Why does pi show up here? | The Gaussian Integral, explained

The Gaussian Integral is a term that describes the area under a normal distribution of mean 1. This value is equal to the square root of pi. In this video, I go over the hidden circle behind this, using a bit of multivariable calculus. Hope you enjoy!

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#calculus #gaussian #integral

[vcubingx]

1:20 e^-(x²+y²) is a 1D function, not 3D. Thanks to u/efihiu for pointing this out!

YES, I know i promised to make a video on Stokes' Theorem and the Divergence Theorem. But, that video was taking a bit too long and I didn't want to leave you waiting for over a month. I had worked a decent amount on both of these videos, so I thought I might as well put this one out. The video on divergence theorem and stokes' theorem should be coming in the next few weeks, so look out for that!

[maxhaibara]

3B1B be like "Why pi is there, and why it's square rooted"

[badjumpcuts]

This was hella cute. Solving it in 3D like that is a beautiful idea, and makes it so much more obvious than staring at the original problem.

[teit-_piyushbhujbal]

Thanks a lot for clearing my doubts. Salute to all mathematicians, physicist and teacher.Really applied maths is beautiful.

[ptorafael]

I remember when I learned to calculate the value of this integral. In 2005, as I took single variable calculus, I learned that exp(x^2) and exp(-x^2) had no elementar primitive. In 2006, in multi variable calculus, I saw how to calculate for this particular case using this polar coordinates substitution, and I just fell in love with it. It is so elegant, and important to get that the integration of the normal distribution over all values is 1.

[uzairm]

Youre channel's finally going to blow up thanks to 3blue1brown's recognition

[weinboyz]

That’s just one of the beauties of multivarisble Calc. I haven’t taken it yet but that is just beautiful thank you for showing me this I’m enlightened!!!

[fahrenheit]

I have one nitpick: the animation that turns the cylinder into a cuboid is quite misleading. I know it's not intended to be the literal transformation, but it confused me for a sec. Though I understand that it might be difficult to accurately animate the unrolling, so I'll let it slide, especially since this video was so well explained. I didn't think I'd understand this integral for a while, since I didn't know how the shell method worked, but your explanation was perfect.

[pendronator]

The hidden circle has been bothering me for a few days now! This was beautifully explained. New subscriber! :)

[matanshtepel]

Beautiful. Thank you. I have seen this done before by extending to the complex plane, but extending another real dimension seems even more intuitive.

[DavidSmyth]

I really liked the animations and how you explained the 2D integration bit but I feel like part of the explanation is missing. You've presented basically the standard proof: start with the integrand f(x), take f(x)f(y) and go to polar coordinates r, theta to evaluate it. Then pi pops out because the limits of integration of theta contain a pi. But this doesn't explain what makes the Gaussian special. Why don't we see pi in almost every integral? After all we can do the trick of taking f(x)f(y) and going to polar coordinates for any function, not just exp(-x^2). The important observation is that for the Gaussian when we do this, the theta dependence of the integrand factors out completely: exp(-x^2-y^2)=exp(-r^2) doesn't have any theta in it! This is a very unusual property of the Gaussian. There aren't a lot of functions for which if you take f(x)f(y) and go to polar coordinates you get something that only depends on r. In general, you get some messy expression involving sin(theta) and cos(theta) which doesn't simplify to anything involving pi.

[MozartJunior]

Somehow I feel like this doesn't get to the core of why pi is there.. I mean, this is the explanation they teach in Calculus courses, but it's still just a trick... unfortunately some math problems are like that.

[diffusegd]

When there is pi, there is circle

Wow

[otesunki]

Got here from 3b1b, you now have a sub ❤

[zigbo]

I can't believe I had just found this today. It's so goddamn elegant it's disgustingly beautiful

[gpowerp]

Very simple and excellent explanation. Thanks for posting.

[alexatg]

Thanks so much it has confused me for like a year and now i finally know why π shows up (subscribed

[augurelite]

I atually remember being blown away learning this in university. thanks for the great video

[kiseryota]

I came from 3 blue 1 brown, i will stay :) Nice video!

[samirelzein]

I love your videos, i think they are underrated, you deserve more views, the ux/ui can be improved, less assumptions made, explaining bottom up, not calling many other theorems and videos etc