© 2023, IJSRMS All Rights Reserved 6 International Journal of Scientific Research in Multidisciplinary Studies Vol.9, Issue.3, pp.06-10, March 2023 E-ISSN: 2454-9312 P-ISSN: 2454-6143 Available online at: www.isroset.org Research Paper Solving Double Integration With The Help of Monte Carlo Simulation: A Python Approach Priyanshi Mishra 1 , Ayush Sharma 2 , Dhananjay R. Mishra 3 , Pankaj Dumka 4* 1,2 Dept. of Computer Science Engineering, Jaypee University of Engineering and Technology, Guna-473226, Madhya Pradesh, India 3,4 Dept. of Mechanical Engineering, Jaypee University of Engineering and Technology, Guna-473226, Madhya Pradesh, India *Corresponding Author: p.dumka.ipec@gmail.com Received: 08/Feb/2023; Accepted: 11/Mar/2023; Published: 31/Mar/2023. | DOI: https://doi.org/10.26438/ijsrms/ v9i3.610 Abstract— In this article, a model of the double integration process has been attempted using Monte Carlo simulation. Both non-rectangular and rectangular domains are dealt separately. Python programming has been used to automate the entire random number-based integration process. Three example problems are used to test the newly created cods, and the results achieved are consistent with those reported in the literature. Keywords— Double integration; Monte Carlo simulation; Monte Carlo integration; Python programming; Integration; Stochastic Integration 1. Introduction Many times, a technological issue produces a system of differential equations that cannot be resolved in closed form, or analytically, hence numerical/probabilistic techniques that approximate the solution are applied [1]–[3]. Numerical methods involve simple arithmetic operations, whether they are based on series expansions or are purely numerical methods that assess the unknown integral at predefined intervals of time. Whereas, in probabilistic approach random variables are used to evaluate the area under the curve or volume of the region [4]–[6]. The results of numerical/probabilistic methods are simply rough approximations of the genuine values. Double integrals are necessary, for instance, to calculate a region's area, the volume below the surface, and the mean value of a two-variable function over a rectangular region [7]. At the same time, it is not that easy to solve double integrals analytically. Hence, a numerical/probabilistic approach is needed to find the answer to a double integral. The multiple- segment trapezoidal rule [8]–[10]can be used to handle the numerical integration. Whereas, Monte Carlo method is a probabilistic approach where random variables are generated in the domain and the probability of occurrence of the points in the domain will return the result [11], [12]. The double integration based on the Monte Carlo approach can be understood by staring with an integration problem as shown in Eq. 1.   2 1 2 1 ) , ( x x y y dydx y x f I (1) The above integration can also be written as the product of area of rectangle formed by rectangle ) , ( 2 1 x x & ) , ( 2 1 y y and the average value of function ) , ( y x f as follows: average x x y y f A dydx y x f I     2 1 2 1 ) , ( (2) where, ) ( ) ( 1 2 1 2 y y x x A     So, the integral might be assessed if there had been a method for determining the average value of the integrand that was independent. The random numbers can be applied in this situation. Consider a list of random values, x and y, that are uniformly spaced apart between ) , ( 2 1 x x & ) , ( 2 1 y y . Simply evaluate ) , ( y x f at each of the randomly chosen points and divide the result by the total number of points to determine the function average, as shown in Eq. 3 [13], [14].    N i average y x f N f 1 ) , ( 1 (3) Now Eq. 3 can be solved to obtain average value of the function and finally from Eq. 2 the value of integration can be obtained. The development of a mathematical technique for computing double numerical integration based on the general Monte Carlo approach mentioned above is the main objective of this study. Even after creating the approach, the process for solving it with pen and paper with high degree of accuracy is not possible. The significance of mathematical programming is now apparent. Mathematical calculations can be automated,