Synopsis for Analysis III: Integration

Overview

In these lectures we define a simple integral and study its properties; prove the Mean Value Theorem for Integrals and the Fundamental Theorem of Calculus. This gives us the tools to justify term-by-term differentiation of power series and deduce the elementary properties of the trigonometric functions.

Learning Outcomes

At the end of the course students will be familiar with the construction of an integral from fundamental principles, including important theorems. They will know when it is possible to integrate or differentiate term-by-term and be able to apply this to, for example, trigonometric series.

Synopsis

Step functions, their integral, basic properties. Lower and upper integrals of bounded functions on bounded intervals. Definition of Riemann integrable functions.

The application of uniform continuity to show that continuous functions are Riemann integrable on closed bounded intervals; bounded continuous functions are Riemann integrable on bounded intervals.

Elementary properties of Riemann integrals: positivity, linearity, subdivision of the interval. The Mean Value Theorem for Integrals. The Fundamental Theorem of Calculus; linearity of the integral, integration by parts and by substitution.

The interchange of integral and limit for a uniform limit of continuous functions on a bounded interval. Term-by-term integration and differentiation of a (real) power series (interchanging limit and derivative for a series of functions where the derivatives converge uniformly).

1. T. Lyons Lecture Notes (online).
2. H. A. Priestley, Introduction to Integration (Oxford Science Publications, 1997), Chapters 1–8. [These chapters commence with a useful summary of background `cont and diff' and go on to cover not only the integration but also the material on power series.]
3. Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis (Wiley, Third Edition, 2000), Chapter 8.
4. W. Rudin, Principles of Mathematical Analysis, (McGraw-Hill, Third Edition, 1976).