Theoretical Framework for Integral Calculus (1st 2021)

Lecture Titles

• 1. Orientation
• 2. Basic Notations, Elementary Functions
• 3. Definition of the Riemann integral
• 4. Examples of the Riemann integral
• 5. Refinements of partitions
• 6. The Cauchy criterion for integrability
• 7. Integrability of continuous and monotonic functions
• 8. Properties of the Riemann integral
• 9. Further existence
• 10. Fundamental Properties for Calculus
• 11. Integrals and sequences of functions
• 12. Improper Riemann integrals
• 13. Riemann sums
• 14. The Lebesgue criterion for Riemann integrability
• 15. Further arguments

Lecture Notes

• 4/22 Definition, Notation
• 4/29 Examples
• 5/06 Refinements
• 5/13 Cauchy criterion for integrability
• 5/20 Integrability of Continuous functions
• 5/27 Properties of Riieman Integral
• 6/03 Further existence results
• 6/10 Fundamenta Theorems1
• 6/17 Consequence of the fundamental theorem 1
• 6/24 Consequence of the fundamental theorem 2
• 7/01 Integrals and sequences of functions
• 7/08 Integrals and sequences of functions
• 7/15 Absolutely convergent improper integrals
• 7/22 Applications

Report

• Give a proof for the integrability of increasing function on a closed interval.
• Or
• Give a detailed proof for Theorem 1.27.
• Chose one of them or both of them.
• Deadline: 30th July, 2021 (23:59)
• Submittion method: By Moodle system.