We typically credit Riemann for his discovery of integrals. However, in school, we never actually learn the actual Riemann Integral as he invented. Instead we learn the "Riemann" Integral, which is actually the Darboux Integral. Why is that? In this video, we uncover the truth behind the two different definition of integrals.

Links:
Formal Definition of Supremum and Infimum
https://en.wikipedia.org/wiki/Infimum...

Why Sup and Inf Always Exist
https://en.wikipedia.org/wiki/Least-u...

Proof That the Lim of Upper Sum = Inf of Upper Sum
https://www.overleaf.com/read/dynwhfq...

Integrability of Monotone and Continuous Functions
https://www.kau.edu.sa/Files/0003377/...

Chapters:
00:00 Intro
01:24 Rigorous Foundations of Calculus
02:06 Different Types of Integration
02:48 Generalized Riemann Sum
04:08 Riemann Integrability
04:49 Failure of Limit
05:24 Non-Integrable Function
06:16 Riemann Integrability of x^3
11:11 Upper and Lower Sum
13:01 Redefining Riemann integrals
13:29 Darboux Integrability
14:49 Darboux Integrability of x^3
16:59 Fatal Shortcomings of the Riemann Integral
17:32 Outro


Corrections:
15:12 The denominator for the sum of cubes is 4