Comparative study of riemann integral and lebesgue integral in calculus

The aim of this research paper is to provide a comprehensive comparison between Riemann integral and Lebesgue integral. Integration is described as the inverse process of differentiation, which is used to determine the original function. Riemann integration is a specific type of definite integral applied to find the exact area under a function graph between two limits in a closed interval. Lebesgue integration, on the other hand, provides a more generalized framework for integration theory. Integrals are essential in mathematical modelling and analysis tools. Studying and comparing the similarities and differences between these two integration methods can help us better understand the essence and properties of integrals, so as to more accurately apply integration methods to solve practical problems. This paper provides a systematic analysis of the basics, definitions, concepts, and properties of Riemann integral and Lebesgue integral. Reasoning, proofs, and examples are consolidated to explain the properties and characteristics of these two integral methods. Finally, this paper explores the strengths and limitations of each integration methods, summarising their advantages and applicability to practical problems and providing insights into their respective computational methods and applicability in different contexts.