Comparar el método Monte Carlo y las fórmulas de Newton Cotes para el cálculo de integrales múltiples

Abstract

Students and professors of Mathematics at the Escuela Superior Politécnica de Chimborazo do not have a support tool for the study and solving of multiple definite integrals that facilitates the understanding and application of the Newton-Cotes formulas as well as the Monte Carlo method through illustrated applications of simulated processes for those cases in which it is not possible to find an analytical solution, Therefore, the aim of this research is to compare the Monte Carlo method with the Newton-Cotes formulas through a theoretical research to estimate multiple definite integrals, as well as simulations using Octave software, which will allow generating a document as a guide to understand the application of these methods. The implemented methodology consists of developing a theoretical research of all the aspects related to the application of the Newton-Cotes formulas and the Monte Carlo method for the resolution of multiple definite integrals and the development of simulated applications in the mathematical program Octave that provide numerical results to make a comparison between the applied methods and establish similarities and differences. As a result of applying this methodological framework, it was possible to establish that the Newton-Cotes formulas are effective for the resolution of multiple definite integrals that consider up to five variables, presenting a margin of error of 5% with respect to the real result. On the other hand, the Monte Carlo method provides greater benefits for cases that present up to 8 variables with a margin of error of 10%. With these results it is concluded that the software tool developed in this curricular integration work provides an educational support for the undergraduate level for the study of multiple definite integral solving techniques for both professors and students.