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Proving the volume and the surface area of a sphere by using integrals
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- Опубликовано: 17 авг 2024
- Proving the volume of a sphere with radius r 0:00
Proving the surface area of a sphere with radius r 6:14
For more calculus tutorials, check out my new channel "just calculus":
👉 / justcalculus
Volume: • Volume of the Solid of...
Arc Length: • Arc Length (formula ex...
Surface Area: • Surface Area of Solid ...
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Happy birthday Waikean!!!
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@@waikeanng145
My pleasure!! Thank you for supporting my contents!!
Happy birthday!!
@@rot6015 thanks!!!
Did u click the video bc of the thumbnail? If so, give this video a like! Thanks!
I clicked because of maths!
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M A I L T I I I I I I I M E
Derivative of volume is surface of sphere,coz:
think u have a sphere with radius R ,and inside of that sphere there another smaller sphere with radius r volume of the big sphere is V=4/3 π R³
volume of small sphere v=4/3πr³
Volume between big and small sphere
ΔV=V-v and width of that volume Δr=R-r
When that width of the volume between 2 spheres becomes soo small it will be surface of sphere,mathematically
lim ∆V/ Δr when Δ r->0 its surface of sphere,and this expresses derivative of V..
please try integration of xcotx dx with limits as upper limit as pie/2 and lower limit is 0
Be careful about the equasion you used:
Y=sqrt(r^2-x^2) has two solutions it has a positive solution and a negative solution. So it would still be a circal. Also you said it was a function. But it is a one to many, so by definition it is not a function. But essentially wouldn't the derivation, if done with the correct tow solutions work out anyways, rather than doing it one way?
One of the greatest mathematical awakenings I received was during a lesson on area and volume. x² and x³, as x increases, volume increases faster than area, and this is why even though you may not look like you've gained or lost weight on the surface (area), your volume has. That made mathematics personally relevant for me at that moment.
The derivative of the volume is the surface area formula 🤔
DLKL that's so cool 😳
With respect to r
@injustice fellow Not quite sure about the mathematical explaination, but if you think of a circle and increase the circumfrence, the area increases (change in amount -> derivatives in play). Same thing goes for the cone from the video. If you increase its (average) radius, the surface area increases. This is again a situation in which a function (the area) depends on a changing variable (the radius), which indicates a differential relationship between them.
If you apply this to the surface area and the volume, you find similar results. This time being that an increase in area, also increases the volume (since everything in circular/round). -> Differentiating the volume (consider the volume as the function) gives the surface area.
Imagine a sphere of radius r then if radius is increased by dr then there is a shell of extra volume dV now as dr is very small the outer surface area and the inner surface area are basically the same say S so the volume of the shell would be equal to the area of a cuboid with base area S and height dr
So dV = S.dr
Or dV/dr = S
@injustice fellow think u have a sphere with radius R ,and inside of that sphere there another smaller sphere with radius r volume of the big sphere is V=4/3 π R³
volume of small sphere v=4/3πr³
Volume between big and small sphere
ΔV=V-v and width of that volume Δr=R-r
When that width of the volume between 2 spheres becomes soo small it will be surface of sphere,mathematically
lim ∆V/ Δr when Δ r->0 its surface of sphere,and this expresses derivative of V..
Integral of surface area in terms of r (4πr²), 4πr³/3 = 4/3πr³ which is volume
oh yeah yeh some straight forward ideas
Careful with notation. You used r for two different things: the fix, constant radius of the sphere, and the variable radius of the slices. They should have different names.
He wrote with different pens
That comment at the start was pretty inspiring. Can't exactly give Such a story, but your videos have helped massively. Keep it up!
Matthew Stevens
I know everyone is unique. thank you!!!
I wish I was in your class.
if you were in his class you will be teaching with him right now .
I wish I was dead
@@MikehMike01 thats a bold statemet
Wow, Toma Kolev's words almost made me cry
This might be my favorite video of yours, I've always wondered how this would work out with integrals
HelloItsMe thank you!!!!
I actually realized on my own about a year ago that the derivative of a sphere's volume is equal to its surface area. What I also realized is that this is not limited to 3D spheres, it also applies to 2D circles. The derivative of a circle's area is equal to its circumference. It seems that in both 3D spheres and 2D spheres (circles) the boundary of the object is equal to the derivative of the interior that is bound within it. It makes me wonder if the same is true for spheres in higher dimensions. Is the 3D surface of a 4D sphere equal to the derivative of the sphere's hypervolume?
it’s so sexy to read this.
One would logically assume this to be the case
I believe this follows directly from the definition of an integral. When an area is changing according to some function, the boundary which is changing is by definition the rate of change of that area, the derivative of said function. This is seen most clearly by looking at the area under a curve.
It's well known that the area under a curve is given by the integral, and taking the derivative of this integral brings you back to the original function that generated the curve. But what is this function but a description of the boundary of the area which is changing? As x increases, the height of the curve at x is the right-most boundary of the area described by the integral, so the area changes according to the height of that boundary.
lets integrate the volume of a sphere then
@@rhombicuboctahedron7811 Ya, that will be πr⁴/3
I dont think that the integration steps for the surface area will be more difficult because....
A TRUE MATHEMATICIAN NEVER CONSIDER A SOLVABLE PROBLEM A DIFFICULT PROBLEM
Ah yeah that's why they officially call some problems "Hard" ("NP-Hard") 😂
Noob: Getting formulas by copying from others.
Normal: Getting formulas by using shapes
Pro: Getting formulas by using calculus.
Engineer: Getting formulas from Google.
@@brianpedigo5519 Engineers = Noob
It's actually far easier to do this instead of the shapes, IMO. One very easy example is the cylinder.
OH MY GOD
THE DERIVATIVE OF THE VOLUME IS THE SURFACE AREA
AAAAAAAAAAA
Wanna know what’s even better?
The integral of the surface area is the volume
@@jcers nobody would ever guess *t h a t*
I did the same thing for a torus as well. It's amazing how these formulas can be derived w/ calc! #yay
Thank you so much bro.......u have developed a way to solve such a lot question...........iin me......❤❤❤😊
Wow. I really appreciate guys that knows numbers ☺️☺️
the derivative of the volume (4/3 *pi*r^3) is the surface area (4*pi*r^2), just like how the derivative of a circle's area (pi*r^2) is the circumference (2*pi*r). Interestingly, for a square the perimeter (4*length) is twice the derivative of the area (length^2) and same with the cube's volume (length^3) and surface area (6*length^2).
Yeah
I really like the following explanation for why you get the volume when integrating the surface area of a sphere with respect to the radius: imagine slicing up a sphere of radius R into many concentric spheres with radii ranging from 0 to R and labeling any specific radius of a tiny sphere r. The volume of the big sphere should be all those differences in volume between adjacent individual spheres added up, right? Two adjacent spheres with radii only a tiny distance apart will have approximately the same surface area, so the change in volume is approximately 4*pi*r^2*dr. So we have a sort of weighted sum of individual surface area values from 0 to R - i. e. an integral of 4*pi*r^2 with respect to r from 0 to R.
Of course, to make this rigorous, you’d have to formulate everything in terms of limits, but I still like this argument because it generalizes nicely into higher dimensions and cuts to the core of what’s going on.
Also, I find Toma’s story really touching. It’s really unfortunate that people have to go through this, and hearing that he managed to overcome this horrible situation and that he’s now doing better feels good. Since there’s a good chance he’s reading this, I’d like to wish him the best of luck and enjoyment in his studies.
I had my exam in volume last week! I searched high and low for any video on your channel and now you make one! BUT STILL, LOVE YOUR EXPLANATION! GUESS I'll Come back during finals.
Your videos are awesome 🙂 thank you for making maths more understandable
We can do it by trig also. The surface area :
thickness of elemental disc = dt
From circle relation, dt = r * dg
( i am using g instead of theta )
Circumference of disc = 2 pi r
But r = R sin g from triangle formed
Surface area = integral (2 pi r dt)
= integral 2 pi R sin g R dg
= 2 pi R^2 integral sing dg from -pi/2 to pi/2
= 2 pi R^2 * 2
= 4pi R^2
Someone pls note 😢😢
So, how come we can approximate the volume using 'cylindrical' discs but not for the surface area ? In both cases we are integrating about a 'curved' shape whose gradient at any point is changing.
The derivative of the volume of a sphere w.r.t. its radius is the surface area. To see why, notice that we could have derived surface area first and then derived volume from it by working in polar coordinates.
The volume of a thin shell of the sphere at a distance r from the centre is dV = S(r)dr
in other words, dV/dr = S(r)
Hey sir, how did u get the expression for dL? Sqrt(1+(dy/dx))dx
& Surface area is dv/dr
taking the derivative of volume results in surface area, i think i did something like this in my calc 3 class
wait no not calc 3
matrices are really annoying I agree matrices are really annoying
@@fanyfan7466 matrices are very usefull in calc 2 like the jacobian matrices are good way to express some nasty derivates
@matrices: That will work for any shape whose volume can be built up by normal (perpendicular) integration of its surface area.
EDIT: I have now verified this for the regular tetrahedron and the cube, as well as a cylinder whose height = diameter of its base, using as r, the radius of the inscribed sphere.
In all these cases, dV/dr = S.
I believe it will also work for the other 3 regular polyhedra; and for any solid whose shape is kept constant as its size changes, using the radius of the inscribed sphere, at least so long as that sphere is tangent to every "face" of the solid.
Fred
Pedro Lopes maybe I’ll like them better when I’m in calc 2 then. Still in calc 1 rn
The derivative of the volume formula is the surface area formula.
Geometrically speaking this is because a really tiny change in the radius can be interpreted as wrapping another surface area around the sphere.
Is it a coincidence that the derivative of the sphere volume is the surface area?...Fascinating!
No, it is not just a coincidence. If you imagine the sphere as a summation of thin spherical shells, each of thickness dr, each incremental shell has a surface area of 4*pi*r^2. Multiply by the thickness, and get 4*pi*r^2*dr as the volume of each thin spherical shell. Add this up, i.e. integrate, and get integral 4*pi*r^2 dr.
As you can see, the integral of the surface area relative to radius, is the volume. The fundamental theorem of calculus tells us that this also implies that the surface area is the radius-derivative of the volume.
You can use this procedure to calculate the space in any sphere of N-dimensions by integrating the previous (N-1) dimension sphere and adding a small piece of sphere into the new dimension (analog to adding a Z component into the circle and making it into a disc)
Triple integrals technique (spherical coordinates) is much easier than the convenient way but I like it though because it’s the principal method.
I noticed that the derivative of the volume respect to r is amazingly the surface area
I was a math major from 1970 to 1974
Back when chalk was still being used!
My professor loved to hammer on the blackboard
Till the chalk splinters flew,
Especially across the cracks between the board sections.
He used a metal cylinder chalk holder
And enjoyed torturing us students
By the screeching it made upon the board
When the chalk was worn down.
These videos bring back many memories of those days
But I wonder what our professor would have
Thought about the blackpenredpen technique??!!
The volume is clearly the integral of the surface area from 0 to r with respect to r. Similar to how the area of a circle is the integral of the circumference from 0 to r. So basically, one must be careful here because it's not gonna be dx but rather dr in that case.
Best teacher you're .... thanks you! Pape Senegalese
i'm not clear on why you can't use a disk, if the slices are infinity small would a cone section and a disk not be the same thing?
Volume can be defined as the sum of all the areas of the spheres an the radius decreases by dr. This is the definition of intergal so intergal of A dr= 4/3 *pi r^3 so differentiate both side to get 4 pi r^2
but how do you know that y=sqrt(r²-x²) in the first place ?
Stiven Andria equation for a circle is r^2=x^2+y^2
How is x of dx same as x of y=root(r²-x²)?
Sir for volume you take cylinders of thickness dx but for surface area you take small frustem of dl why ?
I am not able understand can you please help why we not take the hollow cylinders in the surface area case plz ..
All the heights coming from the circumference must be squeezed into a length equal to radius. Note; from the interval 0-pi/2 , if r is 1 then pi/2 is larger than 1 likewise if radius is 2 then for that interval it's 0-pi and so on.
Actually why do we have to integrate surface of cones? In the dx world „every line is stright”, so every (truncated) cone should just be a cylinder. So it seems enough to integrate cylinder side area.
dS = 2 pi sqrt(r^2-x^2)
Am I wrong?
I mean for the volume, even if dx tend to 0 the radius of the first base is smaller of de radius of the end base : it’s when you “slice the sphere”
Can we just ignore it ?
6:13
I think the surface area integration is harder only because you always do the harder integral at the end.
This is really cool. In fact I was just trying to do this this week!
The problem is finding dL... What if this is one one the wrong calculation that gives right results?
In this kind of task only note the 1st order differences. What if you sum section of cones by infinite sum instead? The dL is like piece of cone, but in this calculation case its piece of cylinder.
Reminds the case lim x->inf f(x)/g(x), where the largest polynomial terms determines the results.
Rsinx is simply reconfigured to Sqrt(R^2-X^2), i.e all of the heights at each radian(x) are rotated 90 degrees and placed and positioned differently.
If you have take dL in surface area but why not in volume
I found on wiki another way to derive surface area if you know volume is by using concentric shells stacked on each other that approximate the sphere.
The way he always smile to the camera that mean he love math so much. :)
2:27 Is not pi*r^2*thickness. Is pi*y^2*thickness. You already used the r for the sphere radius.
At 1:13, 2:27, 3:41, 4:56, 6:10, 7:24, 8:38, 9:52, 11:06, 12:20, 13:34, and 14:48 (so basically every ~74 seconds), the message from the first few seconds pops up for a frame, it is unsettling and catches me off guard, trying to watch the video :/
Edit: this has been fixed since I posted my comment.
Also the calculus connection between the surface area and the surface area of the sphere is the rate of change of the volume of the sphere, so the volume is a primitive function of the surface area of the sphere.
I am very sorry I actually did NOT mean to edit the video like that. I am actually not sure why the video turned out to be that way. I can reupload another one later.
That's weird, it ain't happening to me, I see it the normal way.
I don't see it. Not even on 0.25 speed.
@@diegomullor8605 The reason might be that either your monitor's/phone's framerate is comperatively low or your browser's video player does not support this framerate, those pictures show up for a single frame or so.
@@gergodenes6360 makes sense, I'm watching from the phone and the screen it's 30 fps
Why didn't u use dL in case of volume as well, why dx? ? I didn't understand it from ur video. Please explain clearly.
Bprp, you should help out all of us and do some more videos on solids of revolution. Keep up the great videos!
Volume of the sphere is the integral of its surface. That's the connection.
Do you mind if you explain that "best friend" formula?
x g yes. Check out newest video.
This is probably completley wrong, but I think there might be a pattern for the area, volume, ect. for anything in a circular shape in any dimension, which is this: where d= the dimension, V=(d+1)/3•pi•r^d. In the second dimension, the formula would be (2+1)/3•pi•r^2, which simplifies to pi•r^2. In the third dimension, the formula would be (3+1)/3•pi•r^3. Both of these are the correct formula. Assuming this is correct, the volume of a 4d sphere would be 5/3•pi•r^4
search for volume of balls in wiki
How beautiful is the Integral Calculus! :'D
why dL is equal to square root of (1 +dy/dx)dx?
Thanks u helped me to pass in the exam thank you
Now do it in polar coordinates :D
*Microphone is a ball???*
I remember this formula from geometry honors class way back in middle school. The surface area of the sphere is the derivative of the sphere's volume.
The derivative of the expression of the surface area wrt to radius is the expression for the volume
I will never forget to multiply by the "chen lu"
Why for volume dx works, and for surface area it doesn't?
I'm very curious about the answers you will get... :-)
You can see how we derive the volume and surface area in the description
The differential element, the cone in this instance, no longer has a horizontal small thickness, dx. For the cone, the small thickness is slanted and is represented as dL, which is not as simple as the dx or dy you are used to.
@cypke: dx still works; it's what you have to put in front of it, that's different!
When finding the volume, the volume element is just a "plate," whose volume is a cross-section area times the thickness; dV = A(x)dx.
When finding the surface area, the area element, dS, is a sloped, or beveled surface; the lateral surface, not of a cylinder, but of a frustum of a cone. So calling the slant height of that frustum dL, as bprp does,
dL = √(dx²+dy²) = √(dx² + [ d[√(r²-x²)] ]²) = etc. = r dx/√(r²-x²); and then,
dS = 2π√(r²-x²) dL = 2πr dx
Fred
@@nepraos3151 That can't be right. Such an ellipse has an area of π²r². And there is no ellipse in this problem.
Where are you getting that?
Fred
The Surface area is the derivative of the volume: dV/dr = d/dr(4/3pir^3) = 4pir^2 = S.
That is also true for a circle, but is it true for other things?
Aaa Aaa This video might help you, is not about Volume and Surface area but I think it can help you understand
Your videos are great!
Thanks
something weird is if you take the formula for the volume of a sphere (V=4/3π*r^3) and then take the partial derivative of V with respect to r (𝛿V/𝛿r) you get the surface area of the sphere (SA=4πr²) not sure if there is any connection but thought i would throw it out there
Differentiate the volume expression we get the expression for Surface area
and vice versa (intergrate surface area expression and we get volume)
Very nicely explained! Will you make a similar video for the volume and the surface area of an hypersphere in higher dimension?
Thank you! I found this very useful
Volumes of revolution!!!
whr i can get the expl. for dL
Integral of surface area is volume as int.(r^2)=r^3/3.
The volume is the integral of the surface area with respect to r?
Sir, i dont understand why the height of the integration for volume is dx and not dL? Like you said, if you add up them, it forms a rectangular figure or cylinder right?
Oh okay there’s a video you talked about that thanks i watched it.
can u send the video link@@martrayleigh7988
2:26 the glitch in the matrix
...also at another 15 points during the video. Very annoying!
Someone please help me out to understand why is dL used in SA
感謝老師講解
不客氣 😄
Rotating volumes is probably the toughest topic in Calculus II for me, I have my final exam in two days.
Thx!!! This channel alot help me subscribe!!
great video, i use 2 colours to show math in my classes as well!
I watched your Gabriel's Horn video yesterday and it occurred to me after watching this one: would a one dimensional hyperphere have a volume of 2r and a surface area of zero?
Hey @blackpenredpen ..... Please tell me that for finding surface area why don't we multiply 2πy with dx ???
If I only knew u a few years earlier
Of course! The area is just the derivative of the volume with respect to the radius!
Thank you very much!
Could you explain the logic behind the area of cycloids(3pir^2)
Steve plzzz also do a video on finding centre of masses using double and triple integrals
can you do the proof using parametric equation
The Calculus Relation: The Derivative of the Volume of a sphere with respect to its radius is the Surface area of the Sphere.
well the calculus connection between the surface and the volume kinda looks like a derivative/Integral relationship. if i differentiate the volume in respect to r, gimme a sec V=4/3pi r^3 dV/dr= 4pi r^2. but given i suck at anything that even gets near geometry i have no clue why it should be the derivative. (or the integral. i guess it´s similar to if you integrate a circles circumference over the radius you get the Area inside, which is why i´m not surprised in the slightest (but as i don´t even know why this is the case except ,well that´s what you get for integrating/differentiating with respect to r i wouldn´t dare to confidently say it is even the same connection.)
Incredible as usual!❤❤❤
Then why can you use a disc for volume but not the surface area? I mean in both cases the sides are both ' curvillinear'. Never did get a satisfactory answer for this. Could you possibly elaborate? Thanks for your videos, always appreciated.
With regards to volume you integrate the function pi*r*r over the domain [0-r], so at each point between 0 and r you have an area which either has x or y representing r , if you choose it to be x then you integrating over y and vice versa.
The reason you have to use double integration with surface area is because you don't have a function to integrate over a domain. Remember polar coordinates (rcosx,rsinx) , x in radians here. For every sinx there is a corresponding cosx albeit at different radians over an interval. So if you decide to integrate 2pi*rsinx then you must multiply rsinx by r again since for every height there is an equal horizontal, you will realize this when you look at sine and cosine graphs. Integrate 2pir*r*sinx over interval [0-pi] and you will get 4pi*r*r.
@@singh2702 Thanks for the reply...but I don't see a 'double' integral anywhere in this picture.
@@isobar5857 My mistake but since sine and cosine have a equal one to one correspondence at different radians , either one of them have to be multiplied by r twice over an interval, and then integrated.
@@singh2702 Forgive my ignorance, but I don't see a derivation in this picture that depends upon sine or cosine. I don't see the relevance.
@@isobar5857 How can trig not be relevant when we dealing with circles?
What's the intro Song?
The derivative of 4/3 pi r^3 is 4pi * r^2 but i dont see why ^^'
Its also strange that u can cover a sphere with 4 discs which have the same radius
@@Sharkyeltiburoncio I'm pretty sure he knows that, I think like me, he is also wondering the geometric reason of why that happens
impossible to explain it better than 3B1B, buy obviously you guys didn't even try to watch his video...
@@sergiokorochinsky49 there are an insane amount of videos about this on yt. to blame people for not having watched one of them is beyond ridiculous
Fractal, your opinion is valid, as it is mine. BPRP specifically invites the viewer to watch that video, and leaves the link in the description, so my expectation is not that crazy. And, if there are so many videos on the subject, watch at least a couple and find out the answer for yourself!... Just my opinion
@@sergiokorochinsky49 but why would you assume its the 3b1b video they will find?
I figured out the volume first, I came here just to see the latter part of this video.
Isn't that a frustum instead of cylinder??
What kind of watch do you have? It's looking very interesting
Hey man, just looking at your custom shirts, do you ship to NZ?
True volume / surface integrals:
V = ∭ρ^2*sin(φ)dρdφdθ
S = ∬ρ^2*sin(φ)dφdθ