The Gaussian Integral

In this video, we try to evaluate the Gaussian integral. Featuring some multi-variable calculus, some graphs and my Paint illustrations.

Little background:
I first came across this integral in my quantum mechanics class a couple of months ago (see D. Griffiths' Introduction to Quantum Mechanics, 2nd Ed., Problem 1.3). I thought the way to solve this integral was rather cool, and had it in my list of to-make video for a while now. And here it is.

[GogiRegion]

I love how the video makes you feel smart by hinting at where it’s going so that you figure a lot of it out on your own.

[abhishekbhatia]

Dude integrated so hard that his voice changed at 9:35

[hashimabbas]

No word for this great video.
Spectacular.

[JPiano]

"Might ring a bell"... I see what you did there

[saurabhshukla]

You have transformed the integral into polar coordinates. This is a classic example of how sometimes complex cartesian coordinate integrals can be simplified in polar coordinates.

[gamechep]

I am blown away by the way you simplified it. Pure awesomeness.

[logasimpso]

That is stunningly beautiful. One of the best mathematical explanations I’ve ever seen. Well done, sir

[markolazarevic]

thx a lot man. finally a good explanation

[MichaelMiller-rgor]

This is easily the best video explaining the Gaussian Integral I have ever seen!

[rajendramisir]

I saw Professor Christine Briner from MIT used polar coordinates, double integrals and change of variables to evaluate the Gaussian integral. Now, I see you find the volume under the bell curve by summing tubes(hollow cylinders) whose radii varies from 0 to infinity. I like the geometrical approach to finding the volume of each tube. I find the visual aids intuitive. I think this is a fresh and intuitive way of evaluating the Gaussian integral. Thanks.

[ananthakrishnank]

This is no random math as your channel name suggests. The geometrical proof is always the most elegant way. I can't skip ads for you. Great content!

[nguyenvietdung]

Who else came here after blackpenredpen 's video

[kptib]

my jaw dropped when it came out to be square root of pi!

[bonbonpony]

So in the integral of `e^(-x²)` the simple lack of `·x` is what makes it (almost) impossible to solve?
And the whole idea of translating the problem to polar coordinates it what helps to bring that `·x` (or `·r` in this case) back?

[Bearman]

1:25 "might ring a bell" nice pun haha

[ЧингизНабиев-эг]

OMFG, I hope i could get that video when I started to learn Calculus ... Special Functions(Non-elemental) were such a pain for and after all I just used abstract rules to get them, but omg this interpretation of Gaussian Integral is awesome...

[leonardobarrera]

Fantastic
It is like... you have to use properties of complex numbers to end out with the error funcion
That was very funny to think!!!
Thanks a lot for the video!!!

[Jason-mrtp]

Really great video! Thank you so much for this, I had a really hard time understanding how to integrate wave functions that included this exact integral. Thank you so much!!

[The_Aleph_Null]

Isn't this just using cilindrical coordinates directly into the double integral to solve it and that's about it?

[laman]

I have watched a number of clips on the Gaussian Integral, but I like the practical way this has been explained.