Write a program to calculate double integral numerically.
Example:
Input: Given the following integral.where Output: 3.915905
Explanation and Approach:
- 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
- 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
- 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[].
- 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++
// C++ program to calculate // double integral value #include <bits/stdc++.h> using namespace std;
// Change the function 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 number 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;
} |
Java
// Java program to calculate // double integral value class GFG{
// Change the function 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 number 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 */ |
Python3
# Python3 program to calculate # double integral value # Change the function 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 number 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 |
C#
// C# program to calculate // double integral value using System;
class GFG
{ // Change the function 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 number 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 |
Javascript
// JavaScript program to calculate // double integral value // Change the function according to your need function givenFunction(x, y){
return Math.pow(Math.pow(x, 4) + Math.pow(y, 5), 0.5);
} // Function to find the double integral value function doubleIntegral(h, k, lx, ux, ly, uy){
// z stores the table
// ax[] stores the integral wrt y
// for all x points considered
var z = new Array(50);
for ( var i = 0; i < z.length; i++){
z[i] = new Array(50);
}
var ax = new Array(50);
let answer;
// Calculating the number of points
// in x and y integral
let nx = Math.round((ux - lx) / h + 1);
let ny = Math.round((uy - ly) / k + 1);
// Calculating the values of the table
for (let i = 0; i < nx; i++){
for (let 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 (let i = 0; i < nx; ++i)
{
ax[i] = 0;
for (let 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 (let 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;
} // 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 let lx = 2.3, ux = 2.5, ly = 3.7; let uy = 4.3, h = 0.1, k = 0.15; console.log(doubleIntegral(h, k, lx, ux, ly, uy).toFixed(6)); // This code is contributed by lokeshmvs21. |
Output:
3.915905
Article Tags :
Recommended Articles