Morphocular | How to do two (or more) integrals with just one @morphocular | Uploaded 2 years ago | Updated 1 hour ago
Is there a way to turn multiple, repeated integrals into just a single integral? Meaning, if you, say, wanted to find the second antiderivative of 6x, is there a way to compute it all in one step just using a single integral? Turns out there is! In fact, any number of repeated antiderivatives can be compressed into just a single integral expression. How is that possible? And what does that single integral expression look like?
My old video about Double Integrals: youtu.be/gifQWtTWqEY
The sequel to this video about fractional calculus: youtu.be/2dwQUUDt5Is
A really nice video that derives the gamma function from scratch:
youtu.be/v_HeaeUUOnc
=Chapters=
0:00 - Intro
0:51 - Why Compress Integrals?
2:29 - Analyzing the Problem
3:46 - Visualizing a 2-Fold Integral
5:25 - Deriving the Formula
10:56 - Testing the Formula
12:14 - How Is This Not Impossible?
13:49 - Higher-Order Integrals
15:22 - Application to Numerical Integrals
16:25 - The Gamma Function
===============================
For more on applying Cauchy's Formula to numerical integration, see this paper:
Tvrdá, Katarína & Minárová, Mária. (2018). "Computation of Definite Integral Over Repeated Integral." Tatra Mountains Mathematical Publications. 72. 141-154. 10.2478/tmmp-2018-0026.
researchgate.net/publication/331705202_Computation_of_Definite_Integral_Over_Repeated_Integral
===============================
Want to support future videos? Become a patron at patreon.com/morphocular
Thank you for your support!
===============================
The animations in this video were mostly made with a homemade Python library called "Morpho".
I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:
github.com/morpho-matters/morpholib
Is there a way to turn multiple, repeated integrals into just a single integral? Meaning, if you, say, wanted to find the second antiderivative of 6x, is there a way to compute it all in one step just using a single integral? Turns out there is! In fact, any number of repeated antiderivatives can be compressed into just a single integral expression. How is that possible? And what does that single integral expression look like?
My old video about Double Integrals: youtu.be/gifQWtTWqEY
The sequel to this video about fractional calculus: youtu.be/2dwQUUDt5Is
A really nice video that derives the gamma function from scratch:
youtu.be/v_HeaeUUOnc
=Chapters=
0:00 - Intro
0:51 - Why Compress Integrals?
2:29 - Analyzing the Problem
3:46 - Visualizing a 2-Fold Integral
5:25 - Deriving the Formula
10:56 - Testing the Formula
12:14 - How Is This Not Impossible?
13:49 - Higher-Order Integrals
15:22 - Application to Numerical Integrals
16:25 - The Gamma Function
===============================
For more on applying Cauchy's Formula to numerical integration, see this paper:
Tvrdá, Katarína & Minárová, Mária. (2018). "Computation of Definite Integral Over Repeated Integral." Tatra Mountains Mathematical Publications. 72. 141-154. 10.2478/tmmp-2018-0026.
researchgate.net/publication/331705202_Computation_of_Definite_Integral_Over_Repeated_Integral
===============================
Want to support future videos? Become a patron at patreon.com/morphocular
Thank you for your support!
===============================
The animations in this video were mostly made with a homemade Python library called "Morpho".
I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:
github.com/morpho-matters/morpholib
