one of my favorite things about calculus is that, given a problem like this, it's tricky to finish the problem and not accidentally invent calculus. :D
yeah and like, even archimedes basically used integrals to discover areas and volumes of curves (without knowing that this could become rigorous since the idea was giving lines and shapes weight) and used something extremely similar to epsilon delta proofs to prove the result
@@gabitheancient7664 Shame a bunch greedy imperialistic religious fanatics took over europe, and painted over Archimedes books with fairy tales. Set us back a thousand years at least.
@@gabitheancient7664actually I have heard the view that some of Archimedes’ proofs were so careful as to almost resemble the analysis of the 19th century. He may not have discovered a theorem like this (as he wrote in the Archimedes Palimpsest) but when he proved theorems logically, he was super careful there is no gap in the argument. Maybe because of Zeno’s paradoxes, the Greeks were extremely worried about the rigour of ideas like infinity, so geniuses like Archimedes were very careful to couch their discoveries in infallible logical proof.
@@topdog5252yees, it was amazing fr, he basically proved by contradiction that figures had some areas made an extremely rigorous argument for why it couldn't be less nor greater, showing that any difference could be overcomed by some construction, that's basically analysis if you're curious, in the first minutes of the video "This Is the Calculus They Won't Teach You" by the well-rested dog, he explains archimedes' argument for the area of a circle, if you take some time you can understand it that dude was insane, the ideal mathematician, a great balance of extreme intuition and rigour, my inspiration
@@gabitheancient7664 I agree. He combined perfect logical rigour with a great imagination and came painfully close to some even greater, fundamental results in mathematics and physics - he was aware of the heliocentric model of Aristarchus. Even though so little is known about the man, whoever wrote those books, On the Sphere and Circle, The Sand Reckoner, etc. was a very great man.
Perfect. After 3b1b started his SOME there are a lot of TH-camrs that copy his style but miss the point of his videos. Even him aswell has forgotten his true origins a bit. But you captured it perfectly: a visual proof containing some smart and enlightening perspective, explained very clearly.
I would say Cavalieri’s principal (and much of his work) is a sort of precalculus or proto-calculus rather than separate from calculus just because it was before Newton and Leibniz. Cavalieri’s book influenced John Wallis who’s book Arithmetica Infinitorum, greatly influenced Isaac Newton’s mathematics (Newton took extensive notes from that book and we still have those notes, and when Newton reached the end of the book all this new, profound revolutionary work poured out in his notes and these were some of his new discoveries). Cavalieri was also a student of Galileo who was going to write a text on such methods in mathematics - basically a precursor to calculus. I read that Joseph Louis Lagrange considered Pierre de Fermat to be the true creator of calculus for the fundamental insights he gave, rather than Newton or Leibniz. These ideas of men like Cavalieri, Torricelli, Roberval, even Kepler, and many others like Huygens or Gregory or Barrow, are all sort of calculus ideas, many of them very ingenious, but many of them are ad-hoc methods or extremely hard to use for anything but the simplest or problems, like the method of exhaustion.
Although this method is not as elegant or universal as the relative derivative, the presentation is really good, allowing people paying attention to predict the methods used.
To generalize, area under y=x^n equals to hypervolume of (n+1) dimensional hyperpyramid built by infinitely many n dimensional hypercubes with linearly increasing side length. It is very difficult to express this concept, so correct me if you can.
This is great ! When I was in high school, I tried to find a way to visualize the link between derivatives, primitives and areas, and I noticed that integrating for polynomial functions was like adding a dimension. I was sad that it was not taught in math class, as I found this a very intuitive way to conceptualize derivation and integration. I'm glad I found someone who explained it (and very well !). I'm sure the same visualization would apply to functions like exponentials as well with a bit of work
Cavalieri's Principle basically requires calculus or pseudo-calculus (method of exhaustion), but it's also incredibly intuitive and obvious. So it passes in terms of giving a self-sufficient intuition beyond just "The power rule"
@@npip99 The method of exhaustion - exactly! That is what made it possible, essentially using a more or less unstated version of Cavalieri's principle, to justify Archimedes's identification of the volume of a hemisphere with the difference between corresponding volumes of a cylinder and a cone, both of radius and height equal to the sphere's radius. Archimedes was too honest! If he had acted like the other politicians at the Academy, he would have disguised the way he originally got the result and only spoken out after coming up with the justification for it.
We can also turn the integral into the volume of a solid bounded by octant 1, y = x, and z = x. Each cross section of that shape across the x-axis is an x by x square. There is a way to build an x by x by x cube with 3 copies of that solid.
Cute, but invokes Cavalieri's principle - the only way, I at least, understand that principle is by infinitesimal thinking of ultra thin slices (of the same area) piled up, i.e. integrated in 3D to yield (equal) volumes. So, it's not really 'Calculus-free', but it is rather is based on a principle (Cavalieri) that requires proof by Calculus. That does not detract from its cuteness, though, well done!
Cavalieri's principle originates from the method of exhaustion (MoE), which doesn't involve infinitesimals. MoE involves a reductio ad absurdum, which is based on more-less relation. Thus more-less relation is primitive intuitive foundation of MoE. Infinitesimals make Zeno go berserk. Neusis leads to infinitesimals, so neusis is in contradiction with the primitive intuitive foundation of more-less relation, which e.g. Achilleus and turtle represent.
@@santerisatama5409 Nowadays, mathematicians wouldn't use the word "infinitesimals" at all, since the epsilon-delta definition of a limit, introduced by Augustin-Louis Cauchy, only requires the use of real numbers. Anybody who still uses words like "infinitesimals" or "fluxions" outside of a discussion on the historical development of Analysis would easily be outed as either an antiquarian or a potential crackpot who disregards modern math.
@@alfredomulleretxeberria4239 I'm happy to conceive relational operators as object-independent continuous processes, but I remain unconvinced that non-computational real numbers can do arithmetics and form a field by a mere declaration which is contradictory to basic mereological syllogisms. Based on this, method of exhaustion can be foundationally coherent, epsilon-delta limits not. It seems intuitively possible to define tangents based on circular definition of curvature, independent of coordinate system neusis.
Very exciting! Now I have curiosity about this Cavalieri principle applied to objects with different geometry, like here one object parabolic prizm, other piramid🧐😊!
Idea before watching: I'd consider the area of the triangle (x₁, x₁²), ((x₁+x₂)/2, ((x₁+x₂)/2)²), (x₂, x₂²). Then I will show that the area of that triangle depends only on x₂-x₁, and derive that area T(b) as a function of b=x₂-x₁ Then I would describe a simple binary subdivision process that leads to the expression of the sought area as a limit of successive approximations: ½ - SUM(k, 0, ∞, 2^k * T(2^-k)), and hope that I would be able to give the sum in closed form. I *am* computing an infinite sum, I *am* using successive approximations, but I believe I am not doing calculus (computing an integral), because the process is less general: the "conservation" of T(b) only works for parabolas, not for any other proper curves, so what simplifies to multiplying the triangle area by 2^k (because there are 2^k triangles with the same area) would be a difficult nested sum in the general case.
This is a very good explanation, however while it is true that Cavalieri 's principle predates calculus, the way Cavalieri proved it is not exactly sound, he wasn't as careful as say Archimedes who carefully used upper and lower bounds, even then that was his basic idea, and it is also the underlying principle of the integral, rather than say you're not using integrals i think it would be more correct to say that you're using integration in a geometric sense, and that the fundamental theorem of calculus isn't strictly necessary when things can be geometrically framed, because of the geometric meaning of integration
Just came to mind, can we see Geometric algebra aka Clifford algebra as an application/extension of method of exhaustion? Geometric product is kinda the in-between of inner product and outer product analogues of upper and lower bounds? Based on this intuition, can we construct Geometric algebra independent of coordinate system neusis?
So if we can prove that the number of faces of an n dimensional cube is 2*n, we are done with the generalization as well. It's not too complicated to prove that ig.
No, it's essentially method of exhaustion. The difference is that method of exhaustion does not lead to Zeno-paradoxes; infinitesimal calculus is a neusis method and thus a Zeno paradox.
@@mokouf3 Foundationally method of exhaustion and infinitesimal calculus belong to whole different worlds. It's really not a case of teleological cumulatively linear narrative "Greeks were just naive, now we are smart adults".
I feel like this explanation is "cheating". - The idea of "growing a volume" with a series of planes is basically taking an integral. - I wouldn't expect a pre-calc student to know Cavaleri's principle, considering I have a Master's degree in math and have never heard of it.
You have a master's and never heard of it?! It's usually taught when learning solid geometry, sometimes as early as middle school in the U.S. It's considered proto-calculus, as it doesn't use differentials, those came later. www.khanacademy.org/math/geometry/hs-geo-solids/xff63fac4:hs-geo-cavalieri-s-principle/v/cavalieris-principle-in-3d#:~:text=Cavalieri%27s%20principle%20tells%20us%20that,Created%20by%20Sal%20Khan.
one of my favorite things about calculus is that, given a problem like this, it's tricky to finish the problem and not accidentally invent calculus. :D
yeah and like, even archimedes basically used integrals to discover areas and volumes of curves (without knowing that this could become rigorous since the idea was giving lines and shapes weight) and used something extremely similar to epsilon delta proofs to prove the result
@@gabitheancient7664 Shame a bunch greedy imperialistic religious fanatics took over europe, and painted over Archimedes books with fairy tales. Set us back a thousand years at least.
@@gabitheancient7664actually I have heard the view that some of Archimedes’ proofs were so careful as to almost resemble the analysis of the 19th century. He may not have discovered a theorem like this (as he wrote in the Archimedes Palimpsest) but when he proved theorems logically, he was super careful there is no gap in the argument. Maybe because of Zeno’s paradoxes, the Greeks were extremely worried about the rigour of ideas like infinity, so geniuses like Archimedes were very careful to couch their discoveries in infallible logical proof.
@@topdog5252yees, it was amazing fr, he basically proved by contradiction that figures had some areas
made an extremely rigorous argument for why it couldn't be less nor greater, showing that any difference could be overcomed by some construction, that's basically analysis
if you're curious, in the first minutes of the video "This Is the Calculus They Won't Teach You" by the well-rested dog, he explains archimedes' argument for the area of a circle, if you take some time you can understand it
that dude was insane, the ideal mathematician, a great balance of extreme intuition and rigour, my inspiration
@@gabitheancient7664 I agree. He combined perfect logical rigour with a great imagination and came painfully close to some even greater, fundamental results in mathematics and physics - he was aware of the heliocentric model of Aristarchus. Even though so little is known about the man, whoever wrote those books, On the Sphere and Circle, The Sand Reckoner, etc. was a very great man.
Perfect. After 3b1b started his SOME there are a lot of TH-camrs that copy his style but miss the point of his videos. Even him aswell has forgotten his true origins a bit. But you captured it perfectly: a visual proof containing some smart and enlightening perspective, explained very clearly.
How'd he miss his true origins
They dont copy his style they just use Manim (the python library he developed and uses to animate that he lets others use)
I would say Cavalieri’s principal (and much of his work) is a sort of precalculus or proto-calculus rather than separate from calculus just because it was before Newton and Leibniz. Cavalieri’s book influenced John Wallis who’s book Arithmetica Infinitorum, greatly influenced Isaac Newton’s mathematics (Newton took extensive notes from that book and we still have those notes, and when Newton reached the end of the book all this new, profound revolutionary work poured out in his notes and these were some of his new discoveries). Cavalieri was also a student of Galileo who was going to write a text on such methods in mathematics - basically a precursor to calculus. I read that Joseph Louis Lagrange considered Pierre de Fermat to be the true creator of calculus for the fundamental insights he gave, rather than Newton or Leibniz. These ideas of men like Cavalieri, Torricelli, Roberval, even Kepler, and many others like Huygens or Gregory or Barrow, are all sort of calculus ideas, many of them very ingenious, but many of them are ad-hoc methods or extremely hard to use for anything but the simplest or problems, like the method of exhaustion.
Although this method is not as elegant or universal as the relative derivative, the presentation is really good, allowing people paying attention to predict the methods used.
To generalize, area under y=x^n equals to hypervolume of (n+1) dimensional hyperpyramid built by infinitely many n dimensional hypercubes with linearly increasing side length.
It is very difficult to express this concept, so correct me if you can.
Exactly. Next video will visualize this for 4D case
@@HyperCubist and now I notice that even your name is related to hypercubes!
Using a third dimension, we can turn that integral into the volume of a pyramid.
This is genius, especially with adding the y=ax^2 part to fix the area = volume assumption. Subscribed.
This is great ! When I was in high school, I tried to find a way to visualize the link between derivatives, primitives and areas, and I noticed that integrating for polynomial functions was like adding a dimension. I was sad that it was not taught in math class, as I found this a very intuitive way to conceptualize derivation and integration. I'm glad I found someone who explained it (and very well !). I'm sure the same visualization would apply to functions like exponentials as well with a bit of work
How do you prove Cavaleri's theorem without calculus?
Oof
I was about to ask the same thing 😂
You can't. The goal was more of a gut feeling or intuition on the formula, right?
Cavalieri's Principle basically requires calculus or pseudo-calculus (method of exhaustion), but it's also incredibly intuitive and obvious.
So it passes in terms of giving a self-sufficient intuition beyond just "The power rule"
@@npip99 The method of exhaustion - exactly!
That is what made it possible, essentially using a more or less unstated version of Cavalieri's principle, to justify Archimedes's identification of the volume of a hemisphere with the difference between corresponding volumes of a cylinder and a cone, both of radius and height equal to the sphere's radius.
Archimedes was too honest! If he had acted like the other politicians at the Academy, he would have disguised the way he originally got the result and only spoken out after coming up with the justification for it.
This blew my mind, incredible
This was brilliant, you deserve more recognition!
Units all work out. I think dimensional analysis would have been a good area of investigation.
0:31 Wow. Just yesterday, I was thinking about this exact integral and how weird it felt. What a coincidence.
This is truly what should be taught in precalculus
We can also turn the integral into the volume of a solid bounded by octant 1, y = x, and z = x. Each cross section of that shape across the x-axis is an x by x square. There is a way to build an x by x by x cube with 3 copies of that solid.
yup ... that's what the video is about.
Incredible proof!
That's quite a wonderful way to see it!
a beautiful perspective to a trivial question in calculus. its a bummer you cant have multiple submissions for SoME3!
tesseract jumpscare D:
Cute, but invokes Cavalieri's principle - the only way, I at least, understand that principle is by infinitesimal thinking of ultra thin slices (of the same area) piled up, i.e. integrated in 3D to yield (equal) volumes. So, it's not really 'Calculus-free', but it is rather is based on a principle (Cavalieri) that requires proof by Calculus. That does not detract from its cuteness, though, well done!
Cavalieri's principle originates from the method of exhaustion (MoE), which doesn't involve infinitesimals. MoE involves a reductio ad absurdum, which is based on more-less relation. Thus more-less relation is primitive intuitive foundation of MoE.
Infinitesimals make Zeno go berserk. Neusis leads to infinitesimals, so neusis is in contradiction with the primitive intuitive foundation of more-less relation, which e.g. Achilleus and turtle represent.
@@santerisatama5409 Nowadays, mathematicians wouldn't use the word "infinitesimals" at all, since the epsilon-delta definition of a limit, introduced by Augustin-Louis Cauchy, only requires the use of real numbers. Anybody who still uses words like "infinitesimals" or "fluxions" outside of a discussion on the historical development of Analysis would easily be outed as either an antiquarian or a potential crackpot who disregards modern math.
@@alfredomulleretxeberria4239 I'm happy to conceive relational operators as object-independent continuous processes, but I remain unconvinced that non-computational real numbers can do arithmetics and form a field by a mere declaration which is contradictory to basic mereological syllogisms.
Based on this, method of exhaustion can be foundationally coherent, epsilon-delta limits not. It seems intuitively possible to define tangents based on circular definition of curvature, independent of coordinate system neusis.
that's amazing tbh
Great narration. Thank you HyperCubist Math for your video.
God damn, good job
Very exciting! Now I have curiosity about this Cavalieri principle applied to objects with different geometry, like here one object parabolic prizm, other piramid🧐😊!
i know where this is going through, nice share
Great video. Thank you
Idea before watching: I'd consider the area of the triangle (x₁, x₁²), ((x₁+x₂)/2, ((x₁+x₂)/2)²), (x₂, x₂²). Then I will show that the area of that triangle depends only on x₂-x₁, and derive that area T(b) as a function of b=x₂-x₁ Then I would describe a simple binary subdivision process that leads to the expression of the sought area as a limit of successive approximations: ½ - SUM(k, 0, ∞, 2^k * T(2^-k)), and hope that I would be able to give the sum in closed form. I *am* computing an infinite sum, I *am* using successive approximations, but I believe I am not doing calculus (computing an integral), because the process is less general: the "conservation" of T(b) only works for parabolas, not for any other proper curves, so what simplifies to multiplying the triangle area by 2^k (because there are 2^k triangles with the same area) would be a difficult nested sum in the general case.
Wow excelente explicación
Great content. Which app do you use to make your animations?
Thanks! Geogebra mostly, and some Apple Keynote.
brilliant!!
Great explanation, really nice presentation
This is a very good explanation, however while it is true that Cavalieri 's principle predates calculus, the way Cavalieri proved it is not exactly sound, he wasn't as careful as say Archimedes who carefully used upper and lower bounds, even then that was his basic idea, and it is also the underlying principle of the integral, rather than say you're not using integrals i think it would be more correct to say that you're using integration in a geometric sense, and that the fundamental theorem of calculus isn't strictly necessary when things can be geometrically framed, because of the geometric meaning of integration
Just came to mind, can we see Geometric algebra aka Clifford algebra as an application/extension of method of exhaustion? Geometric product is kinda the in-between of inner product and outer product analogues of upper and lower bounds?
Based on this intuition, can we construct Geometric algebra independent of coordinate system neusis?
i wonder how Newton and Leibniz made this observation about integration
If we convert flat 2 dimensional paraboloid thing into 3d pyramid we can use 3d Volume
VERY nice!
treating the integral as an opperator you can proove that the result without any limits
great proof, but the whole cross section volume thing is basically integration
So if we can prove that the number of faces of an n dimensional cube is 2*n, we are done with the generalization as well. It's not too complicated to prove that ig.
thanks, amazing trick !!
bravo
This is amazing
cool channel
Cavalieri's Principle is still Calculus, essentially
And yet for some reason it's taught in middle school. 🤔
No, it's essentially method of exhaustion. The difference is that method of exhaustion does not lead to Zeno-paradoxes; infinitesimal calculus is a neusis method and thus a Zeno paradox.
It is an early step towards integral calculus, you can see it as pre-calculus like limit.
Not in depth though...even E=mc^2 is seen in middle school
@@mokouf3 Foundationally method of exhaustion and infinitesimal calculus belong to whole different worlds.
It's really not a case of teleological cumulatively linear narrative "Greeks were just naive, now we are smart adults".
I feel like this explanation is "cheating".
- The idea of "growing a volume" with a series of planes is basically taking an integral.
- I wouldn't expect a pre-calc student to know Cavaleri's principle, considering I have a Master's degree in math and have never heard of it.
You have a master's and never heard of it?! It's usually taught when learning solid geometry, sometimes as early as middle school in the U.S. It's considered proto-calculus, as it doesn't use differentials, those came later.
www.khanacademy.org/math/geometry/hs-geo-solids/xff63fac4:hs-geo-cavalieri-s-principle/v/cavalieris-principle-in-3d#:~:text=Cavalieri%27s%20principle%20tells%20us%20that,Created%20by%20Sal%20Khan.
1 NO AUDIO 2 cavalieri principle....zzzzz
What software was used for the examples? Also great video!
GeoGebra. Thanks so much!
@@HyperCubist Awesome!