/sci/ - Science & Math » Thread #11034238

89KiB, 1088x449, Gaussian Integral.jpg

View Same Google iqdb SauceNAO

Gaussian integral equal to sqrt of pi. Why?

Anonymous Sat 05 Oct 17:19:28 2019 No.11034238 View Reply Original Report

Quoted By: >>11034269

I didn't know about this till now and it looks pretty neat but why exactly pi of all numbers?

Does the double integral give you the volume or the area of the Gaussian bump?

Anonymous

Anonymous Sat 05 Oct 2019 17:35:05 No.11034269 Report

Quoted By:

>>11034238
Double integral is volume. A corollary of Fubini's theorem shows that the volume is also the square of the area. This is the most straightforward way to compute the integral and is the go-to method for vector calc students, but there are many other ways if you are having trouble justifying this one in particular.
I recommend Keith conrad's article. 11 different approaches, all yield sqrt pi.

kconrad.math.uconn.edu/blurbs/analysis/gaussianintegral.pdf

Anonymous

Anonymous Sat 05 Oct 2019 17:35:49 No.11034271 Report

Quoted By:

$\int_0^{2\pi}\int_0^\infty re^{-r^2}drd\theta = \int_0^{2\pi}\frac{1}{2}d\theta =\pi$
The double integral gives you the volume under the Gaussian bump, which by construction is the square of the one-dimensional integral.

Anonymous

Anonymous Sat 05 Oct 2019 21:00:38 No.11034750 Report

Quoted By: >>11034761

Thanks anons, I learned a thing today. I try to teach myself some calculus and linear algebra and occasionally I find something oddly interesting like this integral.

Anonymous

Anonymous Sat 05 Oct 2019 21:04:25 No.11034761 Report

Quoted By:

>>11034750
Look up 3blue1brown on youtube to see how pi is found in many series.

Capcode	All Only User Posts Only Moderator Posts Only Admin Posts Only Developer Posts
Show Posts	All Only With Images Only Without Images
Deleted Posts	All Only Deleted Posts Only Non-Deleted Posts
Ghost Posts	All Only Ghost Posts Only Non-Ghost Posts
Post Type	All Only Sticky Threads Only Opening Posts Only Reply Posts
Results	All Grouped By Threads
Order	Latest Posts First Oldest Posts First

Your latest searches