Without integration, why is the volume of a paraboloid half of its inscribing cylinder? (DIw/oI #8)
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- Опубликовано: 16 сен 2024
- Rather than using integration, can we find the volume of a paraboloid? Yes, if we accept a precursor to calculus - Cavalieri's principle. Usually, integration is needed to find the volume of a paraboloid, for example using shell method, but using Cavalieri's principle, and a sneaky little trick, we can find the volume very easily - half of the volume of the circumscribing cylinder!
The idea for this video isn't actually mine, but thanks to Yehuda Simcha Waldman for suggesting the idea of this video! He emailed me about the proof, and I modified it a little bit and adapted it into this video that you are watching here.
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EDIT: At 3:57, the solution of x should be sqrt(a/b-z/b) instead.
Thanks again to Yehuda for suggesting the idea of this video! Here, the paraboloid only means those paraboloids that can be constructed by revolving a parabola around an axis. For other paraboloids, the cross sections would generally be an ellipse instead. I don't have a solid plan for the next video yet, so do feel free to leave a comment for a possible idea.
As with all videos, if you haven't already, do fill in the Google form to log your math levels: forms.gle/QJ29hocF9uQAyZyH6
This is because it makes the planning of future videos better in the sense that I could make the pacing better, and know what knowledge I can assume you know.
That is fascinating! Really elegant, super clear and concise. I love proofs that use Cavalieri's principle for that exact reason, but it never occurred to me that the volume of a paraboloid could be derived so cleanly using it
I didn't expect such clean proof either, which is why I have to share it on RUclips! Glad that you enjoyed it.
@@mathemaniac yes , enjoyed every moment , especially your voice is very catchy
At 2:30, since pi * (1 - z) is linear, you can find the average cross-sectional area by taking the average of pi * (1 - z) evaluated at z = 0 and z = 1 (the height of the paraboloid), which is pi/2. Then multiply by the height of the paraboloid to find the total volume, in this case, pi/2.
Indeed that's an alternative approach!
Your fan from Quora here. Your videos are as beautiful as your answers.
They are making me love math.
Keep up the good work man.
Glad you like them!
Whoooo , you saved my day .
Very intresting and thanks to you for this ( also thanks to Yehuda😅)
Glad that you enjoyed the video!
This viewer who sent the email has a very similar way of math thinking to me. This resonated pretty easily with me
Oh my godddd
This is simple and soooooo elegant and beautiful. I love it!!!
Glad to hear that!
Great site for entertainment and challenging for lovers of maths
I really love this video!! It is so beautiful!
Glad you like it!
Thanks a lot for this. I needed it for my fluid mechanics course.
Keep going!hope your channel go viral
Thank you so much!
holy fuck galaxy brain, thank you for sharing
Thanks for the appreciation!
Spheroid top: 4/6 pi r^2 h
Paraboloid top: 3/6 pi r^2 h
Cone top: 2/6 pi r^2 h
Archimedes did something similar, but he employed equilibrium and center of gravity properties. But Archimedes didn't consider as a true mathematical proof, but as method to discover, to prove these properties he used the so called exhaustation method, which is attributed firstly to eudoxus of cnidus, and it is a rigorous method of mathematical proof. By means of mechanics he not only computed parabolid's volume, but as well, he could figure out area of parabolic sections, volume of spheres and ellipsoids. And it would be interesting if you brought up this archimedes' method of balance, and it's related with this video, since line sections from plane figures or area sections from solids are involved
so, a paraboloid inside a cylinder is the same volume as the space it left
Great video.
3:54 - small nitpick but shouldn't that be sqrt(a/b - z/b)? That way you would have the reverse parabola as z = bx^2, thus x = sqrt(z/b) and the final cross section would equalise out as a/b*pi, thus final volume of the bounding cylinder as (a^2/b)*pi?
Ah yes! Why are you the only person who noticed this after almost two years of this video? However, there is really nothing I can do about this other than editing the pinned comment stating the error.
Try to make your thumbnail look little neat. It seems crowded and cheap. Just suggesting. Your video is great. Appreciated!
Thanks for the comment! I do realise that my thumbnails are not really aesthetic, but I just really don't know how to make a thumbnail properly... I can make a less crowded thumbnail for the next video though. (I don't want to change this one because it might confuse viewers who already watched the video)
This is great, thank you. Galileo used the same principle to show the equivalent areas and volumes of a cone and the cylinder evacuated of a unit hemisphere. But I did not know the name of the principle or its more general application. A paraboloid, and a torus too... Very nice!
I am convinced that the geometry of the cosmos is the dual of a single point coordinatised in this framework. Oh, like a spinor I guess....along those lines. Curves. ❤️
شرح رائع.شكرا.
"We can generalize this argument to all parabolas…"
As Matt Parker showed in a video, there is only 1 parabola. :)
Cavalieri's principle, or something like it, is also required to find the volume of a square pyramid (or any general cone). Three appropriately-shaped congruent square pyramids can fit together to form a cube, so their volumes must each be one third of the volume of the cube. One might hope a clever decomposition like that might exist for all pyramids, or even all polyhedra, but it does not. Dehn's negative answer to Hilbert's third problem proves that in most cases, no such decomposition exists. Therefore, to get a formula for a general square pyramid, we need to use something like Cavalieri's principle to translate the volume of the specially-shaped pyramid to the volume of any other pyramid (or indeed any cone) with the same base and height.
The ancient Greeks did not appreciate this fact, though they used something like Cavalieri's principle implicitly. Two plane figures were "equal" if they had the same area in the modern sense, and the sums of equals were equal as an axiom. But this alone is not enough to justify Cavalieri's principle, because that requires adding infinitely many areas to obtain a volume, which is nonsensical, and properties of finite sums don't necessarily hold for infinite sums anyway. It turns out that it is _impossible_ to compute or even define the volume of most solids without some notion of a continuum--something like the set of real numbers--and either infinitesimals (in non-standard analysis) or limits of sequences of real numbers. In other words, there was nothing approaching a rigorous treatment of volume until Cauchy, even for something as simple as a right square pyramid.
Indeed, to define volume itself, this requires analysis, and in fact, Cavalieri's principle can be thought of as volume being well-defined.
@@mathemaniac Cavalieri's principle is a road toward a definition. Without the real numbers, it still isn't workable on its own.
3:05 Tbh, I didn't see it comin' but that's so cool!!!
Yes!
Hey I'm your new subscriber 💓 from India
Thanks for the subscription! Glad you like my videos!
now with Archimedes method
Very clever!
Cool!!
I haven't got it 2:40 , I don't understand the connection created with the height of the parabolid evaluated from the top of it when it is upside down and the area of 1 of its sections. I don't get the logical step
Juat a question tho..
Doesn't the proof of Cavaleiri's pronciple rely on calculus/analysis?
Or if not, can you explain it in a rigorous way? Cuz I intuitively understand it, which makes me very suspicious of it, thinking that I don't truly underatand where it's coming from
The very definition of volumes rely on analysis, so if you want to be really rigorous, there is no way not to use analysis. Still, Cavalieri's principle is bypassing integration in the sense that you don't have to compute integrals.
Very roughly speaking, to define the volume, you will take cross section at every height. Record those cross-sectional areas as a function of the height. Then the volume is defined as the area under the curve. Of course this is simplifying here, and actually the definition of area still relies on analysis (Riemann's definition). Cavalieri's principle is basically saying that if you have the same functions (cross sectional area as a function of height), you have the same area under the curve.
You can actually think of Cavalieri's principle as just saying well-definedness of the notion of volume as defined very roughly above.
This is really cool
Thanks!
Yeah, but where does Cavalieri's principle come from if not from integration? :)
It is a principle that precedes integration. Doing integration already assumes Cavalieri's principle.
why is the raidus of the cylinder 1
0:30 This is not "forcing students to learn". This is "forcing students to memorize formulas".
YO THIS IS CRAZY
That's why I have to share it!
Can you find the area of a circle using Cavalieri?
Somewhat: Imagine the area of a circle to be made of concenteric rings.
The rings' circumferences increase linearly according to their distance (radius) from the circle focus.
Therefore, the circle can be "unwrapped" to form a triangle with sides r and 2ℼr.
Ok but isnt Cavalieri's principle hidden calculus? Because the proof of Cavalieri's principle should use calculus I believe.
The very definition of volumes rely on analysis, so if you want to be really rigorous, there is no way not to use analysis. Still, Cavalieri's principle is bypassing integration in the sense that you don't have to compute integrals.
Very roughly speaking, to define the volume, you will take cross section at every height. Record those cross-sectional areas as a function of the height. Then the volume is defined as the area under the curve. Of course this is simplifying here, and actually the definition of area still relies on analysis (Riemann's definition). Cavalieri's principle is basically saying that if you have the same functions (cross sectional area as a function of height), you have the same area under the curve.
You can actually think of Cavalieri's principle as just saying well-definedness of the notion of volume as defined very roughly above.
I liked the video, but I'm sorry, by taking these layers you are basically doing a sneaky riemann integration, cant fool me
Its almost as if you went 90% of the way to explaining...but didn't quite nail it. Nothing wins like doing a numerical example.