The Darboux Integral

Darboux Integral Definition
In this video, I define the integral the way it's taught in advanced math courses and which is more intuitive than the Riemann integral. Then I show that continuous functions are Darboux integrable.
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[@bluryourfaceoff]

Huh this is how I was taught integrals, didn't realize they were different from Riemann integrals.

[@drpeyam]

Riemann means random points in your sub interval

[@seanfraser3125]

They’re actually equivalent!

[@CarmeloTLA]

@@drpeyam Most Real Analysis courses (here in Italy at least) teach Darboux Integral calling it Riemann Integral. The first time we talked about the actual Riemann integral, we called it something like "Cauchy sums method".

[@Handelsbilanzdefizit]

Wikipedia: >>>Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral.

[@CarmeloTLA]

@@Handelsbilanzdefizit Oh. Lol

[@slavinojunepri7648]

I just discovered this channel and found it to be an excellent refresher for my calculus and analysis. The proof on the equality of the upper and lower sums of the Darboux integral is neat and elegant. Merci énormément Dr Peyam!

[@drpeyam]

De riennn 😁

[@cassianperera2426]

You are a very good Teacher and a very good lecturer,your explanation is excellent.Thank you.

[@drpeyam]

Thanks so much!!!

[@maximusmadman]

most fun ive ever had watching a math video! this guy is hilarous!!!

[@drpeyam]

Awwwww thank you!!!!

[@luna9200]

You helped me with so much of my analysis course! Now we're onto abstract measure theory and I don't have Peyam to help me through it!! Have you thought about making videos on this?

[@drpeyam]

There’s a video on the Lebesgue integral 😄

[@alieser7770]

Dr. Peyam you are the best! I used tı watch your videos in high school and they are still immensely helpful in college.

[@drpeyam]

I’m so happy to hear that! Thank you 😊

[@mrnogot4251]

I like that you hit the whiteboard when you evoked the monotone sequence theorem lol.

[@masonprice897]

Super helpful!

[@ehess1492]

I actually laughed out loud at 9:18 😂

[@houdarebbouh3149]

You helped very much with my Calculus course thanks to your unique way of teaching. ThankYou!

[@General12th]

This is great! I'm so happy I was able to follow along!

[@juanmolinas]

Greetings Dr!, that was a nice lesson!

[@jyotsanabenpanchal7271]

Woah 😳😨! This is new for me!!

[@darmok3171]

Thanks!

[@drpeyam]

Omg, thanks so much for the super thanks, I really appreciate it! :)

[@phukinho]

Great! Now I can stay home happy ;)

[@skylardeslypere9909]

I didn't know Riemann integration was with random points in the intervals. I was always taught the method with the upper and lower sum and told it was Riemann Integration.

[@skylardeslypere9909]

@@DoctrinaMathVideos i know what Riemann integration is :)

[@IlayShriki]

you are underated

[@drpeyam]

Thank you :3

[@murielfang755]

nice explanation

[@drpeyam]

Thanks!

[@frozenmoon998]

Those of us who love integrals made the effort to watch this whilst it was hidden :)

[@adityaekbote8498]

What how?

[@otaviogoncalvesdossantos862]

Excellent video!

[@Fetrose]

Great presentation.

[@arengolazizian6191]

you are awsome thanks

[@rikthecuber]

@Dr. Peyam How can we differentiate functions of the form x^a where a is irrational? The binomial expansion is defined only for rational exponents, so we cannot use the power rule. Like y= x^(sqrt(2)) Find dy/dx .

[@drpeyam]

You define it as Sup of the derivatives of x^r where r goes through all the rationals less than a

[@rikthecuber]

@@drpeyam I do not know what sup is :(

[@CarmeloTLA]

@@rikthecuber Another way would be by using the limit as x->0 of ((1+x)^a-1)/x that is a (a is any real number). Using some limit algebra in the definition of derivative, you are done.

[@ZipplyZane]

​@@rikthecuber Sup stands for supremum, also known as the "least upper bound." This is the smallest number that is greater than or equal to all numbers in a set. It's kinda like the limit of the maximum value.

[@martinepstein9826]

y = x^(sqrt(2)) = e^(sqrt(2) ln(x)) dy/dx = sqrt(2) e^(sqrt(2) ln(x)) / x = sqrt(2) x^(sqrt(2)) / x = sqrt(2) x^(sqrt(2) - 1)

[@tomasnavarrofebre5876]

Lebesgue integral and mesure next! 🙏🏻

[@drpeyam]

Already done ✅

[@dr.rahulgupta7573]

Many many thanks Dr 3.14159.......m for presenting a nice integral.

[@anthonyjulianelle6695]

In the beginner of the video you talk about subdivide (0, 1) into subrectangles. I think you should say, into, "subintervals".

[@adityaekbote8498]

Soo cool

[@onattanriover]

Appreciate it

[@bamdadtorabi2924]

One question tho. Why do we need f's domain to be [0, 1]? Where does this come into play in the Darboux integral's definition?

[@drpeyam]

Any interval [a,b] suffices

[@SuperYoonHo]

Thank you though i don't understand/...

[@pierreabbat6157]

You've shown that e.g. U15

[@geetathakur445]

Its very similar to the riemann integral defined in walter rudin🤔

[@joluju2375]

Why did Darboux or other mathematicians as well come up with their own definition of an integral ? Is it just an intellectual game, or can it be useful sometimes ?

[@drpeyam]

So the naming is usually done long after their deaths. It’s just that they found for example that the Riemann integral has limitations, so they invent new ones

[@soumyaghosh8823]

Dr.peyam we had an analysis course during my master's so could you make a video on Henstoke integral and one more thing can we extend the proof what you did in measures space

[@drpeyam]

Oh wait, you mean the gauge integral, here it is: kzbin.info/www/bejne/j6rWiYqbbah9jNU

[@vnever9078]

hey peyam love ur videos.......pls make vids on graph theory too.

[@drpeyam]

Thank you!!! But probably not haha

[@vnever9078]

@@drpeyam why? is it an area of mathematics you don't like(to teach)?

[@Errenium]

@@vnever9078 could be outside their comfort zone, or something that is just not conducive to their approach to pedagogy. we all have our limitations

[@dominicjamescunneen1740]

9:17

[@emmazambelli]

❤

[@GhostyOcean]

Hmm, I think it's interesting how you skipped talking about refinements of a partition by using evenly spaced subintervals

[@personxy7443]

if we dont use the partition(1/N),and we just know it has infimum for upper sum,and how do eusure that it converges to the infimum,?is it still monotone?

[@drpeyam]

The result is still true, for this you would use a Cauchy criterion for integrability

[@drpeyam]

people.tamu.edu/~tabrizianpeyam/Math%20409/Lecture%2025.pdf

[@personxy7443]

@@drpeyam thank you,sir!!!~.~

[@timetraveller2818]

its ya boy time traveller back again time travelling proof: profile picture

[@incxxxx]

When the Darboux integrals are useful? Rieman integral is not sufficient?

[@respectpartii6302]

They're useful because they limit the Riemann Integral. The lower Darboux sum is a lower bound of the Riemann Sum and the upper Darbou sum is the upper bound the Riemann sum. And a function is Darboux Integrable if and only if it is Riemann Integrable besides in this case the Darboux sums are equal to the Riemann sum.

[@saroshiqbalbhatti2901]

Module theory book recommend

[@drpeyam]

No

[@akselai]

@@drpeyam bruh

[@saroshiqbalbhatti2901]

@Oily Macaroni what yes

[@rickdoesmath3945]

In this channel we are all analysts and we are afraid of algebraic structures (except for vector spaces, they are cute), so the words you're saying are scaring the hell out of us.

[@rikthecuber]

@@rickdoesmath3945 Meanwhile me being a high school student.

[@sharpnova2]

bruh do a video on Lebesgue integration

[@drpeyam]

Already done ✅