How to do two (or more) integrals with just one

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?

A really nice video that derives the gamma function from scratch:

=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

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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.

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Thank you for your support!

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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:

[jakobr_]

Incredible! Calculus classes often teach a limited view of integrals, only thinking of them as antiderivatives. But now I realize the implications of just how specific the fundamental theorem of calculus is written: Only integrals that fit that pattern are antiderivatives, and there are way more than just those integrals out there!

[classicalmusic]

The animation quality is incredible. For example the transition from a 2D to 3D view at 6:34. So smooth!

[AWellRestedDog]

This was absolutely incredible. The intriguing and seemingly nonsensical question at the beginning (especially from the perspective of someone who only knows basic calculus), the pacing and 3d animation to visualize the intuition behind taking a double integral, and the teaser for the idea of in-between integrals/derivatives. This was mind-blowing to watch and appreciate, both from the perspective of a learner and an aspiring teacher.

[drewduncan]

1:40 Numerically, derivatives are notoriously difficult because computing them involves subtractive cancellation. Integration is much more well-behaved from a numerical standpoint precisely because it only requires summation.

[LinesThatConnect]

Awesome!
This formula felt like black magic when I first saw it (and it still does), but it feels a lot less mysterious now that I can see such a straightforward derivation of the n=2 case.

[alecboyd]

Derivative Compression, the "n"th derivative of a function, can be expressed for some nice functions as an extension of cauchy's integral formula in complex analysis.

this result is highly related to the residue theorem and, consequentially, this yields no (simple) results in the field of fractional calculus, as inputting a fractional n changes the pole in the denominator of the integral to a branch cut, which is not easy (or often even possible) to evaluate. this function also only returns the value at a point, not a function over the entire complex plane.

[madmorto]

This video is amazingly painful.
As a high school student bored with the basic calculus I had been doing in class (volumes of revolution and arc length etc.) I began to play around with trying integrate the volumes of other solids.

I eventually figured out how to tie two integrals together and find the volumes of lots more shapes - like ellipsoids.

It’s really painful to see just how close I was to this amazing formula! I wish I had obtained it myself…

Great video and great animations. Well done!

[qoyote]

you could do five or six integrals, or... just one.

[newgreen]

I found it interesting, that the resulting formula is similar to convolution of x^n with f(x). After thinking about it it makes sense though and gives another way of deriving the formula:

Using Laplace transforms we can transform a function from the time domain into the frequency domain, using s as the new variable. An integration in the time domain shows up as a multiplication with 1/s in the frequency domain, so double integration becomes 1/s^2 etc.

Transforming it back we can however use the fact, that multiplication in the frequency domain becomes a convolution in the time domain.
And what does 1/s^2 correspond to? It corresponds to x. 1/s^n corresponds to 1/(n-1)! x^(n-1).

Plugging this into the formula gives the above result. It's quite nice how in mathematics all roads lead to the same spot.

[cparks]

In practice, we use this technique in reverse to replace an operator norm with a double integral. Also, (16:19) if you're content with working with complex analytic functions, the Bergman kernel allows you to write a derivative as an integral. You can write any power of a derivative as an integral also. I agree though that integrals are very powerful.

[maysaraaljumaily]

Well done! The animation at 6:30 that change the 2D representation to 3D is very smooth!

[chanceroberson]

Just took calculus 3 last semester, this blew my mind. Well done, perfect video

[ralllao]

Wow! Just great educational content! Keep making math learning more interesting & engaging!

[AgentM]

Omg, half integrals? Fractional integrals? Can't wait to get the π's ∫ of a function

[DokterKaj]

So underrated! You explain things so clearly and present a very intuitive visual interpretation. You deserve so many more subs.

[kobemop]

A lot of this stuff appears in physics, especially when we're dealing with phase spaces where you have to expand out momentums and positions (could be in terms of hyperspheres, hypercubes, ..., etc). They also appear when you do higher dimensional Fourier transforms, can occur in special relativity too if your taking account of the fourth dimension (which is usually time), and as well as dealing with characteristic functions. However, I have to say that in physics we write our integrals as ∫d⁴x *which is a neat way of condensing it), and then it comes out as ∫∫∫∫dxdx'dx''dx'''. In terms of phase space where p is the momentum and r is the position, we have ∫d³p∫d³r = ∫∫∫dpdp'dp''∫∫∫drdr'dr'' = ∫∫∫∫∫∫dpdp'dp''drdr'dr'' (which is a six-fold integral). To take it further you can also write ∫∫∫∫∫∫dpdp'dp''drdr'dr'' as one integral ∫dpdp'dp''drdr'dr'', but have many integrals appear out looks nice and aesthetic. Sometimes the integrals derivatives are written with subscript x_1 and so on. These types of integrals appear in quantum optics (where you deal with a lot of things in phase space), quantum field theory, quantum mechanics, and statistical mechanics.

[MrShiggitty]

That was really cool, it was like seeing a u-substitution but in a perfect 3D form. Or like he turned the u-sub into an operator function, again, crazy cool.

It's like this guy understands integration at a level higher than Riemann or Leibniz, well played. Like a version of super calculus?

[robwarriors]

Outstanding video. Really opens up creative thought at a high-school calculus level. Wish I had this sort of direction when I was learning. Instant subscription and can't wait to see what's next.

[prdoyle]

Wow, the visuals are amazing. I literally said "slice it the other way" out loud at 6:55, just as you intended!

[neelanshguptaa]

I just cannot put in words much this helped me understand why limit points for double integrals are the way they are. Just brilliant stuff.