Solving the Gaussian Integral the cool way

(for free) and start solving! My thanks to @maplesoft for sponsoring today's video.

In this video we're going to see a different trick to be able to solve the Gaussian Integral utilizing Feynman's trick of introducing a parameter.

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[bryangough]

Hello Dr. Trefor! I am an undergraduate student who loves your videos a lot. I have been struggling in my current major and really not felt confident in my studies. I think your videos have helped convince me that my passion lies in mathematics instead of the subjects I have pursued so far. Thank you so much for your content!

[caryfitz]

I really liked this. Feynman's technique has always been obscure to me. Your description was revelatory! Thank you very much!

[Valori_]

Hello! I'm currently 12 year's old studying in 6th grade but I'm a very curious person in learning mathematics and physics so I chose you to teach to solve this infamous Gaussian Integral Hope when I get older I'll find a new invention like any other scientist's ❤ Thanks to you for your video's You got a new sub❤

[ntruesdale]

Who in the world is brilliant enough to come up with this?! I feel really smart just being able to understand what’s happening! There’s clever, and then there’s CLEVER. Know your station!

[umairbutt]

This was my first proof of the gaussian that I came across and it blew my mind (I didn't study multi-variate calculus yet).

Does anyone know the original author of this proof?

[ajb]

I have an extreme love for math and I just discovered this channel. This is an absolute goldmine. Thank you Dr. Trefor

[monke]

Interestingly there is another way to compute this integral using single variable calculus, with strong geometric flavor.


Consider F(r)= int _{x^2+y^2<=r^2} e^(-(x^2+y^2))dxdy.

It is not hard to show (manually i.e. by definition of derivative) that

F'(r)=2pi*re^(-r^2).

Integrating from 0 to infinity gives F(infinity)=pi, from here the final result follows easily.

[MurshidIslam]

Shouldn't it be −arctan 1 + arctan 0 + C at 7:26?

[johnchessant]

Wow, up until today I only knew the polar coordinates way of solving the Gaussian integral!

[marylinebentzinger]

Can somebody explain why we introduced y and why when we introduced the new variable y, we set the top border to one?

[aweebthatlovesmath]

Nice video! it's always good to solve a problem in more then two ways. i mean all and all it's not the answers that make math interesting it's the way we do math!

[josafajunior]

7:25 That signal it's wrong? Don't changed because is zero, but should be +arctan(0)?

[cggdpwb]

Thanks, I've done multiple exercises where they give the developed derivative and you have to climb back to the Gaussian integral, and was always wondering how people found this kind of expressions in the first place.

[andrewharrison]

50 years ago I felt that integration was a black art. If anything videos like this confirm it.

Love the way that the annoying C that crops up in any integration turns out to be the part that carries the result from the doable integral to the unknown one.

I will probably watch this again to see exactly when you hid the rabbit in the top hat.

[farhansadik]

I really liked this considering I understoond nothing! I'll come back in 2 years 😪😭😭

[WielkiKaleson]

Cute! As far as I can see, all methods compute a square of Gaussian integral, so they are all related... in their hardness.

[tg]

I remember solving this using power series and differential equations

[airman]

Before the video started: “Feynman technique?”

Video starts: Feynman technique.

[samtux]

Neat approach. But I wouldn't dare to repeat it and would go for the classic switch to polar coordinates.
But, I am not in math (was in chemistry), so this isnot my field.

[foysalsahriar]

Hi, Really nice video! I was wondering if I can help you edit your videos and also make highly engaging shorts out of them.