Journal of Engineering Advancements Vol. 02(04) 2021, pp 203-216 https://doi.org/10.38032/jea.2021.04.006 *Corresponding Author Email Address: toukirac96@gmail.com Published by: SciEn Publishing Group The Periodicity of the Accuracy of Numerical Integration Methods for the Solution of Different Engineering Problems Toukir Ahmed Chowdhury, Towhedul Islam, Ahmad Abdullah Mujahid, and Md. Bayazid Ahmed Department of Mechanical Engineering, Chittagong University of Engineering & Technology, Chattogram-4349, Bangladesh Received: October 05, 2021, Revised: November 30, 2021, Accepted: November 30, 2021, Available Online: December 23, 2021 ABSTRACT Newton-Cotes integration formulae have been researched for a long time, but the topic is still of interest since the correctness of the techniques has not yet been explicitly defined in a sequence for diverse engineering situations. The purpose of this paper is to give the readers an overview of the four numerical integration methods derived from Newton-Cotes formula, namely the Trapezoidal rule, Simpson's 1/3rd rule, Simpson's 3/8th rule, and Weddle's rule, as well as to demonstrate the periodicity of the most accurate methods for solving each engineering integral equation by varying the number of sub-divisions. The exact expressions by solving the numerical integral equations have been determined by Maple program and comparisons have been done using Python version 3.8. Keywords: Numerical Integration Accuracy, Trapezoidal Rule, Simpson’s 1/3 Rule, Simpson’s 3/8 Rule, Weddle’s Rule. This work is licensed under a Creative Commons Attribution-Non Commercial 4.0 International License. 1 Introduction Throughout the entire history of mathematical research, integration is undoubtedly one of the most important mathematical concepts ever conceived. An integral is a mathematical term that defines displacement, area, volume, and other ideas that result from the combination of infinitesimal data. The process of determining integrals is known as integration. Numerical integration is the process of estimating the value of a definite integral from the estimated numerical values of the integrand. If the numerical integration is performed on a single variable, it is called Quadrature, while for multiple variables, it is called Cubature. Scientists and engineers mostly utilize numerical integration to get an approximate solution for definite integrals that cannot be solved analytically. The reasons for which numerical integration is preferred over analytical are:  Although there is a closed form solution, calculating the answer numerically can be difficult.  The integrand f(x) may only be known at a few locations, as determined through sampling.  Many integrals are not analytically evaluable or have no closed form solution. e.g. ∫ −2 3   0  Although the integrand f(x) is not explicitly known, a collection of data points for this integrand is provided. The term ‘Numerical Integration’ was first coined in 1915 in the booklet named A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb. Many academic areas, including applied mathematics, geometry, finance, statistics, economics, and engineering, use numerical integration methods. The available numerical integration methods include Quadrature methods, Gaussian integration, Monte-Carlo integration, Adaptive Quadrature, and the Euler-Maclaurin formula, which are used to compute complex functions. The Newton-Cotes formulas are also acknowledged as the Newton-Cotes quadrature standards or truly Newton-Cotes laws. These are the numerical integration implementation techniques (also regarded as quadrature), especially focused on measuring the integrand at equally spaced numerical analysis factors. The methods are named after Isaac Newton and Roger Cotes. There are two forms of the method for Newton-Cotes; Open Newton-Cotes and Closed Newton-Cotes. Trapezoidal rule, Simpson 1/3 rule, Simpson 3/8 rule, Weddle’s rule and Boole's rule originate from the closed Newton-Cotes formula. On the other hand, Midpoint law, Trapezoid process, Milne's rule is derived from the formula of open Newton-Cotes. The different numerical integration equations are covered in works by S.S. Sastry [1]-[2], R.L. Burden [3]-[4], J.H. Mathews [5]-[6], and others. M. Concepcion Austin [4] was helpful in evaluating different numerical integration producers and addressing more sophisticated numerical integration techniques. Gordon K. Smith [5] made contributions to this discipline through his analytic study of numerical integration and a collection of 33 papers and books on the subject. Rajesh Kumar Sinha [6] attempted to evaluate an integrable polynomial without using the Taylor Series. Gerry Sozio [7] examined a comprehensive overview of different numerical integration methods. J.Oliver [8] explored the multiple evaluation procedures of definite integrals using higher-order formulas. A. Nataranjan and N. Mohankumar [9] compared several quadrature methods for approximating Cauchy principal value integrals. D.J. Liu, J.M. Wu, and D.H. Yu [10] explored the super convergence of the Newton-Cotes rule for Cauchy principal value integrals. Romesh Kumar Muthumalai [11] attempted to calculate the inaccuracy of numerical integration and differentiation, and he also developed several formulas for numerical differentiation by divided difference. Md. Mamun-Ur- Rashid Khan [12] devised a novel technique to numerical integration strategies for uneven data space. In the realm of applied mathematics, numerical integration has a wide range of applications, particularly in mathematical physics and computational chemistry [13]. It is also employed in population estimation [14], medical picture reconstruction [15], and physics [16]. Chapra SC showed in Applied Numerical Methods with MATLAB [17] the application of numerical methods to solve problems in engineering and science.