Why We Never Actually Learn Riemann's Original Definition of Integrals - Riemann vs Darboux Integral

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

Why Sup and Inf Always Exist

Proof That the Lim of Upper Sum = Inf of Upper Sum

Integrability of Monotone and Continuous Functions

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

[varno]

I think the important thing about the reiman integral is it "feels" more general and natural. So proving that the Darboux integral is well founded and equivalent to the reiman integral is an important logical step, even though once the proof is done, you dont ever want to look at it again. The knowledge that they are equivalent is in itself valluable.

[markborz]

1) Riemann never intended to use his definition for calculations. This is one of those typically theoretical definitions, very generell and useful for existence proofs and such.
2) In 1868 Riemann was already 2 years dead.
He found his legendary definition around 20 years earlier as a young student in Göttingen.

[waarschijn]

Riemann's definition generalizes to the Gauge Integral, where the max width δ isn't the same for all intervals, but depends on the tag, so it's really a function δ(x), called a gauge. All Lebesgue- and indefinite-Riemann-integrable functions are gauge integrable. It has apparently found use in evaluation of some path integrals in Quantum Mechanics.

[NaHBrO]

Maybe highschool, but not in uni. My analysis class taught the Riemann integral first, then the darboux integral. We have some useful theorems to prove, almost never used original Riemann definition, just like you said.

[infastka]

There's a misprint on 15:12 - there must be "4" instead of "3" in the denominator of an expression for sum of cubes.
I actually learned both integrabilities in university and can say that this is a really cool and interesting video! Keep it up 👍

[fedebonons]

Nice video!
You should do a follow up on Lebesgue integration next, it would be epic!

[maxfred]

So why do I hear Darboux‘ name the first time today if their method is more powerful?

[FreeGroup]

when you understand the construction of the riemann integral, you understand the construction of the lebesgue integral when you yearn the basics in measure theory

[Mau-vzqo]

Very nice video!
To my mind, a strong advantage of the Darboux variant is that you need only check one specific sequence of partitions. In contrast to that, in the Riemann variant you had to keep the partitions (and tags) arbitrary. Just wanted to mention this since you used it (of course) but didn't actively point it out.

[Eknoma]

We learn Riemanns definition of integrals in Calculus 1 mate...
And then immideately learn Darboux integral.
Then in any further analysis course, you learn about Lebesgue integrals.

[maxfred]

Please do a Video about Lebesgue integrals :)

[Basilisk-tuud]

I actually did dadboux integrability proofs in my freshman calc class. My math dept was on some crazy shir lmao

[cparks]

Using the Darboux definition is more-or-less just applying the squeeze theorem to the Riemann definition.

[HADN]

Yet another "you never learned this in school" topic that we DID learn in HS... Thank you Bélabá, you were the best math teacher we ever had!

[caspermadlener]

Defining integrals based on a definition would be a bit unmathematical anyway, as you always want to use an axiomatisation instead.

Darboux has the first rigorous definition, equivalent to the following three/four properties.
1. The integral of a constant function is...
2. The integral of f(x) from a to b plus from b to c is the integral of f(x) from a to c.
3a. If f(x)≥0 for a≤x≤b, the the integral of f(x) from a to b is greater or equal to 0.
3b. If f(x)≤0 for a≤x≤b...

The cool part about axiomatisation with inequalities is that you don't have any limits. I guess it cool, but you would want to use limits in practise anyways, since you basically only care about cases of equality.

[jonathanewebster]

There is flexibility in Riemann integral when integrating from the definition. For instance you can integrate sqrt(x) from 0 to 1 by choosing x_i = (i/n)^2 for i = 1, ..., n as the partition points. Now, Δx_i = (2i + 1)/n and sqrt(x_i) = i/n. So, this non-regular partition results in a sum that can be evaluated. The Darboux definition on a regular partition results in evaluations points for which the sum cannot be evaluated directly because we have no general exact formula of sum of sqrt(i).

[bigshrekhorner]

I've actually learnt about both Riemann and Darboux integration, although the arbitrary partitioning method was attributed to Darboux, not Riemann. We've also formalized it using the Lower and Upper sums and their convergence to the same number from the get go. Never seen the arbitrary rectangle point/height method before.

[Stars]

Nice one 🎉 Good topic to discuss...But I think need more like this

[williamdavis]

Great video! Would you consider covering the Ito and Stratonovich integrals? They are very interesting topics!

[bjornfeuerbacher]

Newton and Leibniz did _not_ invent the idea to approximate the area by rectangles and refining the approximation by using more and more rectangles. That idea existed centuries, if not decades, before them. E. g. Fermat, Pascal and Cavalieri, among many others, used that idea extensively.