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:
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\left[{\frac {f(a)+f(b)}{2}}\right]}](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-623137118f9f8dada1650f0efe52439b_l3.png "Rendered by QuickLaTeX.com")
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 ]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-e74526c7e8666ca4c834df4485ac631e_l3.png "Rendered by QuickLaTeX.com")
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 = 0xn = 1 # Number of grids. Higher value means # more accuracy n = 6print ("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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