Program to calculate Double Integration

Write a program to calculate double integral numerically.

Example:

Input: Given the following integral.
 \int _{3.7}^{4.3}\int _{2.3}^{2.5}\sqrt{x^4+y^5}\:dxdy 
where
 f(x, y)=\sqrt{x^4+y^5}\ 
Output: 3.915905

Explanation and Approach:

  1. We need to decide what method we are going to use to solve the integral.
    In this example, we are going to use Simpson 1/3 method for both x and y integration.
    To do so, first, we need to decide the step size. Let h be the step size for integration with respect to x and k be the step size for integration with respect to y.
    We are taking h=0.1 and k=0.15 in this example.
    Refer for Simpson 1/3 rule
  2. We need to create a table which consists of the value of function f(x, y) for all possible combination of all x and y points.
    x\y y0 y1 y2 …. ym
    x0 f(x0, y0) f(x0, y1) f(x0, y2) …. f(x0, ym)
    x1 f(x1, y0) f(x1, y1) f(x1, y2) …. f(x1, ym)
    x2 f(x2, y0) f(x2, y1) f(x2, y2) …. f(x2, ym)
    x3 f(x3, y0) f(x3, y1) f(x3, y2) …. f(x3, ym)
    …. …. …. …. …. ….
    …. …. …. …. …. ….
    xn f(xn, y0) f(xn, y1) f(xn, y2) …. f(xn, ym)

    In the given problem,

    x0=2.3
    x2=2.4
    x3=3.5
    
    y0=3.7
    y1=3.85
    y2=4
    y3=4.15
    y4=4.3
    
  3. After generating the table, we apply Simpson 1/3 rule (or whatever rule is asked in the problem) on each row of the table to find integral wrt y at each x and store the values in an array ax[].
  4. We again apply Simpson 1/3 rule(or whatever rule asked) on the values of array ax[] to calculate the integral wrt x.

Below is the implementation of the above code:

C++

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// C++ program to calculate
// double integral value
  
#include <bits/stdc++.h>
using namespace std;
  
// Change the funciton according to your need
float givenFunction(float x, float y)
{
    return pow(pow(x, 4) + pow(y, 5), 0.5);
}
  
// Function to find the double integral value
float doubleIntegral(float h, float k,
                     float lx, float ux,
                     float ly, float uy)
{
    int nx, ny;
  
    // z stores the table
    // ax[] stores the integral wrt y
    // for all x points considered
    float z[50][50], ax[50], answer;
  
    // Calculating the numner of points
    // in x and y integral
    nx = (ux - lx) / h + 1;
    ny = (uy - ly) / k + 1;
  
    // Calculating the values of the table
    for (int i = 0; i < nx; ++i) {
        for (int j = 0; j < ny; ++j) {
            z[i][j] = givenFunction(lx + i * h,
                                    ly + j * k);
        }
    }
  
    // Calculating the integral value
    // wrt y at each point for x
    for (int i = 0; i < nx; ++i) {
        ax[i] = 0;
        for (int j = 0; j < ny; ++j) {
            if (j == 0 || j == ny - 1)
                ax[i] += z[i][j];
            else if (j % 2 == 0)
                ax[i] += 2 * z[i][j];
            else
                ax[i] += 4 * z[i][j];
        }
        ax[i] *= (k / 3);
    }
  
    answer = 0;
  
    // Calculating the final integral value
    // using the integral obtained in the above step
    for (int i = 0; i < nx; ++i) {
        if (i == 0 || i == nx - 1)
            answer += ax[i];
        else if (i % 2 == 0)
            answer += 2 * ax[i];
        else
            answer += 4 * ax[i];
    }
    answer *= (h / 3);
  
    return answer;
}
  
// Driver Code
int main()
{
    // lx and ux are upper and lower limit of x integral
    // ly and uy are upper and lower limit of y integral
    // h is the step size for integration wrt x
    // k is the step size for integration wrt y
    float h, k, lx, ux, ly, uy;
  
    lx = 2.3, ux = 2.5, ly = 3.7,
    uy = 4.3, h = 0.1, k = 0.15;
  
    printf("%f", doubleIntegral(h, k, lx, ux, ly, uy));
    return 0;
}

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Java

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// Java program to calculate
// double integral value
class GFG{
  
  
// Change the funciton according to your need
static float givenFunction(float x, float y)
{
    return (float) Math.pow(Math.pow(x, 4) + 
                        Math.pow(y, 5), 0.5);
}
  
// Function to find the double integral value
static float doubleIntegral(float h, float k,
                    float lx, float ux,
                    float ly, float uy)
{
    int nx, ny;
  
    // z stores the table
    // ax[] stores the integral wrt y
    // for all x points considered
    float z[][] = new float[50][50], ax[] = new float[50], answer;
  
    // Calculating the numner of points
    // in x and y integral
    nx = (int) ((ux - lx) / h + 1);
    ny = (int) ((uy - ly) / k + 1);
  
    // Calculating the values of the table
    for (int i = 0; i < nx; ++i)
    {
        for (int j = 0; j < ny; ++j) 
        {
            z[i][j] = givenFunction(lx + i * h,
                                    ly + j * k);
        }
    }
  
    // Calculating the integral value
    // wrt y at each point for x
    for (int i = 0; i < nx; ++i) 
    {
        ax[i] = 0;
        for (int j = 0; j < ny; ++j)
        {
            if (j == 0 || j == ny - 1)
                ax[i] += z[i][j];
            else if (j % 2 == 0)
                ax[i] += 2 * z[i][j];
            else
                ax[i] += 4 * z[i][j];
        }
        ax[i] *= (k / 3);
    }
  
    answer = 0;
  
    // Calculating the final integral value
    // using the integral obtained in the above step
    for (int i = 0; i < nx; ++i)
    {
        if (i == 0 || i == nx - 1)
            answer += ax[i];
        else if (i % 2 == 0)
            answer += 2 * ax[i];
        else
            answer += 4 * ax[i];
    }
    answer *= (h / 3);
  
    return answer;
}
  
// Driver Code
public static void main(String[] args)
{
    // lx and ux are upper and lower limit of x integral
    // ly and uy are upper and lower limit of y integral
    // h is the step size for integration wrt x
    // k is the step size for integration wrt y
    float h, k, lx, ux, ly, uy;
  
    lx = (float) 2.3; ux = (float) 2.5; ly = (float) 3.7;
    uy = (float) 4.3; h = (float) 0.1; k = (float) 0.15;
  
    System.out.printf("%f", doubleIntegral(h, k, lx, ux, ly, uy));
}
}
  
/* This code contributed by PrinciRaj1992 */

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Python3

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# Python3 program to calculate 
# double integral value 
  
# Change the funciton according
# to your need 
def givenFunction(x, y): 
  
    return pow(pow(x, 4) + pow(y, 5), 0.5
  
# Function to find the double integral value 
def doubleIntegral(h, k, lx, ux, ly, uy): 
  
    # z stores the table 
    # ax[] stores the integral wrt y 
    # for all x points considered 
    z = [[None for i in range(50)]
               for j in range(50)]
    ax = [None] * 50
  
    # Calculating the numner of points 
    # in x and y integral 
    nx = round((ux - lx) / h + 1
    ny = round((uy - ly) / k + 1)
  
    # Calculating the values of the table 
    for i in range(0, nx): 
        for j in range(0, ny): 
            z[i][j] = givenFunction(lx + i * h, 
                                    ly + j * k) 
          
    # Calculating the integral value 
    # wrt y at each point for x 
    for i in range(0, nx): 
        ax[i] = 0
        for j in range(0, ny): 
              
            if j == 0 or j == ny - 1
                ax[i] += z[i][j] 
            elif j % 2 == 0:
                ax[i] += 2 * z[i][j] 
            else:
                ax[i] += 4 * z[i][j] 
          
        ax[i] *= (k / 3
      
    answer = 0
  
    # Calculating the final integral 
    # value using the integral 
    # obtained in the above step 
    for i in range(0, nx): 
        if i == 0 or i == nx - 1
            answer += ax[i] 
        elif i % 2 == 0:
            answer += 2 * ax[i] 
        else:
            answer += 4 * ax[i] 
      
    answer *= (h / 3
  
    return answer 
  
# Driver Code 
if __name__ == "__main__":
  
    # lx and ux are upper and lower limit of x integral 
    # ly and uy are upper and lower limit of y integral 
    # h is the step size for integration wrt x 
    # k is the step size for integration wrt y 
    lx, ux, ly = 2.3, 2.5, 3.7
    uy, h, k = 4.3, 0.1, 0.15
  
    print(round(doubleIntegral(h, k, lx, ux, ly, uy), 6)) 
      
# This code is contributed 
# by Rituraj Jain

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C#

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// C# program to calculate
// double integral value
using System;
  
class GFG
{
  
// Change the funciton according to your need
static float givenFunction(float x, float y)
{
        return (float) Math.Pow(Math.Pow(x, 4) + 
                            Math.Pow(y, 5), 0.5);
    }
      
    // Function to find the double integral value
    static float doubleIntegral(float h, float k,
                        float lx, float ux,
                        float ly, float uy)
    {
        int nx, ny;
      
        // z stores the table
        // ax[] stores the integral wrt y
        // for all x points considered
        float [, ] z = new float[50, 50];
        float [] ax = new float[50];
        float answer;
      
        // Calculating the numner of points
        // in x and y integral
        nx = (int) ((ux - lx) / h + 1);
        ny = (int) ((uy - ly) / k + 1);
      
        // Calculating the values of the table
        for (int i = 0; i < nx; ++i)
        {
            for (int j = 0; j < ny; ++j) 
            {
                z[i, j] = givenFunction(lx + i * h,
                                        ly + j * k);
            }
        }
      
        // Calculating the integral value
        // wrt y at each point for x
        for (int i = 0; i < nx; ++i) 
        {
            ax[i] = 0;
            for (int j = 0; j < ny; ++j)
            {
                if (j == 0 || j == ny - 1)
                    ax[i] += z[i, j];
                else if (j % 2 == 0)
                    ax[i] += 2 * z[i, j];
                else
                    ax[i] += 4 * z[i, j];
            }
            ax[i] *= (k / 3);
        }
      
        answer = 0;
      
        // Calculating the final integral value
        // using the integral obtained in the above step
        for (int i = 0; i < nx; ++i)
        {
            if (i == 0 || i == nx - 1)
                answer += ax[i];
            else if (i % 2 == 0)
                answer += 2 * ax[i];
            else
                answer += 4 * ax[i];
        }
        answer *= (h / 3);
      
        return answer;
    }
      
    // Driver Code
    public static void Main()
    {
        // lx and ux are upper and lower limit of x integral
        // ly and uy are upper and lower limit of y integral
        // h is the step size for integration wrt x
        // k is the step size for integration wrt y
        float h, k, lx, ux, ly, uy;
      
        lx = (float) 2.3; ux = (float) 2.5; ly = (float) 3.7;
        uy = (float) 4.3; h = (float) 0.1; k = (float) 0.15;
      
        Console.WriteLine(doubleIntegral(h, k, lx, ux, ly, uy));
    }
}
  
// This code contributed by ihritik 

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Output:

3.915905


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