Why is the volume of a cone 1/3*pi*r^2*h? Here's a proof with the disc method! Calculus basics
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- Опубликовано: 3 окт 2024
- Why is the volume of a cone 1/3*pi*r^2*h? Let me show you a proof with the disc method! This is a must-know integral application that you will encounter in your Calculus 1 class and AP Calculus AB class. Many students don't like the disc method but I think it's one of the coolest topics in your calculus class! Subscribe to @bprpcalculusbasics for more calculus tutorials!
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Isn't this limit the same as the definition of derivative? ruclips.net/video/C440uWSzFGg/видео.html
If you want clear cut explanation , its here ruclips.net/video/WsQQvHm4lSw/видео.html
correction ruclips.net/video/WsQQvHm4lSw/видео.html
This proof is special to me because my Physics teacher 15 years ago asked us to calculate the volume of a cone for extra credit and this is how I did it. It was one of the first times I answered such an open question using calculus all by myself.
No you didn't. In order to solve this, you need to know the trick .You can't intuitively come up with the idea that you can work with 3D objects in a 2D cartesian plane. Never .This formula wasn't derived originally from this method. So it's just a trick .People know that it works but probably don't know why it works.Or at least you had to have the knowledge of 'disc' method.
@@study-i7bMaybe he or she is just smarter than you and they actually did do it.
@@study-i7b ..i beg to differ... im absolutely sure this is how he/she solved it... its a methodd that works for every rotational volume... no matter how complex it is... in some cases its the only way to solve it mathematically..
...this was also something i myself came up with long before rotational volumes was even thought.. so yes im absolute certain he /she figured it out...
..its also what sets some student apart..
there is two types of students, one who know why, and one who always get the right asnswer, its the once who know why that change the wourld (to quote myself)
in a nutshell V = πy^2dx; y=lim(F'(x) * F(x) )
@@Patrik6920 so you didn’t have the idea of "Rotational volume " right?Wow you're smart.You're so different from other students. I guess you could do Algebra without learning how to add, subtract or multipy.
@@siamsama2581 wooh!what a smart guy he is!He could do cartesian geometry without having the knowledge of slopes.
People like you don’t progress in life.Move on and sit with pen and paper and try to figure out if you're also smart.
the way you change markers without people realising is amazing, man
Thanks!
It's over 60 years, since I did anything like that at school. But I had a hunch, that calculus would do it. It was easy to follow.
My elementary school math teacher (who was strict but also very wholesome and encouraging) took a cone and a geometric cylinder, both with same radius and height, and poured three cones of liquid into the cylinder. This was a reason good enough, for he spilled no water.
This is a really clean proof.
Excellent tutorial. I like how you make your explanation simple, entertaining and easy to understand. Thanks.
Once you understand the integral, you realize that any 3D shape which converges to a point at one end via a constant slope (straight lines) has a volume which is 1/3 of an equivalent prism extended from the base (i.e. doesn't converge).
god bless you for mastering mathematics and being so clear and pedagogical in your approach while teaching students !
Video Idea :
How to take elements and integrate them, it will really help us in physics
Hu Tao?
@@capt.price1419 yes, it's me Hu Tao, Funeral Business was not going well so Mr.Zhongli sent me to high school, It's so boring here, here no fire butterflies 🦋 🔥
Sir,Excellent explanation . Thank you.
You can also proof the general equation of pyramid (1/3)BH with calculus. Notice the cross section area from different heights are similar to its base, if you put the invert the pyramid (vertex at 0 and the base at H) the volume of the pyramid is
∫ B(h/H)^2 dh from 0 to H. It becomes
(B/H^2) ∫ h^2 dh from 0 to H
= (B/H^2) (1/3)(H^3)
= (1/3)BH
I was 🤯 when I figured out, years ago, that a cone was a pyramid with a circular base.
Your formula makes no assumptions about the shape of the base, so you may as well generalize this further:
Given you have a two-dimensional figure where the area is known. Now imagine you create a three-dimensional figure by carrying the edges of the 2-D figure to a point P not in the plane of the 2-D figure. For example, if the 2-D figure is a circle, you'll get a cone; if the 2-D figure is a hexagon, you'll get a hexagonal pyramid, etc. Then the area of the 3-D figure is going to equal Bh/3, where B is the area of the base and h is the perpendicular distance from the plane containing the 2-D figure to the point P.
And now, what the heck, why stop at three dimensions? Given a 3-D object with a known volume which is turned into a 4-D figure by connecting it with a point P in the fourth dimension, then the 4D-volume of the 4-D figure is Vh/4, where V is the volume of the 3-D figure and h is the perpendicular distance from the space containing the 3-D figure to the point P in the fourth dimension. (I know we can't visualize this because we're stuck in three dimensions, but the math is valid, so the formula must be correct!) And, we can even generalize this for any n-dimensional figure turned into an (n+1)-dimensional figure by connecting it with a point P in the next dimension. The n-Dimensional volume will be Vb/n.
Yeah the cross-sectional method is really useful.....
👍👍👍👍👍❤
that's it? so I guess selling icecreams is better
I told you Calculus was sweet.
selling icecream IS INDEED better
I taught my kid the formula before he learned calculus. Essentially I told him that the volume is defined so that after cutting it to thin slices, if each slice has the same area for two objects, the two volumes are the same. Then I can teach him the formula by reducing the computation to that of a tetrahedron cut from a cylinder with a square base, which can be shown have volume being one third of the cylinder.
@6:25 the consistency of the compound curve s-shaped arrows across uncountably many BPRP videos is satisfying. isn't it?
Would you please explain how the volume of a sphere can be established.
Revolution volumes with integrals are so cool. It's a real pity we did not have time to teach it in my school.
You should have learned it in calculus II
@@Eric-xh9ee The spanish education system is just built differently...
It's bad, I'm not gonna lie.
@@Dantido Huh weird. This is a pretty core concept so it's good to know.
@@Eric-xh9ee Yeah, it's a real shame. We only went over limits, differentiation, undefined integrals, function areas with defined integrals, and some theorems.
I guess it's to leave space for matrixes and geometry, which is important, sure, but it's much less fun and more hard work-oriented than analysis in my opinion.
I actually ended up looking up more stuff about calculus myself. It's so damn interesting and fun, definitely the best part of math for me.
@@Dantido yours is good compared to ours lol. they don't teach matrix here, also integral is removed from the curriculum too. it is really shame. i think these are not overwhelming or ''hard'' if taught good enough.
If the Volume of the Cone V = 1/3 Pi r(2)h,where h is the height of the Cone,
and r is the radius of the Cone,therefore the Volume of the Cylinder whose
height is h and radius r,would be Pi r(2)h.The remaining Volume of the
Cylinder would be 2/3 Pi r(2)h by simple subtraction between Cylinder
and Cone.
Assumes the proof. But true.
When I was in secondary school, I wanted to know but my teacher skipped it saying you won't understand now,
Now I know it 😏
Honestly I don't think you need integration to prove this volume formula. I'm sure they'd figured it out many hundreds of years ago already and calculus is a kind of modern overkill here
calculus' foundation originated over 5000 years ago in the moscow papyruts or something. I forgot the name but I do remember they were moscowian @@jellymath
Excellent explanation of integration . .
my method before watching vdo....
a cone can be formed by discs of reducing radius one upon another...so integrating [πr^2 ]dh from h=0 to h=H where r/h=r/H by similarity of triangle...so u got the volume
Since the Triangle is 2 dimensional, the area of the Square is divided by 2. (1/2*a*h)
Since the Cone is 3-dimensional, the Cylinder volume is divided by 3. (1/3*pi*r^2*h)
A n-dimensional cone will be divided by n
got me with that missing 1/h then BAM fixed, what a relief
How I wish you are my math teacher during my school days. Thank you.
Always blaming the teacher even if you are stupid
Why don't you blame your parents
Go back from where you came 😂😂😂
Very good. Thanks
Or you can take a right angle triangle with h, r and l. And take area of that triangle along the perimeter of the base circle.
Nice application of high school mathematics. I still remember how astonished I was when I learnt this in my 3rd year :-)
Even better, get the area of the triangular cross section and integrate that from 0 to pi
integral of x^2 (base of any pyramid and cone) = (x^3)/3, assume C is 0, I think this is the simplest explanation?
Can someone explain why we can’t just say it’s bh/2 * pi
Since the area of a slice is a triangle, we do that in rotation of pi times to get the area no?
Very good video thank you man
Don't need to draw it on a graph and I use this method to derive the volume of any shape including tetrahedron and n sided solids
Just wondering... Can you calculate the volume by rotating the triangle along one if its side? The limt would be from 0 to 2pi
I think it's safe to say, seeing some of the ego-induced commentary on "rotational volume" assumptions and fortuitous problem solving gambits, etc:
I always thought systematic analysis of "x" along the axis for the 3D or "x cubed" problem is precisely what calculus and its differential/integral theorem is all about.
I see it. The proof is necessary, and so pedestrian. And I'm a philosopher with, um, a philosopher's more general "survey" interests.
Helps me to philosophically reflect on why I have never particularly relished the company of disaffected amateurs in any capacity, or egotistical number-smiths, anyway.
Thank the gods for a profession turned to clarifying bullshit into useful language and objects of relevance.
Kindly clarify once again how you are writing y= (r/h)x by using simple relevant examples?😊
Thank you
y=r, x=h , this implies that y/x = r/h , from this you can derive y = (r/h)x
Thanks!
why do I get the wrong formula when I use pythagorean theorem to get the value of y in term of r and x? When i used that i get the function pi(r^2-x^2)
Finally. Calc 3 along with analytic geometry.
This is wonderful.
Hey, math-man, greeyings from Istamboul, Turkiye. Please, for next: how to calculate the volume of sphere? Thanks beforehand…
Human know this fact long long before the calculas invented. Can you tell us How they did it ?
If the cone is not symmetrical, this method does not work. Although one can argue a “shifted” discs cone is equivalent to the symmetrical one. Can you do another video showing cone’s volume is generally using the same formula, please.
Beautiful explanatiom
How about the general formula to the volume of a n-dimensional simplex in the next video?
Wouldn't you say that 'r' varies along the horizontal axis?
how?
That was pretty awesome.
But why does (1/2rh) Multiplied by 2πr not work? Area of the triangle revolve around 1 circle, sounds right to me. Idk how to prove it wrong
When you rotate the triangle to sweep a circle, the outer edges of the triangle trace a larger circle than the inner, and thus account for a greater portion of the volume. Since the triangle is thinner at the outer edges, the smaller parts of the triangle contribute more to the swept volume. Meaning the volume is less than the 1/2 method you propose. If you do the integral, you realize the correct factor is 1/3.
You must integrate multiplying gives you the volume of a wedge instead
how did he found that y=(r/h)*x ?
2:34 why that need to times x?
So how did they do this without calculus?
ok when the height of the cone is perpendicular to the circle...
what happens if the top of the cone is inclined by any angle...?!
The same logic of infinitesimal cylinders and triangle would not work in case of SURFACE AREA. Expect a new video on that😊
Beautifuil proof.
I love it!
School taught what is formula
But never told WHY
Thats why I got 35 /100 in mathematics+geometry
And 90+ in Other Subject.
Mobile weren't invented those days
Why ⅓[πr²h] and not [πr²h]/3
It is literally the same, even in terms of clarity it makes absolutely no difference.
@@Ninja20704 But why do people choose to use ⅓πr²h instead of πr²h/3 when dividing by 3 makes more sense for most people?
Troll question!
@@channelbuattv I mean, we do the exact same with like the area of triangle (1/2)*b*h and also volume of sphere (4/3)*pi*r^3. I would believe it is because writing the fraction shows the relationship between the cone and cylinder that the cone is 1/3 of the cylinder (same as with a triangle and rectangle)
so we use the space efficiently. putting everything over small number would leave unusable space
at 2:30 you failed to explain the slope component variables individually. Where does the x come from in the scenario?
its Function when you come h from x axix the equ=r.h/h=r its our y axis
I am genuinely surprised he made me understand in just one viewing! 😯
i didn't understand the y=(r/h)*x part
The line has a slope of r/h (rise/run) and a y-intercept of 0. So writing it as y=mx+c we get y=(r/h)*x + 0 -> y=(r/h)x
it's "the curve" . it also has the +b part equal to 0, y=mx+b
@@jamescollier3 yeah ik i just forgot that the derivative is just a ratio rise over run and i was thinking why shouldnt m be like a normal number a for example
So I'm looking for an answer: why is cone volume 1/3 of cylinder volume?
Do you know why the volume of a pyramid is height*width*length/3?
Well you've come to the right place. Watch the video and find out.
Well, explain.
PEOPLE WHO MAKE: CONES, CYLINDERS,
ETC. AMAZE ME BECAUSE THEY CAN
VISUALIZE END. PRODUCT.
PRESENTED BY ENGINEER
Can you add a translation in Arabic?
As a student from high school I want proof by higher school math rule. I don't want integration and derivatives, i don't understand it
yes , you can find without integration ... euclides book 12 proposition 10 ... based on props before ... anyway in order to understand it is a bit a reasoning alike integrals .... actualy it is based on the fact that you can split up piramide in two smaller piramides of the same shape and two prisms ... , that smaller piramides you can do the same tric .... finaly you get a sum of piramides and prisms ... the greeks say that you can split it up as far as you want and that the sum of the small priamides becomes neglectable ... the other sum is kind of infinite sum in the sence of 1/4 + (1/4)^2+ ( 1/4) ^3 ... which leads to 1/3 ... , anyway the greeks say it a bit different , just to avoid the infinite sum like this ...
a more intuitive approach is to construct a piramide of two piramides of the same shape and two prisms ... if you assume that the two little piramides have 1/8 of the volume of the big one ( since every distance is devided in two ) and the two litlle prisms have the 1/8 of the volume of the surounding prism of the big piramide then you can avoid infinite sum by putting it in an equation where snales bytes its tail .... like 1/8 P + 1/8 P + 1/8 Pr + 1/8 Pr = P and this leads to Pr/3 = P
the approach of euclides book 10 is more or less the following ...o first they proof that the volume of piramide is proportional to its base , and proportional to its height ( not explicitely but indirectely as prisms are also proportional to their base ) .... then they cut 3 piramides out of a prism and proof that these piramides have the same volume ... the most difficult proof where infinity is a bit avoided is the proof where it is stated that the volume of a piramide is proportional to its base ... once proved for piramides , the step to cones is a bit already a sum of very small piramides which in a way seems a bit calculus ...
@@josleurs4345 that's pretty genius, thanks for your reply it was very helpful
Take a cylinder same as cone dimension fill the cone with water, amount of water to fill the cylinder is 3 times of cone volume. Volume of cylinder is pi x r^2 x h
using solid of revolution is much easier
You are amazing
How did Archimedes figure it out? He did not have calculus.
I shall remember this!
Yes, good. But, can you use the disc method and no calculus? Use only geometry, algebra and concept of limit.
Because that's what the integral of a circle as the radius goes to zero reduces to?
Still doesnt answer why it equals that. Only proves WHAT it equals. Completely different question.
Yes, a cone is a circle whose radius gradually reduces to zero. You've restated the question instead of answering it.
Doing volumes with calculus makes you feel bad for the poor old Greek codger who had to arduously come up with the formula by other means.
Is it a coincidence that it is 1/3 of the volume of the cylinder of same radius & height.
It's this "lawful regularity" that kills me.
How'd it all git like that?
Now I want a cone-shaped pie 🥧😮😅 with white icing 🥵 yummie
Your English channel is better than Chinese. English is much better used to interpret math than Chinese
Is there any other way except integration?
No, it can only be done with intrigation
I have no idea why it equals that. Probably is no reason, tbh. Why does there even have to be a reason?
1/3 Pi r²h formula = 9th grade
1/3 Pi r²h formula proof = 12th grade 😂
As a retired engineer. I do not care how to prove any formula. It has been proven for ages and I just use it in my former job.
Your marker hoarding shelf is full lol.
This is not really a challenge... Try without integral
Adakah penerjemah ke bahasa Indonesia
Nice
The volume of every pyramid is V = Ah/3, where A is the base area and h is the height.
I love it when these guys say they can prove a mathematical problem when the proofs are not even theirs especially not by any Chinese
Now, give the proof for centre of gravity is at 1/4 h
🇮🇳🇮🇳
这口条还真敢
You will also blame the video because hou understood zilch
Mh! That 1/3 came from no where
Sir plz in English
I started losing my shit trying to figure out why h^2 wasn’t in the denominator at the end 😂😂
Stop with your cursing. It is ignorant and needless.
cant you see?? like, its literally on the whiteboard
@@NaradaFox he made a mistake and corrected it later. I paused it before he made the correction trying to figure out what i wasn't getting, but in the end it just turned out that he made a mistake
@@robertveith6383 Bro it's the internet who cares if i curse
@@NaradaFoxDamn ignorant comment. Not everyone is perfect as you in maths.
Prove it without use of calculus
fill it with water you will get xlitter
To wszystko OK, ale
Matematyka to logika plus rachunki. Czy potrafisz wyprowadzić ten wzór tak by zrozumiał to uczeń, który ma 11 lat.
Po co strzelać z armaty do wróbla !
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That's all OK, but
Mathematics is logic plus calculations. Can you derive this formula so that a student who is 11 years old can understand it ?
Why shoot a sparrow with a cannon !
Because it's exactly 1/3 of a cylinder? End of video..
But why?
👍❤🙏🙏🙏
get out of here