[OLD] TABLE OF CONTENTS Preface Preface for the Instructor Chapter 1. Newton’s Original Integral 1.1. Beyond the original Newton integral 1.2. Larger exceptional sets 1.3. Our version of the Newton integral Chapter 2. Covering Relations 2.1. First mean-value theorem for integrals 2.2. Riemann sums 2.3. Riemann sums constructed from the derivative 2.4. Full covers 2.5. Cousin covers 2.6. Cousin covering lemma 2.7. An application of the Cousin lemma Chapter 3. The Integral 3.1. Towards a definition of the integral 3.2. Formal constructive definition of the integral 3.3. Relation with Newton’s integral 3.4. Ignoring a sequence of points 3.5. The convention for ignoring sets of points 3.6. Linear combinations 3.7. Additivity 3.8. Comparing Riemann sums 3.9. Necessary and sufficient condition for integrability 3.10. Integrating continuous functions 3.11. The indefinite integral 3.12. Differentiating the integral 3.13. A comparison test 3.14. Summing inside the integral Chapter 4. Descriptive characterization of the integral 4.1. Modern theory 4.2. Negligible sets 4.3. Ignoring negligible sets 4.4. Growth of a continuous function on a set 4.5. Key constancy lemma 4.6. Descriptive characterization |