Особливості застосування чисельних методів в обчислені визначених інтегралів

Abstract

The article discusses the features of the application of numerical methods in the calculation of a definite integral using the classical mathematical apparatus of quadrature formulas for rectangles, trapezoids and Simpson's (parabolas). The use of numerical integration is critically important, since it allows solving mathematical problems that are impossible or too difficult to calculate analytically using the classical Newton-Leibniz formula. Based on the results obtained, it is proven that Simpson's formula provides the highest accuracy of calculations compared to the formulas for rectangles and trapezoids for the same number of interval divisions. Simpson's formula can be applied to evenly spaced nodes in the case of an even number of subintervals and an odd number of nodes. Simpson's formula has the fourth order of accuracy. The calculation error of the formulas of the left and right rectangles has the first order of accuracy. It is larger than for the formula of the middle rectangles due to symmetry violation. The formula of the middle rectangles and trapezoids has the second order of accuracy. The accuracy of the quadrature formula is characterized by the order of the residual term relative to the degree of the integration step. The quadrature formula is considered to be more accurate, the higher the order of its residual term. In addition, the residual term of the quadrature formulas depends on the integration step. A theoretical and practical implementation has been applied, which is optimal for solving applied problems where analytical finding of the initial is impossible or too complicated. In order to qualitatively perceive information and assimilate it, the Maple V R4 computer mathematics system was used for calculations. The use of the Maple V R4 computer mathematics system allowed not only to automate cumbersome calculations, but also to obtain a visual visualization of the integration process. The use of such systems as Maple, Wolfram Mathematica and MATLAB allows not only to automate the computational process, but also to visualize the dynamics of approximation, which is critically important for analyzing the convergence of algorithms.