What is the area under an arc of a cycloid curve?
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- Опубликовано: 16 сен 2024
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Why does the area under a cycloid curve equal to 3 times the area of the circle used to trace out that curve?
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Programs used:
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- Processing
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These videos can make anyone feel like they have really taken in the beauty of mathematics after watching
I think beauty of euclidean geometry, there's a non-euclidean beauty is not yet represent in the video yet.
Or, going by symmetry at 1:48...
Total area of the rectangle: 2r * 2πr = 4πr²
Area of unlabeled regions: 4πr² - 2πr² = 2πr²
Area of bottom region: 2πr² / 2 = πr²
Area under curve: 2πr² + πr² = 3πr²
How you reach 2r and 2pi r in rectangle ?
Can anyone elaborate please..
@@fahadahmed8172 the width (diameter) of the circle is 2r, and since the point is back at its position after the circle rolled, it have traveled as far as the circumference of the circle, 2πr.
The point shifts 2r vertically, and 2πr horizontally.
Edit: adding things here, the rectangle is added to make a clear image
Who needs integration?
Could just be me but this seems easier because you dont need the equation for the curve
It's always well explained, and the simulation increases this feeling that you have understood, now if you look up to this problem, you will find it way easier just because of your explanation, thanks
increases*
@@JorgetePanete thanks
@@SimonWoodburyForget That was true when I was younger, but since i'm french and studying in something called prepa, (you should look what's the purpose of prepa), we are doing that kind of stuff and even more in maths, physics, and ingeneering sciences, but i feel you it's way to sad to see how the system is working, clearly a shame, that students will just remember some stuff a passionate teacher would have provided them in some lessons, and the knowledge will stop there
I love your work, this is nothing short of brilliant on every level
0:38 was just soooo mesmerizing!! Excellent use of visualization.
I’m not that advanced in math, but this helped me understand quite nicely. I love the smooth animation and calm music, by the way.
Lena DeathBlossom thank you:) glad you enjoyed it.
The simplicity, the animation, the lofi music. I love everything about this!
omg this is perfect i derived this formula in calculus literally 2 days ago
Beautiful application of Cavalieri's principle. It would take me way longer to come up with something this clever than it would to just integrate it, but I do enjoy seeing these kinds of arguments.
I'm loving your channel, you don't just explain it, you make it seem obvious.
I've been obsessed with tautochronous curves recently. Thank you for this!
Great video! The animations really helped and you gave enough time so it could really be understood before moving on. Thank you for such great content
Just from the video, it’s not so obvious to me that the diagonal line at 1:53 would cut the dark portion in half. It should be stated that the green curve is just the blue one rotated by 180° (by construction).
Besides that, excellent video as always!
Giovanni Bracchi That part is clear I believe because that line was from another circle rolling on the "ceiling", so its the same curve ad the other one on the right
@@noa_1104 It's perhaps clear to some, but that's not the point that Giovanni is making. The point it that it's a step left out, and either a) you are attempting to prove this result in which case all steps are required or b) you are attempting to teach someone how to arrive at this result, in which case each step must be covered.
Aaron Sherman Oh yeah, agree with that. I just think it could have been clearer earlier then, that the other curve is basically mirrored, maybe when it appears
That's just an exercise left for the viewer :v
I really appreciate you for the videos you make which simplifies the toughest concepts into an easy and comprehensible way .
Thanks a lot:)
Congrats for 50k subs. You deserve more. Your videos have top notch quality.
In school we learn parabolas, ellipses and other things and currently I have no idea how to solve them or something and we had already a lot of lessons. Your video is 3 minutes and 22 seconds long and I understood it perfectly. Now tell me if problem with school learning is in me or somewhere else🙂 Very good vid, you earned a like and a sub.😊
In school you learn general things to solve all kinds of problems, related in this case to parabola etc.
This video, although very good, interesting, very well made and intriguing
Only showed you one particular solution to one particular problem. It did it very well, but what you leaned know is the solution to this particular problem.
While in school what they try is to teach you a general understanding so that you can come up with your own solutions to any problem.
Ita different, school is not doing a bad job. Keep on learning and trying and you'll get there
Wonderful Animation
Interesting as always!
Thank you so much sir. Your concept is very clear and well explained. You are so much great.
Brilliant explanation.
That is the most beautiful way of thinking about it that I have ever seen.
One of my favorite channels.
SillySOSedge
Elegant work once again! :)
Rahul Upadhyay Thank you very much.
Finally something I can easily understand :) I think one step might be missing, some explanation why the hypotenuse divide that shape in two, while it looks and feels like it should be halfs, sometimes in math it might not be, at least we know they have the same length, not sure about the curvature. Guess it's based on same tangent angle per step?
Brilliant work
Darshan Gupta thank you:)
way better then the original. congratz
Great video as always, never I would imagine finding this area.
But in 1:48 wouldn't it be better if you called the blue area x (it's equal to the reddish area, because red area + brown area = brown area + blue area = area under the cycloid) so 2r^2π + 2x = 4r^2π, x= r^2π? Anyway this video is still awesome.
beautiful and very informative.
when you understands the reasoning for the arguments then only you are truly able to appreciate the idea and animation.
Amazing!
It is always a good day when Think Twice uploads something.
Atomic Compiler
Think Twice
I'm not sure what happened with the latest video (that's after this), but I was so excited! 😫
Sho Am sorry.. i made it public by accident, it’s not fully finished yet. Should be up in few days:)
@@ThinkTwiceLtu that's reassuring. At least RUclips didn't block it
Mind blown !
wow, great video, think twice! these proofs are all so elegant, they make me feel very satisfied
Beautiful, as always :)
scares009 thanks!
Hey Think Twice, i think you should do an animation for the nine point conic (it’s the generalised version of the nine point circle)
good one
how the fuck did u became one of my favourite math channels with only 22 videos??!?!?? I ❤️ u
Really under rated channel
Wow, awesome as always !
Jérémie Herard thanks a lot!
Your videos are so good!
Excellent. Bravo. Well done.
Wow; simple after the fact, but non-obvious when you approach the problem.
Hey, I love this so much! I would like to support you, can you have skillshare as a sponsor next time?
Wow, elegant, both solution and your video :)
:)
Can't wait for some more quality content
Please make a video on simple harmonic motion .
Shortly 3/4 of a rectangle that touches the radius of the circle. Incredibly useful for calculating area under parabola without using integrals.
I'm concerned about two little snags here - the first is I feel the proof that the cycloid enclosures at 0:39 have area 2π could be a bit stronger, and the assumption that the diagonal through the enclosure at 1:53 divides it cleanly in half seems a bit loose.
Still, beautiful visual proof.
this is great but towards the end after finding the leaf-like shapes, you could have done -pi^2 r^2+r pi
2:20 You can simplify further. π=3 so this equals (π*r)^2.
This is an amazing proof!
Fascinating
So elegant
Very good video make more
I still watch RUclips thanks to videos like this
Bravo. Thanks for that.
Wow, this is pure beauty
So beautiful, as everything you upload, thank you
Neat !
Beautiful 👏👏👏
Thank you Thanks so much
So nice, So good
The Bests for you & your Family
Thank you
Amazing video
Hmm it has the same area as the total surface area of a hemisphere. Wicked but beautiful
Ugh...
SOO GOOD!!!
Jhon Lawrence Bulosan
i;m just glad i found you... came here from 3b1b
3(pi)r^2 is also the total surface area of a hemisphere of the same radius which I believe is kinda interesting too...
Wow never knew it works this way
Yo i actually liked the advert too
Wow, amazing 🎉🙂🙏
great video!
Luiz Felipe Garcia thanks!
That was beautiful!
its always so clever
Next this guy is gonna proove Collatz' conjecture using two triangles...
Is this a dejavu? :P
Keep the awesome work! Didn't expect a video so early!
Nuno Mateus Thanks man~
Yes, he made another vid on this months before this, but the proof wasn't good enough to upload publically. I think it was only this moment that he has found a more rigorous proof (rigorous enough to upload).
I’m so relaxed
This is a beautiful explanation! I only have one question: how are you sure about the statement made at 0:40? Is there a mathemathical way to prove this?
It's called cavalieri's principle!
Question about statement at 0:45, why if that statement true, the area of two shape is equal
what software do you use to make your animations?
Couldn't you also take integral from start to end point if given the function or 2 times the integral from 0 to b assuming it's an even function.
there's another proof, based on some ideas of calculus, but without nasty integrals and with geometry. here's a drawing: www.desmos.com/calculator/vqxt2uyiwm
basic Idea: break the area under the cycloid into thin triangles, abundant a third of them and be left with trianles that together make a circle, with radius pi.
you can break the path of a point moving on a cycloid to a straight forward movement and a circular one. the black is forward, the blue is circular (sorry it's not a circle, I can't draw that good). Connect the points on the cycloid to the corresponding on the ground (orange, corresponding here means the one on the groung after t seconds). add some more lines as done in the drawing (red). the parallelogram is nearly twice the area of the closest triangle, because of similarity of angles and length of lines. combining the blue-based triangle, because it's a circular movement, you get a circle that it's area is pi, multiply by 3 and you get the answer. can you animate this? it's a cool proof as well
nice proof:)
How were you able to create these animation?
OH MY GOD NO WAY
Holy heck wow very nice
please make a video on appolonius theorem
great suggestion:) I'll add it to my list
So if the whole figure is 3 pi r^2, that means at 1:39, the bottom shape has the same area as the other two shapes.
Nice animation
Saty Veer thank you:)
Really nice video, though I had to mute the music so I could concentrate. That beat was WAY too much...
Always applied Greens theorem
For this area calculation but I never tried thinking this way
Just bcoz of lack of any this type of animation tool
Would you say he first animated it and then saw the connection?
Can anyone tell me how to calculate it using calculas? Even a hint would do!!! Thank you
En el caso hipotético de que el universo sea infinito y haya infinitos multiversos, ¿podría existir una Tierra o un planeta plano y hueco?
Do a video of this but with a 3D structure :P
I probably would have tried to do it by integrating the function of the height of the point on the circle. This is simpler though.
Cycloid. It reminds me of Brachistichrone.
Oh wow, you are supported by 3b1b :o
That was beautiful as fuck.
yes please!
ronP __ :)
The thumbnail made me think it was a michael penn video lol