Chapter 9 - Riemann Integration

Abstract

Chapter 9 is about the Riemann integration. It starts with the definition of proper Riemann integral and presents Darboux and Riemann criteria for integrability. It is proved that continuous functions and functions of bounded variation on closed bounded intervals are Riemann integrable. Lebesgue characterization of integrable functions is presented. Properties of proper Riemann integral are discussed. Mean value theorem for integrals, fundamental theorem of calculus, integration by parts, and change of variable formula are proved. Picard-Lindelöf theorem on existence and uniqueness of solution for differential equations is given. The Riemann integral depending on a parameter is considered and different theorems on interchange of limit and Riemann integral are discussed. The chapter ends with improper integrals.