Exploring error estimates of Newton-Cotes quadrature rules across diverse function classes

This in-depth study looks at symmetric four-point Newton-Cotes-type inequalities with a focus on error estimates for numerical integration. The precision of these estimates is explored across various classes of functions, including those with bounded variation, bounded derivatives, Lipschitzian derivatives, convex derivatives, and others. The research synthesizes and extends existing knowledge, providing a nuanced understanding of how error bounds depend on the characteristics of integrated functions. Through a systematic review of seminal works, the study contributes to the practical application of numerical integration techniques, offering insight for researchers and practitioners to make informed choices based on the specific features of the functions involved.