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Trapezoidal Rule for Approximate Value of Definite Integral

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In the field of numerical analysis, Trapezoidal rule is used to find the approximation of a definite integral. The basic idea in Trapezoidal rule is to assume the region under the graph of the given function to be a trapezoid and calculate its area. 
It follows that:
{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\left[{\frac {f(a)+f(b)}{2}}\right]}
For more accurate results the domain of the graph is divided into n segments of equal size as shown below: 
 

trapezoidalrule2


Grid spacing or segment size h = (b-a) / n. 
Therefore, approximate value of the integral can be given by: 
\int_{a}^{b}f(x)dx=\frac{b-a}{2n}\left [ f(a)+2\left \{ \sum_{i=1}^{n-1}f(a+ih) \right \}+f(b) \right ]

C++

// C++ program to implement Trapezoidal rule
#include<stdio.h>
  
// A sample function whose definite integral's
// approximate value is computed using Trapezoidal
// rule
float y(float x)
{
    // Declaring the function f(x) = 1/(1+x*x)
    return 1/(1+x*x);
}
  
// Function to evaluate the value of integral
float trapezoidal(float a, float b, float n)
{
    // Grid spacing
    float h = (b-a)/n;
  
    // Computing sum of first and last terms
    // in above formula
    float s = y(a)+y(b);
  
    // Adding middle terms in above formula
    for (int i = 1; i < n; i++)
        s += 2*y(a+i*h);
  
    // h/2 indicates (b-a)/2n. Multiplying h/2
    // with s.
    return (h/2)*s;
}
  
// Driver program to test above function
int main()
{
    // Range of definite integral
    float x0 = 0;
    float xn = 1;
  
    // Number of grids. Higher value means
    // more accuracy
    int n = 6;
  
    printf("Value of integral is %6.4f\n",
                  trapezoidal(x0, xn, n));
    return 0;
}

                    

Java

// Java program to implement Trapezoidal rule
  
class GFG
{
    // A sample function whose definite
    // integral's approximate value  
    // is computed using Trapezoidal
    // rule
    static float y(float x)
    {
        // Declaring the function 
        // f(x) = 1/(1+x*x)
        return 1 / (1 + x * x);
    }
      
    // Function to evaluate the value of integral
    static float trapezoidal(float a, float b, float n)
    {
        // Grid spacing
        float h = (b - a) / n;
      
        // Computing sum of first and last terms
        // in above formula
        float s = y(a) + y(b);
      
        // Adding middle terms in above formula
        for (int i = 1; i < n; i++)
            s += 2 * y( a + i * h);
      
        // h/2 indicates (b-a)/2n. Multiplying h/2
        // with s.
        return (h / 2) * s;
    }
      
    // Driver code
    public static void main (String[] args)
    {
        // Range of definite integral
        float x0 = 0;
        float xn = 1;
      
        // Number of grids. Higher 
        // value means more accuracy
        int n = 6;
      
        System.out.println("Value of integral is "+
                           Math.round(trapezoidal(x0, xn, n) 
                           * 10000.0) / 10000.0);
    }
}
  
// This code is contributed by Anant Agarwal.

                    

Python3

# Python3 code to implement Trapezoidal rule
  
# A sample function whose definite 
# integral's approximate value is 
# computed using Trapezoidal rule
def y( x ):
      
    # Declaring the function 
    # f(x) = 1/(1+x*x)
    return (1 / (1 + x * x))
      
# Function to evaluate the value of integral
def trapezoidal (a, b, n):
      
    # Grid spacing
    h = (b - a) / n
      
    # Computing sum of first and last terms
    # in above formula
    s = (y(a) + y(b))
  
    # Adding middle terms in above formula
    i = 1
    while i < n:
        s += 2 * y(a + i * h)
        i += 1
          
    # h/2 indicates (b-a)/2n. 
    # Multiplying h/2 with s.
    return ((h / 2) * s)
  
# Driver code to test above function
# Range of definite integral
x0 = 0
xn = 1
  
# Number of grids. Higher value means
# more accuracy
n = 6
print ("Value of integral is ",
     "%.4f"%trapezoidal(x0, xn, n))
  
  
# This code is contributed by "Sharad_Bhardwaj".

                    

C#

// C# program to implement Trapezoidal
// rule.
using System;
  
class GFG {
      
    // A sample function whose definite
    // integral's approximate value 
    // is computed using Trapezoidal
    // rule
    static float y(float x)
    {
          
        // Declaring the function 
        // f(x) = 1/(1+x*x)
        return 1 / (1 + x * x);
    }
      
    // Function to evaluate the value 
    // of integral
    static float trapezoidal(float a, 
                       float b, float n)
    {
          
        // Grid spacing
        float h = (b - a) / n;
      
        // Computing sum of first and 
        // last terms in above formula
        float s = y(a) + y(b);
      
        // Adding middle terms in above
        // formula
        for (int i = 1; i < n; i++)
            s += 2 * y( a + i * h);
      
        // h/2 indicates (b-a)/2n. 
        // Multiplying h/2 with s.
        return (h / 2) * s;
    }
      
    // Driver code
    public static void Main ()
    {
          
        // Range of definite integral
        float x0 = 0;
        float xn = 1;
      
        // Number of grids. Higher 
        // value means more accuracy
        int n = 6;
      
        Console.Write("Value of integral is "
          + Math.Round(trapezoidal(x0, xn, n) 
                        * 10000.0) / 10000.0);
    }
}
  
// This code is contributed by nitin mittal.

                    

PHP

<?php
// PHP program to implement Trapezoidal rule
  
// A sample function whose definite
// integral's approximate value is 
// computed using Trapezoidal rule
function y($x)
{
      
    // Declaring the function
    // f(x) = 1/(1+x*x)
    return 1 / (1 + $x * $x);
}
  
// Function to evaluate the
// value of integral
function trapezoidal($a, $b, $n)
{
      
    // Grid spacing
    $h = ($b - $a) / $n;
  
    // Computing sum of first
    // and last terms
    // in above formula
    $s = y($a) + y($b);
  
    // Adding middle terms 
    // in above formula
    for ($i = 1; $i < $n; $i++)
        $s += 2 * Y($a + $i * $h);
  
    // h/2 indicates (b-a)/2n.
    // Multiplying h/2 with s.
    return ($h / 2) * $s;
}
  
    // Driver Code
    // Range of definite integral
    $x0 = 0;
    $xn = 1;
  
    // Number of grids. 
    // Higher value means
    // more accuracy
    $n = 6;
  
    echo("Value of integral is ");
    echo(trapezoidal($x0, $xn, $n));
      
// This code is contributed by nitin mittal
?>

                    

Javascript

<script>
  
// Javascript program to implement Trapezoidal rule
  
// A sample function whose definite
// integral's approximate value  
// is computed using Trapezoidal
// rule
function y(x)
{
      
    // Declaring the function 
    // f(x) = 1/(1+x*x)
    return 1 / (1 + x * x);
}
    
// Function to evaluate the value of integral
function trapezoidal(a, b, n)
{
      
    // Grid spacing
    let h = (b - a) / n;
    
    // Computing sum of first and last terms
    // in above formula
    let s = y(a) + y(b);
    
    // Adding middle terms in above formula
    for(let i = 1; i < n; i++)
        s += 2 * y(a + i * h);
    
    // h/2 indicates (b-a)/2n. Multiplying h/2
    // with s.
    return (h / 2) * s;
}
  
// Driver code
  
// Range of definite integral
let x0 = 0;
let xn = 1;
  
// Number of grids. Higher 
// value means more accuracy
let n = 6;
  
document.write("Value of integral is "+
               Math.round(trapezoidal(x0, xn, n) *
               10000.0) / 10000.0);
                         
// This code is contributed by code_hunt
      
</script>

                    

Output: 

Value of integral is 0.7842


References: 
https://en.wikipedia.org/wiki/Trapezoidal_rule



 



Last Updated : 11 Sep, 2023
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