PMAT 42363: Theory of Riemann Integration

Course Code    : PMAT 42793

Title                   : Theory of Riemann Integration

Pre-requisites   : PMAT 12253

 Learning Outcomes:

At the end of the course the student should be able to

1. identify the Darboux integrability considering the convergence of upper and lower Darboux sums.

2. make use of first and second Cauchy’s to decide the integrability of a function

3. compare Riemann and Darboux integrals of a function and discuss the properties of Riemann Integrable functions

4. prove and apply Intermediate Value Theorem for Integrals, Dominated Convergence Theorem and Monotone Convergence Theorem, and distinguish the Fundamental theorem of calculus and Second Mean Value Theorem for Integral

5. categorize various types of improper integrals based on the locally integrability and make use of properties of improper integrals

6. appraise improper integrals of nonnegative functions using given tests and determine absolute and conditional convergence of improper integrals

7. identify appropriate change of variables to simplify improper integrals.

 Course Content:

Darboux Integration: 

• Upper and Lower Darboux sum, Darboux Integrability, Properties of darboux Integrability, First and Second Cauchy criterion for Integrability.

Riemann Integration: 

• Riemann sum and the Riemann integral, Relationship between Darboux Integrability and Riemann Integrability, Properties of the Riemann integrability, Intermediate Value Theorem for Integrals, Dominated Convergence Theorem, Monotone Convergence Theorem, Fundamental theorem of calculus, Second Mean Value Theorem for Integral, Change of Variables.

Improper Integrals: 

• Improper Integrals of first and second kind based on locally integrability, Improper integrals of unbounded functions (at left end, right end, both end and an interior point) with a finite domain of integral, Comparison test, Limit comparison test, Cauchy test, absolute and conditional convergence, convergence of beta function. Improper integrals of bounded functions with infinite domain of integrals, convergence at infinity, Comparison test, Limit comparison test, Absolute and conditional convergence, convergence of Gamma function, Abel’s test, Dirichlet’s test.

 Method of Teaching and Learning : A combination of lectures, tutorials and presentations

Assessment :  Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. 1. 1.  Ross, K.A. (2nd Ed., 2015). Elementary Analysis. The Theory of Calculus, Springer.

      2. Trench, W.F. (Hyperlinked Edition 2.04 December 2013), Introduction to Real Analysis, Library of Congress Cataloging-in-Publication Data.

      3. Widder, D.V. (2nd Ed., 2012). Advanced Calculus. Courier Corporation.