A Comparative Study of Numerical Approximation Methods for Integration: The Trapezoidal Method and Simpson's Method As A Model

Abstract

The research evaluates the accuracy and efficiency of two prevalent numerical integration approaches which include the Trapezoidal Rule and Simpson’s Rule within numerical analysis. A structured numerical model served as the research method to divide the integration interval into equal subintervals. The study used linear, quadratic, exponential and trigonometric functions as representative examples to compare numerical integral approximations against exact analytical values for evaluating method accuracy.

The study performed absolute and relative error analysis using different subinterval counts (n=4,8,16) to create a fair comparison between the methods in identical conditions. The research data confirmed Simpson's Rule delivers more accurate results for non-linear functions because it reduces errors at a rate of  while the Trapezoidal Rule achieves  accuracy. The Trapezoidal Rule stands out as the simpler method which provides better flexibility compared to other rules when subinterval numbers are restricted or irregular.

The optimal method selection requires an evaluation of function behavior together with desired precision level while taking numerical efficiency and algorithmic stability and implementation ease into account. The research suggests combining different approaches in challenging situations to achieve accurate results while managing computational expenses.