Trapezoidal Rule for Approximate Value of Definite Integral

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:

$\int _{a}^{b}f(x)\,dx\approx (b-a)\left[{\frac {f(a)+f(b)}{2}}\right]$
For more accurate results the domain of the graph is divided into n segments of equal size as shown below:

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 evalute 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 evalute 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 evalute 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 evalute 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 evalute 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 
?> 

Output:

Value of integral is 0.7842

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

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C++

Java

Python3

C#

PHP

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