Integral with a limit

[@drpeyam]

Link to the math blog: www.math3ma.com/blog/dominated-convergence-theorem Dominated Convergence Theorem: kzbin.info/www/bejne/o4ayk3iQf5p_iNk sin(x)/x: kzbin.info/www/bejne/gWGWlaCZpdV0aq8

[@carlosgiovanardi8197]

www.amazon.com/Equations-Mathematical-Physics-Samarskii-Tikhonov/dp/0080102263 www.amazon.com/Collection-Problems-Mathematical-Physics-Dover/dp/0486658066

[@route66math77]

The mighty Dominated Convergence Theorem strikes again, awesome! :-)

[@guitar_jero]

Better than just getting the answer and say a wizard did it :)

[@route66math77]

@@guitar_jero As I always say, "It's not magic, it's logic!" :-D

[@insouciantFox]

When guessing, guess π. When in doubt, Euler did it.

[@OuroborosVengeance]

This is pure poetry

[@arvindsrinivasan424]

If you decide to take the “hard” route and integrate first, the integral gives pi*n*(1-e^-(1/n)) [I integrated this using Residue Theorem and recognizing that the real part of (e^(i*x/n)-1)/(i*x/n) is equal to sin(x/n)/(x/n) ] Taking the limit as it goes to infinity requires taking the limit of an indeterminate argument, but is quite easy to show it gives pi

[@charlesbromberick4247]

that´s what I would have tried

[@elephantdinosaur2284]

Nice solution Arvind! I tried my luck with n*exp(iz/n)/z(z^2+1) directly but had issues doing the small loop around the origin because of the pole there. Your trick circumvents the issue nicely. Regarding the final limit I like to expand exp(-1/n) = 1 - 1/n + O(1/n^2) and then just substituting in. Saves all the messiness with l'hopitals rule.

[@violintegral]

You could also use Feynman's trick to evaluate this integral as an alternative to Residue Theorem

[@blightedcrowmain8236]

i like how he says ty for watching at the beggining

[@pbj4184]

Now that you say it, only Dr. Peyam does that 🤔 I never noticed it but now I realized I always know it's a Dr. Peyam video whenever I hear those words. 11/10 on branding, Dr. Peyam!

[@thedoublehelix5661]

7:25 lmaoo

[@byronvega8298]

Ah yes, the fundamental theorem of engineering

[@AnkitSharma-ef7md]

Dr. Peyam is high on mathematics. I loved the approach. Coolest professor I ever have seen. Wished you were my post graduate professor.. 😁😁😁😁😁

[@tejpalsinghantal8987]

Mandatory comment to help with the algorithm

[@umerfarooq4831]

Great video informative , interesting and nonetheless entertaining especially the "some kind of people"

[@LatestBinge]

Love your videos! I'm an 8th grader but thanks to you I start to understand and love complex math!

[@danielaorozco9995]

feels like magic :0 thanks a lot doctor!

[@joelschwed7177]

There is no limit to your wisdom good sir...

[@yogeshwagh2849]

Awesome 🙌🏻

[@TheAustrianAnimations87]

This video got randomly recommended to me. Nice, I've learned something new.

[@RalphDratman]

I enjoy mathematics but this is one of those moments when I become very aware that I could never become a mathematician. I did become an engineer (of computer-related hardware and software), but unofficially. Now that I'm retired, I wonder how I ever managed to do any of that!

[@shivaudaiyar2556]

Thanks for such a great content with love from India

[@rounaksinha5309]

Happy mathematics day Perfect integral for Mathematics day

[@IoT_]

7:26 I'm an engineer and I don't understand why you didn't allow to put equal sign between sine function and x on your math channel 🌝

[@123ucr]

If I were a math professor like you, I would put this problem in a Calculus 2 Final exam or an AP Calculus BC exam. Heck, it might even be an extra credit problem. Why? For starters, because you would have to probably use L'Hospital's rule and you definitely need your trigonometry skills to solve this problem. Second of all, this problem's solution can be messy like a chili cheeseburger and takes a lot of time to solve (at least 9-10 minutes). A typical Calculus 2 problem takes like 5-7 minutes to solve. Third and finally, this is one of the hardest Calculus 2 problems that I have seen in my life.

[@UltraMaXAtAXX]

Well, except a calculus II student wouldn't really know about the DCT.

[@123ucr]

@@UltraMaXAtAXX --- I guess it looks like a Calculus 2 problem. That is why I would say it should be an extra credit problem on a Calculus 2 final. Is it possible for a Calculus 2 student to solve this problem? Short answer, yes. But, it is difficult for a Calculus 2 student to solve? Short answer, also yes. In reality, it is actually an advanced calculus or a real analysis problem or at least an Honors Calculus 2 problem.

[@Chariotuber]

great video as always dr peyam! this really brings me back to when I took a mathematical physics class, which basically ended up being a class on functional analysis. good times lol

[@comingshoon2717]

gracias Doctor Peyam... ahora ya sabemos que los límites pueden ingresar dentro de una integral sin problemas jejej ... saludos desde 🇨🇱🇨🇱

[@deadfish3789]

You did this proof entirely backwards, but I liked it

[@deadfish3789]

Also I knew you could swap it if f_n-->f uniformly, but I'm assuming this is a weaker condition, and therefore applicable in more circumstances

[@JSSTyger]

Dr Peyam is like that midnight snack while the wife is sleeping.

[@vishalmishra3046]

Simply change integration range from a to b and you get the result | atan(b) - atan(a). Now replace a and b with ∞ and -∞ to get π/2 - (-π/2) = π. Simple.

[@drpeyam]

You’re completely missing the point of the video, this is precisely how you shouldn’t evaluate the integral

[@vishalmishra3046]

@@drpeyam For a moment let's assume no specific value of a and b except that both are real and a < b. Is there any combination of (a,b) for which the value of the integral is not ( atan(b) - atan(a) ) ? Given no exceptions, the value of the integral remains accurate for all combinations of such a constrained pair (a, b) including when -a and b are extremely large (and approaching infinity). This is clearly a more generalized solution that is also applicable to the specific case of (a, b) -> (-∞, ∞) in the video. Right ?

[@drpeyam]

Again, not the point of the video, there are some integrals for which you cannot simply pass to the limit like n times indicator of (0,1/n). The point is not the (a,b) values but whether you can simply put the limit inside the integral

[@toby6225]

Dr. Peyam: uses Dominated Convergence Theorem Engineers: we don't do that here

[@Frandahab]

Physicist here, we don't do that either xD

[@lucho2868]

9:42

[@rounaksinha5309]

Happy National Mathematics day to everyone

[@jamesbentonticer4706]

Is it just on India? Or is it national math day in usa as well?

[@rounaksinha5309]

@@jamesbentonticer4706Hii! James it is just India. IN USA and all over the world Math day is celebrated on 14 March and I have also read somewhere that another Math day or same kinda stuff is observed on 15 October in USA

[@jamesbentonticer4706]

@@rounaksinha5309 okay thanks for the info. I thought I'd make sure before I went and said happy math day to everyone lol

[@frozenmoon998]

If there is a Peyam, there is a way!

[@adam_elm_5680]

im happy when I get a notif of a new video!

[@mathjitsuteacher]

Hi Peyam, great video as always! There is a little mistake. The inequality -x

[@drpeyam]

It is true actually. sin(-pi/2) = -1 which is between -pi/2 and pi/2

[@mathjitsuteacher]

@@drpeyam I believe you wanted to use sin(x)

[@robertgerbicz]

You also made a mistake, in the video -x

[@drpeyam]

Yeah I meant to say for positive x. In any case doesn’t matter since we’re taking absolute values

[@jiteshsingh3473]

Make video on value of evolution of 'e'

[@strmandola5484]

Please calculate integral 1/a + cos(X) !

[@heliocentric1756]

7:26 😂😁 What type of people are you referring to?

[@gisopolis77]

engineers

[@nanangmuhammad2067]

I like this trick. Really can we put limit inside integral?

[@afrolichesmain777]

If the conditions that were shown are satisfied, you can swap the integral and limit signs. However, if you know that fn converges to f uniformly, you can also swap the limit and integral signs.

[@ajiwibowo8736]

Doc, sometimes I do innequalities of rational function involving absolute value in it with the number test on number line. If mathematician they wont do test like that right? They must be make it in some cases or so. How did you do if theres some innequalities like that doc? May you give us how is your mathematical method on solving that kind of problem?

[@charlesbromberick4247]

smart guy

[@michaelz2270]

You really dominated in that video.

[@drpeyam]

LOL

[@DELTASERPENT]

Great teaching Payamji. Have you written any books? Maths made easy!!!!

[@drpeyam]

I should!!!

[@레온하르트-r2p]

Lebesgue Integration Theory is very fun

[@arturcostasteiner9735]

Grreat! The dominated convergence theorem comes from measure theory, right?

[@drpeyam]

Yes

[@carlosgiovanardi8197]

GREAT! Thanks for sharing. if possible, can you make videos on pde´s (parabolic, elliptic, hyperbolic) focusing on examples and their solutions? do you know the books of Budak, Samarsky and Tijonov?

[@drpeyam]

There are two PDE playlists, check them out! Also nothing beats Evans’ textbook

[@carlosgiovanardi8197]

@@drpeyam thank you for your answer. i know your playlists. appreciate but this not what i am looking for. anyway, will go after your reference.

[@dipayanguhapatra3580]

Please integrate x square times of tan2x dx x^2.tan2x dx

[@iabervon]

I say that, once you've proven what the derivative of sin x is, you can use l'Hopital's rule to find the limit of (sin x)/x if you forget what that value was. I think the first time I saw the derivation of the derivative of sin x, that limit wasn't called out as a lemma, and we just proved it again later when we needed it for something else.

[@randomlife7935]

According to blackpenredpen, using l'Hospital's rule on the limit of (sin x/x) is wrong because to determine the derivative of sin x, the limit of (sin x/x) is used, thus making the proof circular.

[@iabervon]

@@randomlife7935 If you've gotten to the point of being able to use the derivative of sin x in arbitrary problems, there's no reason you shouldn't be able to use it for the limit of (sin x)/x. Of course, if you're allowed to use the derivative of sin x in arbitrary problems, you should just be able to use the fact that the limit is 1, because you proved it in class along the way, but you're not necessarily going to memorize and reference every true statement you've established. L'Hopital's rule is the easiest way to reprove it from the table of derivatives, if you need it for some other limit, like in this case. bprp's point is that, if the question on the test is "prove the limit of (sin x)/x", you can't use l'Hopital's rule, because the implied context of that question is that we haven't yet proven anything that we used that limit to prove. But if the question on the test is something new, you can use everything we've seen in class, and you don't have to use the original derivations if you need values you've forgotten.

[@tesla5374]

Waaaw nice

[@yacinedjalil2355]

Intégrale indéfinie comme lim quand x tend vers infini

[@reogreggeen5738]

How do we properly justify the application of the DCT to an improper Riemann integral to begin with? Is this not an issue?

[@drpeyam]

Not really an issue, here we’re doing a Lebesgue integral actually, and the DCT applies to (improper) Lebesgue integrals

[@reogreggeen5738]

@@drpeyam Ah! Okay, thank you for the clarification on this - much appreciated. Have a good one!

[@kanewilliams1653]

wow I have never heard of the Dominated Convergence Theorem. Make a video on it!! :=)

[@drpeyam]

Already done ✅

[@kanewilliams1653]

@@drpeyam Oops.. should have looked at your comments.. will watch it now!!

[@zedisnotded8136]

I don't understand why you would put the limit inside the integral when you need to know the limit of the inside function before anyway.

[@ςγτε]

Like-dislike ratio is ∞ (infinite) now. Don't make it less, everyone !

[@pbj4184]

The domain of 1/x is R - {0}!! How dare you not respect that??? Division by 0 is meaningless!!!! Aaaaargh!!!!!!!!!!!!

[@ςγτε]

@@pbj4184 That's why be positive with your limit ! 😉

[@arvindsrinivasan424]

@@pbj4184 in certain contexts this may be true, however you can define 1/x on the Riemann sphere including infinites...

[@pkvlogs5078]

What if one adopts to clash this with fourier transformations or EULER integrals...???..☘🙂

[@Kdd160]

That looked super horrible in the beginning but the answer was awesome 🤩🤩 pure math is my fav 😍❤

[@bouch7643]

You could use L'Hopital theorem too because when n goes to infinity, lim of f (n) = Sin (x/n) is equal to 0 and lim of g (n)= x/n is equal to 0. Then lim of f (n)/g (n) is equal to lim f'(n)/g'(n). I did not do thd calculus but this should works. Thoughts?

[@drpeyam]

The main point is why you can put the limit inside the integral 🙃

[@bouch7643]

@@drpeyam it is obvious because integrals are additions technically.

[@bouch7643]

@@drpeyam sorry I just saw you mebtionned L'Hopital at 2:25. To burn steps in maths videos I use the youtube 10 fast forward feature a bit too much.

[@Grassmpl]

You completely ignored the fact that the integrand has a singularity at x=0.

[@drpeyam]

No it’s removable, no big deal

[@Grassmpl]

@@drpeyam yeah but you didn't justify that when finding the upper bounded "g" function.

[@yilmazkaraman256]

can i use dominated convergence theorem for changing differentiation and integration?

[@drpeyam]

Actually yes, you write the derivative as a difference quotient and use the DCT. Check out my video on the dominated convergence theorem

[@yilmazkaraman256]

@@drpeyam thanks. I will check your video

[@JoshuaHernandez8a]

Engineering be like: this is an internal product of a Dirac distribution at 0 and an arctangent derivative

[@FT029]

2:42, I'm wondering why you can't use L'hopital's rule to evaluate lim as y goes to 0 of sin y / y. great explanation of the dominated convergence theorem! reminds me a bit of another similar one, arzela's theorem

[@shivaudaiyar2556]

No we cannot use lopithals rule

[@FT029]

@@shivaudaiyar2556 Why?

[@shivaudaiyar2556]

@@FT029 we cannot use lopithals rule it's explanation is in one of the vedios of blackpenredpen watch it

[@shivaudaiyar2556]

@@FT029 kzbin.info/www/bejne/o4vMgZevfd6IrKc

[@shivaudaiyar2556]

@@FT029 watch this video

[@mohammadabdulla8601]

for some kind of ppl you put equal but not on this channel 😂😂😂😂😂😂😂😂

[@drpeyam]

?

[@mohammadabdulla8601]

@@drpeyam because some engineers use the approximation sinx = x for small x.I thought you are talking about this point.

[@enzy8022]

I can’t help it, but you look so much like Kyle from Nelk cool vids though

[@arnoldvillodas4997]

👍👍👍👍👍👍👍👍👍👍👍

[@davidepierrat9072]

gud stuf

[@aryadebchatterjee5028]

15th viewer 3rd like