$5 Integral Vs. $500 Integral

[@rbdgr8370]

Tibee's integral: what they teach us in the class Yen's integral: what comes in the examination

[@memelord9965]

this felt too real

[@kintarooe7745]

Jens :)

[@lucile8591]

Sadly true :')

[@seljukturk8627]

Jens'

[@veridianr2490]

@@memelord9965 why you dont changing your dp? Is your movement still on?

[@kenmolinaro]

34 years ago my grade 10 math teacher pointed out an easy way to remember the value of e to many digits and I have never forgotten it: 2.7 (easy part) 1828 (this rhyming bit repeats) 1828 459045 (right triangle angles) 2.718281828459045

[@christianschultz9012]

Ken Molinaro SOH some old hippy CAH caught another hippy TOA trippin on acid

[@Luffy-yz9gj]

2+3=5

[@brunobaran9840]

3.141592653589

[@satacombre]

Christian Schultz it’s funny because I’m french and i learnt it with « Casse toi » ( CAH SOH TOA ) which means « go away » and also my theacher was named Schultz ! :)

[@brunobaran9840]

i remember only 2.71828182845904523536028747135266249775724709369995

[@shurjoaunibar]

A bit of Tibees' calm, soothing voice with Flammy's humour. A perfect mix, something we didn't ask for but we all needed.

[@blackpenredpen]

Wow, I just woke up and saw this! Is today Xmas???

[@SoapFX]

At least it feels like it is :D Now imagine a new video of Papa flammy and you :^)

[@_DD_15]

Ha ha ha isn't it? :) what's black pen red pen doing here tho?

[@tibees]

blackpenredpen we will have to include you in the next one ;)

[@CHStriker]

Integral of Oreo from 0 to infinity plx

[@dandoe2461]

yes

[@omarathon5922]

10x better than buzzfeed

[@dramawind]

Cancer is better than Buzzfeed.

[@omarathon5922]

no it's 16

[@walkastray007]

5x^2 better than BuzzFeed

[@Phenom98]

0x10^99 is still 0.

[@DrMrPersonGuy]

This video has nothing to do with buzzfeed, you only said that to feel better about yourself because you don't actually care about math. But it's funny that you mention buzzfeed, considering how clickbaity the title on this video is.

[@NerologicalAdventure]

Flammable : Hey, we should make a collab video together Tibees: Okay sounds good. What should the video be on? Flammable: I only make videos on integration

[@Zzznmop]

anstatic he has some awesome analytical and quantum mechanic videos, as well as theory/proof, complex analysis, the list goes on...

[@marcioamaral7511]

Flammable Maths More maths than physics but you're still right

[@axelcastillo7432]

Flammable Maths Hey! ^^ Do you know how to convert a complex number re^jø form to polar form?

[@daniarevalo9880]

:V I got tha reference

[@adarshtiwari8603]

@@axelcastillo7432 e^jΦ can be written as cosΦ+jsinΦ

[@XxDiego006xX]

16:42 He made a mistake, the 2nd term of the perfect square binomial has to be multiplied by 2

[@sodiumaluminatevelocity4614]

yeah bro... but later at 19:00 he just calls that integral as J and eventually it isnt required ...but kudos to you for finding the error :)

[@koenth2359]

4:53 'Plus our boi C' .... papa flammy is so contageous!

[@jeannelson8403]

Top 10 anime crossovers

[@phonkey]

I would never spend $500 on an integral, not even for a special occasion.

[@46pi26]

This is a collab that the universe should be proud to host.

[@samuelmcdonagh1590]

I understand the second one but the first one seemed non-intuitive, could someone explain?

[@dionsilverman4195]

It's a step-by-step derivation of the Taylor/Maclaurin series, without a priori knowledge. Clarification: We know eˣ is 1 at x = 0. Since eˣ is continuous, for deviations from x = 0, say x = 0 + ∆x, it must be e⁰ + f(∆x) for some function f, which she then elucidates incrementally, recursively.

[@2funky4u88]

@@dionsilverman4195 I think that was a joke, not sure though

[@epicmorphism2240]

It was a joke

[@smitagrawal8689]

You surely are expecting "LOL" in the replies

[@soutriksarangi5580]

@@dionsilverman4195 r/whoosh

[@JorgetePanete]

Lemma is for lemma explain later

[@amiramnoamdoron]

lol

[@umar11ism]

Lemma what?

[@NCarolina4455]

Lambda?

[@alephnull4044]

lol

[@1234Needforspeed]

@@NCarolina4455 Ligma

[@somadityasantra5572]

"e^x was created by mathematicians to make calculus easy". Madam you are truly insightful. Thanks a lot for sharing with us.

[@epicmorphism2240]

Somaditya Santra this isn‘t true. e was came first up in the double interest

[@epicmorphism2240]

@human guy Facepalm emoji really? Don’t be arrogant - the funny thing is that you are wrong. But why should I explain it to you? An intelligent person wouldn’t react that way even if my statement was wrong.

[@epicmorphism2240]

@human guy Actually the discussion is unnecessary. But I‘m pretty sure that mathematicians didn’t just think of the number e being 2.7... and then created the function e^x because it has nice properties in analysis. You see that doesn’t make sense. I’m sure that Bernoulli found the (let me call it idea) of the number e, then Euler came and found the series for e and could easily see that for the more general function e^x the derivative is the same. So it came naturally and the simple statement that the function e^x, which contains the number e has been created *because of* it‘s properties is wrong.

[@Misterlikeseverythin]

My head runs out of RAM when doing complex analysis.

[@destinyovbiebo8988]

You just gotta add more RAM.

[@killerKoala08]

Download more off the internet bruh

[@timothyaugustine7093]

Buy more RAM lol

[@younes7nv]

@@timothyaugustine7093 no you have to download ram man

[@Axel-hr5kk]

i need to upgrade my thinking rig my processor is too outdated and can only do high school algebra

[@SoapFX]

Papa and Toby

[@TSANOOvlogs]

That was sick. Haven't taken complex analysis yet so it's amazing seeing something like this worked out in front of me.

[@bumpty9830]

Complex analysis is surprisingly beautiful.

[@vru6431]

@@bumpty9830 I must undoubtedly agree with you my friend along with other Mathematicians.

[@Wasabiofip]

Ha! I thought it was funny how much more complicated than the engineer's way: factor the complex function, look up the Laplace transform in your table like a civilized person, then apply the integral, inverse Laplace back and you're on your way ;)

[@colinjava8447]

Those complex logs are a pain though, its something like ln z = Ln(|z|) + i argz from what I remember. And the domain of ln z is that infinite spiral thing, I'd have to learn it again to understand it properly.

[@smoothtriston6203]

How is it amazing? If you haven't even seen complex analysis yet, 90% of it probably sounds like complete gibberish.

[@davgregeric]

Cauchy's residue theorem was the neatest thing I ever studied in maths for physicists. But the $500 dollar integral was way beyond my budget! Great stuff, two subscribes.

[@cedric2803]

I like the 'C boi' that you have to add just in case :D

[@heatranlux]

I wish mainstream channels were all like yours : pure knowledge, relaxing voice and a bit of humor.

[@mateuszadamowicz770]

16:55 he lost 2 before ln(x)i*pi

[@pabloespanagutierrez7034]

Glad to see you noticed too ^^

[@pabloespanagutierrez7034]

But it doesn't change the final answer as it is the imaginary part, so it's okay ;)

[@pabloespanagutierrez7034]

@@darthcrossfader aha you were faster than me x)

[@ZweiZombies]

It's difficult to keep your coefficients together when solving such a beautiful integral, who wouldn't have made this mistake? :')

[@marvinlin6063]

I scrolled down to the comment section to see if someone noticed the error and I saw your comment haha^^

[@aaron-kirk]

"So, where are our poles? They are part of Western Europe" LOL IM DEAD

[@user-nr3qb1mc7x]

The second one made "our mind explode to infinity"

[@cavver3523]

limit of mind going to infinity

[@laxminarayanbhandari855]

The complex analysis method is harder. It can be done much easily using the derivatives of beta function.

[@drfpslegend4149]

Yes, the theorems you learn in Complex Analysis are very powerful when it comes to solving real valued integrals.

[@ukaszduleba1668]

I've found your channel by an accident about two months ago. And here's the thing - I don't really enjoy science of any kind. I'm kind of a poet, a daydreamer, romantical soul. I'm absolutely terrible at maths, physics, science etc. (not becouse they're hard for me, I've just never wanted to learn them, as I didn't enjoy them). But I watch many of your videos since I've found you, and every one of them is so damn fascinating! Even if I don't understand something becouse of my poor knowledge at certain topic, it's still so great to hear how you talk about this. That's becouse you enjoy what you do, and you do it with passion. So, I just wanted to say, that you do a great job with this channel and it's amazing how you make people interested in what you love. Keep it up!

[@justfunandgames3984]

She herself couldn’t keep a straight face 😂😂😂😂😂😂😂😂

[@igorkhavkine]

Pro-tip: for (2*pi*i)*f(z) = (1/3) (log(-z)^3+pi^2*log(-z))/(z^2+1), the jump (bottom to top) of f(z) across a branch cut along the positive real axis is exactly log(x)^2/(x^2+1). Hence, the second integral is I = int_C f(z) dz, where the contour C goes counter-clockwise around the positive real axis. Close the contour with a large circle and sum over both residues to get the answer. No other transformations needed.

[@andrewhone3346]

Yes, he didn't mention that a branch cut is needed for the complex logarithm. The keyhole contour is definitely much neater.

[@PaulGuevara]

No Cs left behind; we never leave behind our hang-about boi C.

[@upandatom]

You're so cool

[@hamiltonianpathondodecahed5236]

how come no one is recognising you

[@NKomarov]

I cannot refuse clicking a video with such a pretty face

[@irfanjames]

You too.

[@yashkrishnatery9082]

@@hamiltonianpathondodecahed5236 because she does not have that verified tick mark. ( Which KZbin stole from Twitter)

[@asificare7985]

@@NKomarov ig but her voice is very soothing and almost has a healing effect

[@anapaulinacabrera]

I just love how he gets that exited when he is doing math, TOTAL RESPECT

[@smallswole]

Hey guys! Sweet video! So I'm going to be super nit picky and point out a small mistake in the calculation for the contour integral. When doing the binomial expansion at @16:45, the middle term should be 2 ln(x) * i * pi. Fortunately, taking the real part at the end of this problem swept this mistake under the rug! Keep these videos coming. I very much so enjoyed having the residue theorem dusted off from the back of my brain. Cheers and happy mathing! :D

[@vVToxicTearsVv]

I was going to point that out. Fortunately I scrolled down to see if anyone else already had. Good catch.

[@ricardofranco7419]

Both nice. I remember taking complex analysis and just really enjoying it. I don’t think I ever saw the derivation of e nor did I learn about the history so that was really cool (particularly how this all directly relates to ODEs, PDEs, and other math fields). Awesome work!

[@TheMultipower47]

This title really put a smile on my face

[@joelbiffin]

What an amazing walkthrough of contour integration! Not surte my university lecturer managed to be so articulate over an entire lecture series!

[@masteryoda1748]

6:02 Beethoven 2018

[@houssamassila6274]

Directly subscribed to Flammable Maths. I've rarely seen as satisfying maths video on youtube as this one. the flow is like poetry! reminds me of the good old days....

[@grandmastergyorogyoro532]

Woow.... Do consider making a basic course on differential/Integral Calculus. It was good to see

[@sohail9157]

Int(f(z))dz=int(closed region)f(z)dz+int(limits -RtoRf(z)dz=2(pi)i*(sum of residues at interior poles) We try to show that intergral closed region vanishes as z~infinity hence no need to find all other three intergrals....

[@velfyz]

I have never expected this... This is just great

[@michaelsommers2356]

Nobody expects the Spanish Integration. Our chief weapon is the residue theorem, and Jordan's lemma. Our TWO chief weapons are ...

[@Scy]

"I'm here to solve intregrals and drink milk. I've just finished my milk."

[@Melomathics]

By the way, Tibees, you should definitely collab with Looking Glass Universe.

[@diegotejada55]

This is my new favorite thing. Please do $1 ODE vs $1000 ODE Ty and papa bless

[@howardlam6181]

$800000 FPDE

[@Gbzildo]

PLEASE

[@m_riatik]

oh my god, crossover of the century

[@ThePrimevalVoid]

"He's our friend that we just have to remember not to leave behind." I feel attacked.

[@damianlin4052]

7:10 It's pronounced "elegant" as in "gif". ;)

[@TheTdw2000]

You pronounce it elejent?

[@lynettemojica6503]

I thought gif was pronounced "gif" like gift but without the T.

[@franciscofragoeiro5229]

Do you mean elejant? Learn to write

[@Dekross]

@@lynettemojica6503 like jift? I'm joking but you get the point someone could take this to infinity.

[@devabask3348]

@@lynettemojica6503 hiii

[@sagarmajumder7806]

Your explanation is better than your solution ❤️🙏🙏🙏

[@chorthithian]

I'm no mathematician so a few parts of the second integral went over my head, but I would definitely love to watch a slightly longer video that goes a little deeper into the guts. Its gorgeous and is indeed pi3/8.

[@xsquaredthemusician]

this is the math content i have been craving

[@Jaokku]

WHY WAS THIS SUGGESTED TO ME. WHY IS THIS A THING. WHY IS THIS SO GOOD. WHY DO I LOVE THIS.

[@charlesmoncul4724]

16:54 you missed the 2 in front of the double product of ln and i pi a^2 + 2ab + b^2 = (a+b)^2

[@tonk6812]

charles moncul yes...😁😁..simply mistk..

[@charlesmoncul4724]

Ara ?

[@Prophetic_Piano]

I'm glad I wasn't the only one who saw it! Thankfully it wasn't needed for the real part of the solution

[@matthewjeffery3686]

Amazing video that shows both the beauty and simplicity as well as the complexity and brilliance that can come from the calculus!

[@TheZabbiemaster]

This is amazing, you'll always be my favourite physics channel!

[@SylComplexDimensional]

Keep doing these analysis videos, Tibees!! Orthogonality, dot product and things! ♥️🐻✨

[@JupiterJones1996]

A SEASONED INTEGRAL TASTER. oh my goodness i love you. hahahaha

[@stevethecatcouch6532]

17:20 Here's a usage note for non-native speakers of English who might be reading this. In the American dialect of English, the word "guys" has, since at least the '70s, been non-gender specific when so indicated by context.

[@morganmitchell4017]

Also, in British English, "guys" is almost never gender specific.

[@enriqueortiz7942]

This really happened. OMG.

[@yummyzerg]

It was a little crazy but Yen is actually really good at explaining that. I just finished the calculus sequence for my engineering degree and I got most of it - of course, I didn't get to Residue and all that jazz, as my last calc course finished with stokes and divergence theorems, and I couldn't ever do this myself, but I could follow without my head spinning. +1.

[@abrarshaikh2254]

This integral is also equal to 4 time Dirichlet beta series(3)....

[@idahogie]

Who knew the opposite of ASMR would be equally mesmerizing.

[@46pi26]

"Jordan's Lemma" is probably the most ambitious DLC of 2018

[@GainsGoblin]

"my manager will personally fire me" lmao

[@matron9936]

The first integral is like: „assuming the necessary assumption“

[@ggarcia3237]

No one can be this beautiful and smart

[@spaghetti1383]

Flammy forgot a 2 when he expanded (ln(x) + i pi)^2. Luckily, it didn't affect the final result because J was zero so 2J was still zero.

[@kaitlinarnold5105]

Someone else caught it! I was looking for this in the comments

[@AntonioKowatsch]

I like this format. You should do more videos like this.

[@Melomathics]

I'm starting Complex analysis in 3 weeks!

[@librecolombe7376]

Good luck I advise you to revise - trigonometric laws and properties

[@HansensUniverseT-A]

R.I.P

[@howardlam6181]

How did it go?

[@Reclamalo]

@@howardlam6181 RIP.

[@matthewstevens340]

Oh. My. Word. This is beautiful

[@crazyharajuku1]

That moment when you ask yourself is physics really for me

[@jacobharris5894]

MrGoodkat I’m torn between both. Math used to just be a means to an end for me. Now I like it almost as the physics itself. Don’t get me wrong math is hard to understand but there is few things more satisfying for me when I can get a difficult math problem done and it actually makes sense to you. The beautiful thing about it I think is with enough time and brainpower or computation power on your hands you can use it to model or prove any problem whether or not it is even representative of what you perceive as reality. The more you learn you about what math can do and the more you get familiar with it the more powerful it feels. This last summer break I used the disk method for solids of revolution to find the volume of a vase in my house. The longest and most tedious part was finding a good equation to model it which I did from taking a picture of the vase with a ruler and pixel scaling it. I used some java program I had never really used before and used it to plot points for a polynomial and thankfully it gave me the x and y coordinates. I then had to put all of these data points in a good graphing program (the only one I was really familiar with was logger pro so I used that). There were some complications with the thickness of the vase and were I wanted the origin to be for the graph (for those that don’t know many programs have the origin be in the top left corner for some reason). Once all of the points were in exactly the for my two curves (one for the inside and one for the outside) all I had to do was use logger pro to find the curve of best fit, I found that a quintic function had the best correlation (about 99.5%). Then finally I was able to set up the integral and use my ti-84 calculator to solve it (I think it approximates it numerically by Riemann sums or trapezoidal rule or something like that). Then the last thing was converting the number from pixels to inches to cm cubed (which is the same thing as a milliliter) Then to test to see how accurate I was I filled the vase with water and poured it into the biggest measuring cup I had (multiple times). I added all of the measurements together to get my total measured volume. Then I calculated the percent error between my measured volume and calculated volume. This wasn’t 100% representative of the actual % error because there was some spillage (the vase was very heavy). However there wasn’t that much compared to the amount of water in the vase so I believe it is still pretty accurate. I found the % error to be a little more than 1% (I was going for 5% or lower). I was overjoyed. I’m not saying this story to brag (you have no way of knowing if I’m actually telling the truth or not).but to make the point that math is powerful. Was what I did very practical or even the best way of solving the problem? Probably not but it was a good experiment that taught you how to deal with the cards you are dealt and what you know. It was a pretty humbling experience because I used the equations, methods and technology of countless hardworking programmers, scientists and mathematicians. Kepler and newton would have killed to have these resources when they were trying to calculate the volumes of wine bottles. Sometimes you might feel like you simply don’t know enough to do a project you want to do but you can always try to simplify it and make assumptions to use what you know. Then when you learn more you can come back and do something with it later. You may move on to different projects and other areas of your life You may never do it but it feels refreshing to have that possibility in your life.

[@juanvolvoturbo]

You won't be re-inventing the wheel.

[@Filmbert]

I dont understand the most of that because as a german i am not good at "mathematical english" You're a lovely lady, i love watching your videos and listening to the things i understand :-)

[@atrimandal4324]

Papa Flammy ❤️ Oh boi, this is a crossover for the ages

[@jonathanvictorpajaresmoran3126]

the perfect combination , smart and beatiful !!!

[@NikitaKarciauskas]

The video just showed me how to rank up from silver to global elite but in real life.

[@arkadiptadas4148]

The way you explain your videos just puts a big smile on my face :)

[@gamingmaster4422]

Complex integrations are somewhat easy than that of normal integration

[@Photo650D]

That were some spicy integrals! And beautiful braid you have there :)

[@farissaadat4437]

I did not expect that title in my sub feed.

[@kyletrom2622]

Flammable Maths fan checking in. Happy to see the collab!

[@GrandMoffTarkinsTeaDispenser]

I don't even watch anime but... *Top 10 anime crossovers*

[@閲覧アトム]

I have never watched this kind of videos. This video is so cool and useful.

[@orikane1]

in 17:00, I think the numerator is lacking a multiple of 2 for the ln(x)*I*PI expression. Shouldn't it be 2*ln(x)*I*PI? Great video btw, learned a lot! Thanks!

[@TheGreatJRB]

I saw that too. Maybe just put the 2 inside, i(pi) ln (x^2)

[@sodiumaluminatevelocity4614]

yeah bro... but later at 19:00 he just calls that integral as J and eventually it isnt required ...but kudos to you for finding the error :)

[@Impedancenetwork]

I took complex analysis and he is right. It is awesome.

[@LynnXternal]

Omg two of my favorite math KZbinrs! _Squee!_

[@cremefrais]

i am grateful for this content .

[@sophiarubens54]

I've taken AP Calculus and two terms of multivariable calculus at Dartmouth college and no one ever explained e^x in a way that made that much sense-I was always told that e^x having itself as a derivative was a wacky coincidence instead of a design choice. I also learned that the Maclaurin Series for e^x is Σ from n=1 to infinity of (x^n)/n! but this was simply stated as a fact to memorize instead of a requirement in order for the infinite limit of the derivative to equal its counterpart for the function. I have to say I smirked a bit when e^x was presented as the $5 integral, but consider my mind blown!

[@rhodexlx1659]

So glad I went through complex analysis to understand the full joke of the second integral

[@91722854]

Sounds like you had a pretty special and intimate relationship with the boy C and that writing it out after integration was almost comparable to completing a masterpiece of artwork.

[@paulallen579]

This video gives me chills, I like it so much. 🙂

[@rekiepimie]

I’m currently studying a master in history and I have no clue whats going on, but the tapping on the bord and their voices are really calming and nice:)

[@abdulrahmann.9024]

There you have it, calculus, basically exploites equations (our ability to plug in values) and the idea of limits and infinitesimals to come up with logical models of absolute intricacy describing relative rates change. It all obeys logic. Through the centuries people have been able to progressively design creative methods to exploit every bit of quantitative logic. Calculus is just a result of very in depth record and progress.

[@lindaaa3299]

Stuff I watch during procrastination :)

[@isaac.zeitgeist]

21:18 "We're done ma boys and girls and everything else in between" LMAO

[@dannyboy900102]

Well at least he acknowledged enbys 🤣🤣

[@garitube09]

He is like one great musician playing his masterpiece! Wow!!!

[@Zzznmop]

Fappabl... I mean flammable maths is bae, he lifts infinity bois then integrates things.

[@umutcanakdag2624]

This is by far the funniest video topic i have ever seen in entire KZbin history. I'm just not gonna watch the video in case of it fails to satisfy its name. This is the risk that i cannot take.

[@dhivyashivaniraviselvan1666]

THOSE 3 SECONDS OF LIEBESTRAUM AT 21:08

[@MrAssassins117]

Both integrals are beauty for itselfs

[@thephysicistcuber175]

Did you also just prove that the integral over positive reals of ln(x)/(1+x^2) is 0?

[@JLConawayII]

I definitely need a refresher on complex analysis.

[@hydropage2855]

He made a mistake near the end where he didn’t multiply the middle term by 2 when he expanded the squared binomial. But he got lucky because the imaginary part didn’t matter in the end

[@m35926]

Don't watch this when you first wake up in the morning

[@mariomuysensual]

Why

[@svader9680]

Too late

[@stewartzayat7526]

@@mariomuysensual just don't... You will regret it if you do.

[@pvloff-l9n]

It's not that complex. Wait....

[@michaelkyzar8215]

>woke up >checked KZbin >Saw this vid >saw this comment >I have no regrets

[@YounesLayachi]

Holy INTEGRAL I can die in peace now I can't believe this happened :D I like both of you's channels and content