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

The sequel is here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-2dwQUUDt5Is.html

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!

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

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.

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.

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.

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.

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

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

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.

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.

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.

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

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!

you forgot the constant c

Most excellent! I've wanted to see this for so long and was frustrated about it, ever since seeing the formula in the first pages of a text on integral equations. Thank You so much!

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

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

👏🏼👏🏼👏🏼 Very well done. You've made a pretty abstract concept very accessible and easy to understand.

Wooow!! This channel is so underrated. I really enjoy your content, Thanks a lot.

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.

Beautifully explained! Thank you!

This was amazing. It opened up a new horizon for me. Thank You.

Legendary explanation I have waited for to explain in visual terms double integrals. New insights that bring nostalgia of first learning integral calculus and the FTC itself. I am giving a Subscribe and thank you.

Yay a new video from this math youtuber! Keep up the good work

How do you not have 100 times as many subscribers? Your content is of the best math on RU-vid!

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

I’m curious, can you prove Cauchy’s formula for repeated integration by using a 2-to-1 integral formula repeatedly? So, something like proof by induction. Or does it give a different uglier result?

Really outstanding video, every question I had as I was watching you answered two seconds later

Thank you for making calculus seems so easy and appealing

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.

This was beautiful stuff - blew me away on a Saturday night!

This was fantastic! Keep it up!

I was going to ask about that last part just before it apeared, great introduction to fractional derivatives and integrals

ive never seen a person do a math cliffhanger lol

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?

Nice cliffhanger at the end ;) waiting for a sequel!

Awesome video! Thank you!

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

oooh another really fun derivation from this is the subfactorial complex extension, making use of the subfactorial property that is most similar to normal factorials (!n=n*!(n-1)+(-1)^n) you just have to modify the formula slightly to change the nonderivative term of the partial integration to match this property instead of the factorial "n!=n*(n-1)!" and then it turns out you get complex numbers in return (makes sense since the formula contains -1^(z))

Thankyou! I learned a lot! Very good!

Amazing video and visualizations

Just beautiful and poetic

Very interesting! I like it how you used 2*Pi, Pi, sqrt(2), Pi/2 and e and so one at 13:11.

The music and visuals are giving me nostalgia

Fun fact: to integrate (t-x)(f(x)), unless f(x) is a polynomial or some easy function, we would need to integrate x*f(x) using integration by parts, which requires us to find the double integral of f(x)

Nice video, thank you !

Very fascinating!

Great video, keep going!

3:40 Well, I suppose using the laplace transform would be a way of integrating without integrating: ℒ{ ∫∫ 6x dx dx } = 6 ℒ{x} / s² = 3! / s⁴ = ℒ{ x³ }

Super explanation 👌

Magnificent Meticulously Innovative Method ascertain the truly ideal Teacher

Also, another interesting thing is contour integrals. A contour integral is an integral that integrates over a boundary in any dimensions, unlike normal integrals, which only integrate in their respective dimensions. To differentiate the differentials of contour and normal integrals, we use s's rather than conventional letters, so dx -> ds, dA/dxdy/rdrdθ -> dS, dV/dxdydz/rdrdθdz/ρ^2*sin(φ)dρdφdθ -> dC (for cellular volume). Also, ds, dS, and dC are magnitudes of the dr, dS, and the dC vectors: dr = , dS = , dC = . These differentials are crucial in finding arclength, surface area, and cellular volume. For example, the arclength of a given function in 2-space is given by ds, which is √(1+y'^2)dx, √(x'^2+y'^2)dt, and √(r^2+r'^2)dθ. Surface area follows a similar pattern. The most common formula for surface areas are dS = sqrt(1+∂x+∂y)dA = |∂s x ∂t|dsdt, and 2πrds, from solids of revolution. Because of that, we can derive the formulas for cellular volume being sqrt(1+∂x+∂y+∂z)dV = |∂r x ∂s x ∂t|drdsdt, and 2πrdS. To evaluate contour integrals, you need to make sure your differentials are in magnitude form. Sometimes, they would already be in such form, like when you are trying to find the mass of a spring or a solid lamina, but most of the time, you are given a vector field and are trying to compute the circulation or the flux across a given path or surface, in this case, you must dot the vector field with the normal to turn it into a magnitude. You also need to parametrize the surfaces as well. However, there are some useful formulas: Fundamental Theorem: If a vector field is conservative, meaning it is the gradient of a function, a line integral is just taken as an ordinary antiderivative, as the differential of a multivariable function is just the gradient dotted with the dr vector. Divergence Theorem: The flux of a given vector field is proportional to its divergence and its interior content: ∯ F ∙ dS = ∭div F * dV, ∰ F ∙ dC = ⨌div F * dH Stokes' Theorem: The surface integral of a given vector field is proportional to its curl and its circulation: ∮ F ∙ dr = ∯ curl F ∙ dS

Looks like this is really just replacing the "nested" integrals with other integrals that just happen to have a well known form (areas of rectangles, rectangular prisms, etc.). Reminds me of a video about the circumference of an ellipse having no closed-form constant w.r.t. axis, but a circle does - pi, but pi is just a special constant we've come to recognize and could easily define another constant for other ellipses

Holy.. Your's math is super fun!!!!

Thanks for this knowledge :> Hopefully I can use it when I encounter multiple integrals in my self study journey

This formula for repeated integration also appears when expressing the remainder of a Taylor polynomial approximation in the "integral form" - because, unlike the Peano, Lagrange, or Cauchy forms, it uses an integral

beautiful video

Amazing quality content

You made my day

There is a way to compute iterated derivatives using matrix diagonalization. Since derivatives are regular invertible linear operators, they can be encoded as a diagonalizable matrix. Using eigenvalue decomposition to compute powers of this matrix and representing functions as vectors with real coefficients in a function space is equivalent to computing iterated derivatives.

16:20 you may want to have a look into Leibnitz's theorem. It's a method for taking the nth derivative of a function. It splits the function into parts and takes the derivative of both parts n times then combines them together.

Brilliant!

Clicked for the impossible task. Stayed for the amazing explanation

Oh boy! Fractional calculus here we come!

Such a fantastic way to talk about calculus! Am I right that this principle applies only to functions dependent on a single variable e.g. f(x) but it would be different for multiple variable function e.g. f(x,y)?

Actually theres formulas for n-th derivatives for p much every function including chains, it's just usually a sum or product up to n, which isn't really expandable for non integer n, however they are p beautiful if you asked me!

At 12:44 you mentioned that the integrand is not the inner integral. This is because we are switching the limits of the integration, ie the inner integral goes out and the outer one comes inside. This is also what you did geometrically by slicing the integral a different way.

👍👍I think i remember learning this formula in 'calc 3', multivariable calc

It can also be thought as taking the integral with the greens function for (d/dx)^n operator

Noice broooo keep making videos with moree quality over quantity

Mark my words it's only a matter of time before this channel blows up

Awesome!

In terms of Laplace transforms: Let * operator denote convolution. Recall the definition f(t) * g(t) = int f(t-x) g(x)dx. Then, int (t-x) f(x) dt = f(t) * t t = u(t) * u(t) So, int (t-x) f(x) dt = f(t) * u(t) * u(t) Since convolving by the unit step function u(t) is the same as integrating, int( (t-x) f(x) dt = int^2 f(x) dx. This video actually explains the relationship between the functions and their Laplace transforms.

Damn you really ended it with a plot twist. Fractional integrals and thereby opening doors to Fractional derivatives.

6:34! Brilliant!

You know ; I just HATED it when he closed with a teaser about the Gamma- function and its role in computing a multiple integral. But that's the way to get more subscribers I suppose. And in my case : it WORKED. So I just subscribed and will DEFINITELY be back and recommend this channel to any math-geek I talk to. Keep up the good work: "Aux Revoir "......

When I thought thta 2:30 part problem, I think we can just make this as general form where: ∬6x dx dx can be expressed as k*∬x^n where n =1, k=6 for example, if we take one integral as in our mind, we see that it grow up power n with +1. Also we have to take division k/(n+1) * ∫x^(n+1) dx we see how it end up, we can write basic form multi-integral as formula: (k / (∏(n+i,i,1,t)) )*x^(n+t) where k=constant, n=x's power on starting, t=number of integrals. for example, if we have ∭∭14 sqrt(x) dx^6 . We can write this as: 14/((n+1)(n+2)(n+3)(n+4)(n+5)(n+6)) * x^(n+6) where n =1/2 ((14)/(∏(n+i,i,1,t)))*x^(n+6)|n=((1)/(2)) and t=6 ▸ ((128*x^(13))/(19305))

Is this alternate way to think of nested integration related to how Lebesgue integration is considered superior for higher dimensions (input or output), as it tends to generalize cleaner than Reimann integration?

Interpreting things with geometrifying trigonometry give meaningful visualization for Cauchys repeated integrals

Great video by the way. Is using Reimann-Louiville's fractional derivative/integral considered as integral and differentiation compression? The fractional construction generalizes the differential operator, such that the differential operator of the order n defines derivation done n number of times, and -n defines integration done n number of times. The formulation is a single integral and it makes use of the generalization of the aforementioned factorial: the gamma function. Not only does this formulation take into account both integration and derivation, but also any differential operator that is of the order of any real number - even fractional integration and derivation

I have an old HP15C that does numerical integration, and the manual has a whole section on dealing with cumulative error.

I definitely learned about that in university, but completely forgot - or didn't understood back then.

Nested Integral : You can't beat me, I'm compressible. Derivative: You're right I can't. But he can. (d2y/dx2)

Video starts around 3:40 Edit: turns out it's actually an amazing video ♡

Is this related to stoke's theorem? (not the surface integral to integral of divergence of a function over a volume, but the more general one) From what I understand it's a generalization of this "integral reduction" we see with the fundamental theorem of calculus, divergence theorem and the "little" stoke's theorem taught in a first course on vector calc.

17:45 This is a good lead in to your next video, fractional derivatives / integrals

The expression under the transformed single integral reminds me a lot of a Lebesgue integral in a way, being a value of the function times the measure of the interval at which it takes that value, is there a connection here in some way to integration with respect to measure?

Before watching video: cauchys integral formula for the nth derivative... and u can use negitive numbers to mean integrals.. but you gotta use gamma formula and complex numbers. But it works After watching video.. no cauchys integral formula is equivalent to the one you showed

@Morphocular heres one thing about differentiating.. it can go below 0 and integration can go above 0. why don't we regularly perform these functions? why do we infact simplify to cancelation as a rule? x^0 goes to # 1, but X^-1 is right there as a possibililty for the next differential. simarly the reverse for integration. why dont we go past these arbitary lines? Or is there something im missing?

Very cool! I should have guess that factorial would show up. Still annoys me that the continuous version most people use is the gamma function, and not the actually equivalent pi function. (But I love gamma for Four 4s, since it makes 6 super easy to get, among other things.)

A quick tip I learned in calculus classes that I think is good for anyone taking multivariable calculus: For an integrand of 1, 1 integral means 1 dimension of measurement. So a triple integral with 1 as the integrand measures volume. For any other integrand, n integrals means n+1 dimensions of measurement. So a double integral with a non-1 integrand measures volume. Sometimes its easier to define the shape in the integrals' limits, sometimes its easier to define the shape with less integrals and an integrand. Though not a rigorous statement in the least, it should still help those who just "need to pass". (Helpful for tutoring more than anything)

This video was a trip.

What about the case where the lower bounds of the nested integrals aren't all the same? If they're related, e.g., a_2=2a_1, I assume your line y=x in the animation looks something like y=2x, or maybe y=x/2. But what about the case when the lower integration bounds are arbitrary (including a_2

Nice

the cliffhanger at the end was better than most netflix shows ngl

i saw the beauty of math thanks a lot

so thinking about aplications in physics i dont assume there is any neat trick to make it work for integrating over spheres for example since those are different variables ?

the audacity you have to end this video on a cliffhanger!! Anyways, it was a really interesting video, thank you very much!!

O(1) formulas are incredibly powerful.