This fourth volume, Advanced Numerical and Algorithmic Approaches to Irresolvable Integrals, is aimed at students, researchers and engineers faced with analytically intractable integrals. Through a rigorous approach, it presents error bounds, maximum precision estimation and calculation of the optimal number of intervals required. Each method is illustrated by an algorithm, a Python implementation and exercises applied to various fields (biology, finance, telecommunications, etc.). Combining mathematical theory, computer tools and case studies, the book offers a clear methodological framework for evaluating and mastering the accuracy of numerical methods. This volume completes a four-volume series dedicated to analytical and numerical integration, offering a structured, exhaustive and resolutely applied vision of integral calculus in science and engineering.