O Estudo de Integrais Múltiplas e suas Aplicações
Raissa Sousa Ferreira
UFERSA
Ivan Mezzomo
Palavras-chave: Integrais múltiplas, Integração numérica, Aerogerador.
Resumo
Multiple integrals are an essential tool in engineering and the exact sciences, widely applied in determining physical quantities such as volume, mass, center of mass, and moment of inertia of three-dimensional solids. The formulation of these integrals allows the mathematical representation of real systems; however, analytical solutions are not always feasible due to the complexity of the functions involved or the geometry of the integration region. In this context, numerical integration emerges as an efficient alternative, providing reliable approximations through iterative methods and successive discretizations. This study aimed to conduct a comparative analysis of four classical numerical methods applied to the evaluation of triple integrals: the Trapezoidal Rule, Simpson’s 1/3 Rule, Simpson’s 3/8 Rule, and Gaussian Quadrature. These methods are primarily used for planar figures, and in this work, they were adapted for three-dimensional geometries. The study was based on an engineering application involving the determination of the volumes of components of the Enercon E-82 EP2 E4 wind turbine, whose main parts were geometrically modeled as solids of revolution: the nacelle as an ellipsoid, the tower as a truncated cone, and the rotor as a paraboloid of revolution. The modeling of the functions was performed using the real-scale dimensions of the components, and the three-dimensional visualization was developed in SketchUp software, ensuring accurate representation of the adopted geometries. The numerical simulations were implemented in Python (version 3.13.2) and executed on a computer equipped with an Intel Core i5 processor, 8 GB of RAM, and the Windows 10 operating system. Different domain subdivisions (n = 30, 60, 120, 360, and 600) were applied to evaluate the performance of each method concerning accuracy and computational cost. The performance of each method was assessed based on the relative error, using analytical solutions of the corresponding triple integrals as a reference. The results indicated that all methods provided adequate approximations, with Simpson’s 3/8 Rule standing out for its lower relative error and higher stability, achieving errors on the order of 10⁻⁸ in determining the nacelle volume. Gaussian Quadrature achieved similar performance only for finer subdivisions, while the Trapezoidal and Simpson’s 1/3 Rules showed intermediate accuracy. Furthermore, computational cost was observed to increase proportionally to the cube of the number of subdivisions (n³), due to the three-dimensional nature of the integral. In conclusion, the study confirms the effectiveness of numerical integration in solving multiple integrals, highlighting Simpson’s 3/8 Rule as the most accurate and stable method among those evaluated for all tested subintervals, being recommended for applications involving the calculation of volumes and physical properties of complex solids.