample function f(x) = 1/(1+x^2)

def y(x):

    num = 1;

    denom = float(1.0 + x * x);
 
    return num / denom;
 
# Function to find the integral value
# of f(x) with step size h, with
# initial lower limit and upper limit
# a and b

def WeedleRule(a, b):

    # Find step size h

    h = (b - a) / 6;

    # To store the final sum

    sum = 0;

    # Find sum using Weedle's Formula

    sum = sum + (((3 * h) / 10) * (y(a)

            + y(a + 2 * h)

            + 5 * y(a + h)

            + 6 * y(a + 3 * h)

            + y(a + 4 * h)

            + 5 * y(a + 5 * h)

            + y(a + 6 * h)));
 
    # Return the final sum

    return sum;
 
# Driver Code

if __name__ == '__main__':

    # lower limit and upper limit

    a = 0;

    b = 6;
 
    # Function Call

    num = WeedleRule(a, b);

    print("f(x) = {0:.6f}".format(num));
 
# This code is contributed by sapnasingh4991

// C# program to Implement Weedle's Rule

using System;

class GFG{
 
    // A sample function f(x) = 1/(1+x^2)

    static float y(float x)

    {

        float num = 1;

        float denom = (float)1.0 + x * x;

        return num / denom;

    }

    // Function to find the integral value

    // of f(x) with step size h, with

    // initial lower limit and upper limit

    // a and b

    static float WeedleRule(float a, float b)

    {

        // Find step size h

        float h = (b - a) / 6;

        // To store the final sum

        float sum = 0;

        // Find sum using Weedle's Formula

        sum = sum + (((3 * h) / 10) * (y(a)

                                    + y(a + 2 * h)

                                    + 5 * y(a + h)

                                    + 6 * y(a + 3 * h)

                                    + y(a + 4 * h)

                                    + 5 * y(a + 5 * h)

                                    + y(a + 6 * h)));

        // Return the final sum

        return sum;

    }

    // Driver Code

    public static void Main()

    {

        // lower limit and upper limit

        float a = 0, b = 6;

        // Function Call

        float num=WeedleRule(a, b);

        Console.Write("f(x) = "+num);

    }
}
 
// This code is contributed by AbhiThakur

Javascript

<script>
 
// Javascript program to Implement Weedle's Rule
 
// A sample function f(x) = 1/(1+x^2)

function y(x)
{

    let num = 1;

    let denom = 1.0 + x * x;

    return num / denom;
}

// Function to find the integral value
// of f(x) with step size h, with
// initial lower limit and upper limit
// a and b

function WeedleRule(a, b)
{

    // Find step size h

    let h = (b - a) / 6;

    // To store the final sum

    let sum = 0;

    // Find sum using Weedle's Formula

    sum = sum + (((3 * h) / 10) * 

        (y(a) + y(a + 2 * h) + 5 * y(a + h) + 

            6 * y(a + 3 * h) + y(a + 4 * h) + 

            5 * y(a + 5 * h) + y(a + 6 * h)));

    // Return the final sum

    return sum.toFixed(6);
}
 
// Driver code
 
// lower limit and upper limit
let a = 0, b = 6;
 
// Function Call
let num = WeedleRule(a, b);

document.write("f(x) = " + num);
 
// This code is contributed by divyeshrabadiya07
 
</script>

Output:

f(x) = 1.373448

Article Tags :

DSA

Mathematical