Some people have commented that they don’t understand the point or that this solution is far too complex, since the torus is just a cylinder bent into a full circle. It’s a good thing to recognize that similarity and then form a hypothesis that their volumes would therefore be the same. But to automatically assume that as fact without first proving it is a dangerous thing. There are many examples in mathematics of things that seem obvious but end up being very counterintuitive. One example is the paradox described in my “Belt Around the Earth” video. I don’t really expect anyone to actually use this method if they genuinely need the volume of a doughnut. This is just a fun proof that demonstrates many important concepts in calculus. Thank you all for watching and for all the supportive comments.
@shikyokira30652 жыл бұрын
My first thought was that too, but after giving a second thought, I realize that the inner circle have smaller volume than the outer circle, hence you can't just think of it like a typical cylinder The correct way of viewing it is a cylinder with 1 side higher than the other, it would therefore won't be a cylinder And using this logic, u can actually find out the volume. Since the outer side of a torus has more volume than the inner side of the torus, we can just use the mean of both Rs (inner R and outer R), and use the mean R to get the mean R perimeter. This is the reason the big R in the video is in the middle So the breakdown of the volume is z=2πr V=πr²*z z is perimeter of the mean R. Exactly like the formula in the video
@marktolands96922 жыл бұрын
hey LearnPlaySolve Im sorry for posting that comment against you at first I thought this was a like one of those click bait 10,000,000 view videos but you actually did a really good job explaining the integral calculus of solving volumes when thier solutions might not be so intuitive. Keep it up!
@LearnPlaySolve Жыл бұрын
lol no worries! Thank you so much for watching and commenting. 🙂
@ebrahimudaipurwala37536 ай бұрын
I was working with tori for a math paper and I must say I have not found a derivation that is this well explained! Kudos 👏
@chalkao5071 Жыл бұрын
Beautifully derived! Thanks! I have not worked on this before, but from general volume formula (Base area * Height), if we cut the donut and straighten it up in cylinder shape, the Base area becomes the area of a small circle where as the Height that passes through the center of the donut cylinder becomes the circumference through the center of the donut. That seems to finalize the donut formula in short in seconds.
@mostafaelsokkary21583 жыл бұрын
great animation and thanks for your help
@LearnMathwithMrJerry2 жыл бұрын
Thanks for the video. I searched for many videos and this one is the easiest to get!!!!Love it
@dan-lh3px3 жыл бұрын
U could also use Area of a semi circle for the integral of sqrt(r^2-x^2) with limits -r to r
@Laahustaja3 жыл бұрын
Thanks a lot, helped me tremendeously with a similar issue on integrating on a shape that is not touching x-axis.
@div_072 жыл бұрын
x-axis? how bout you touch some grass? /jk
@ridwan66952 жыл бұрын
Bro I loved your explanation 🥰
@Chalisque2 жыл бұрын
It had me wondering why it is exactly this nice 2πr*πr² which would seem like a naïve first guess. Then it occurred to me to think in terms of a bunch of circles, and noting that the circumference of a circle is a linear function of its radius. Thus, for each circle of radius R+r, there is a corresponding circle of radius R-r, whose combined circumference is 2π(R+r+R-r)=4πR. Essentially, we can treat the torus as if it is all concentrated exactly on the circle of radius R at the centre of the torus, and get the right answer.
@JatPhenshllem2 жыл бұрын
How did you put that square sign on your r in 2.pi.r*pi.r?
This was the most complex explanation I ever saw, for a very simple problem. Just cut the donut such that you will get a Cylinder with a base area of pi*r^2 and a length/height of 2*pi*R.
@LearnPlaySolve2 жыл бұрын
It's easy to say that in hindsight, because of the result. This proves why that formula works.
@redtoxic87012 жыл бұрын
Yeah, the thing is to also prove that it works that way. Because visual representation isn't always enough, sometimes it can lead you to wrong formulas
@smalin2 жыл бұрын
The inside (minimum) radius of a torus, Rmin, is less than R, and the outside (maximum) radius, Rmax, is greater than R, so we know that the volume of the torus has to be between (pi*r^2)*(2*pi*Rmin) and (pi*r^2)*(2*pi*Rmax), but why would we expect it to be exactly (pi*r^2)*(2*pi*R)? If you can explain that in a simple but convincing way, then I will agree that it is a very simple problem.
@zerglingsking2 жыл бұрын
@@smalin yeah I had the same thought before starting the problem. I thought the different perimeters outside and inside the circle would have an effect on the volume so I integrated horizontal slices of the torus: rings of different width, which gave me the same result in the end.
@joonatan0033 жыл бұрын
Very helpful :)
@Ambigious2 жыл бұрын
Why am I watching this, I already know this from way back... But btw good explaination
@r.guerreiro1407 ай бұрын
Thank you :)
@reguret29762 жыл бұрын
underrated channel
@LearnPlaySolve2 жыл бұрын
Wow thank you very much!!
@Ireikes10 ай бұрын
Can you explain why you can change the limits like you do at 3:10? Is it literally just because r was used as the limit for the initial integral in terms of x?
@zerglingsking2 жыл бұрын
It's funny I did it the other way around : I integrated "vertically" rings of area 4*pi*R*(sqrt(r^2-y^2)) where y varies between -r and r and i get the same result! I thought integrating along the circle would maybe cause problems since the "speed" at which each side of circle move but I guess not 😅
@Bonzeaux_Bleuxgrene Жыл бұрын
That's actually very cool out-of-the-box thinking! 😃
@pitbull_cruel3 жыл бұрын
Very helpful video, thank you. May I use some of the images and drawings from this video for a school project? They are the best ones I've found so far and it would help me a lot.
@LearnPlaySolve3 жыл бұрын
Thank you for your kind words. I would be honored if you used it. Go right ahead!
@pitbull_cruel3 жыл бұрын
@@LearnPlaySolve, thank you very much. How should I quote you?
@LearnPlaySolve3 жыл бұрын
Here’s the link to an article I found on how to properly cite youtube videos: chat.library.berkeleycollege.edu/faq/166951
@pitbull_cruel3 жыл бұрын
@@LearnPlaySolve thank you again
@DerLiesl Жыл бұрын
Now I want to know how to calculus the volume of a donut made from donut (let's call it a tube torus)
@Sg190th9 ай бұрын
Would this also kind of like help that integrating the circumference gives you the area?
@Mathematics_tv2 жыл бұрын
You are genius
@TristanEllison3 жыл бұрын
Great video! I need to calculate a volume for a sector not a full circle. Example would be a pizza slice from point R to 1pm and 2pm (on a clock face) then rotated around the x asix just like the torus. Any idea for that formula or where I could look for help?
@LearnPlaySolve3 жыл бұрын
Thank you very much! That’s an interesting problem, and it sounds like a fun one to solve. You would have to separate it into two separate integrals at the point where the straight line of the sector (radius) intersects the curved part (arc). It’s kinda hard explain in just words. I might make a video for that kind of problem in the future.
@TristanEllison3 жыл бұрын
@@LearnPlaySolve www.dropbox.com/s/i38t3g3ugupfbhd/Circle%20Sector%20Torus.PNG?dl=0 That's a quick sketch I've just made, in case my technical terms about pizzas wasn't clear :) I can easily calculate the volume with CAD but I want to be able to do it with formulas. Your video is the best explanation I've found with the step by step workings, but I'm stumped on how to apply it to my problem.
@TristanEllison3 жыл бұрын
Pappus-Guldinus Theory, that's what I need. I can calculate the area, find the centroid point rotate it into its correct position then just do V = A 2pi y where y is the centroid distance from the x axis
@LearnPlaySolve3 жыл бұрын
To be honest, I’m not familiar with the Pappus-Guldinus Theorem, but I’m excited to learn something new about solids of revolution, especially if the theorem provides a faster and easier way of finding their volumes. So thank you for bringing that to my attention.
@TristanEllison3 жыл бұрын
kzfaq.info/get/bejne/kLemYMh3nZaqcWQ.html&ab_channel=MichelvanBiezen A series of videos by Michel van Biezen.
@maholly2893 Жыл бұрын
hi, sir. I have a question. What's the purpose of doing Trigonometric Substitution in the integration process?
@LearnPlaySolve Жыл бұрын
If your integral is an algebraic expression that "resembles" a trigonometric identity, then you can perform a substitution which turns it into a trigonometric expression. This allows you to exploit certain properties of whatever identity it is, and rewrite it in a way that is far easier to integrate. I hope that helps.
@Jasmine-ww7jl3 жыл бұрын
Hi, thank you for the helpful video! Can you explain why dx=rcos theta d theta at 2:53?
@LearnPlaySolve3 жыл бұрын
Thank you for your question. Anytime you solve an integral via substitution, you must remember to also change the differential. In this case, since we substitute x with rsin(theta), we take the derivative of both sides of that substitution to get a new differential. x becomes dx, and rsin(theta) becomes rcos(theta)dtheta. Now we have something in terms of theta to replace the dx. Remember, the derivative of sine is cosine. I hope this helps.
@Jasmine-ww7jl3 жыл бұрын
@@LearnPlaySolve got it, thank you!
@matr1x_glitch3 ай бұрын
I SAY WE MAKE DONUTS SQUARESSSSSSS
@dave_lawrence2 жыл бұрын
Hmmm, if I have a hosepipe of internal radius r and it is laid in a straight line of length 2.pi.R, then its volume would be 2.pi.R.pi.r.r - what your result is. If I arrange the hosepipe in a circle so that its end touches its start, the outer edge travels more than 2.pi.R going around the circle and in inner edge less than 2.pi.R (actually 2.pi.(R+r) and 2/pi/(R-r)). It seems that the lesser volume of the inner section (less than R from the centre of the circle) is exactly compensated by the extra volume from the section greater than R from the centre of the circle. A lucky coincidence?
@LearnPlaySolve2 жыл бұрын
Probably not a coincidence. However, what if you made that assumption before you really knew whether it was true or not. How would you prove it? Many things in math may seem obvious but turn out to be counterintuitive.
@dave_lawrence2 жыл бұрын
@@LearnPlaySolve I love maths for this reason. Next postulate: is the volume of the hosepipe always 2.pi.R.pi.r.r, independent of it's shape?
@LearnPlaySolve2 жыл бұрын
That's a good question, and it really depends on what you mean by r. But, if by pi*r*r you mean the area of the cross section, then yes.
@sinexitoalmiedo Жыл бұрын
Hi, at minut 3:51 you factor out the 1/2 but what happen whit the one "1" before cos(u) du??
@LearnPlaySolve Жыл бұрын
Yea I can see how that's a little confusing now that you point it out. I probably should've put parentheses around the (1+cosu). The differential du always belongs to every part of the integral, not just the last term.
@sinexitoalmiedo Жыл бұрын
@@LearnPlaySolve mmmm, i mean, if you factor out 1/2 of "1 + (cos u)/2" the resull is 1/2(2 + cos us). But you factor out 1/2 of "1 + (cos u)/2" and you let as "1+ cos u" I m sorry, that realy confuses me. It something that i am not seeing why
@LearnPlaySolve Жыл бұрын
You don't factor it out of the (1+cosu) because it was never distributed to those terms. The 1/2 is only with the du, not the cosine. Its just a matter of moving the 1/2 from the end of the integral to the beginning.
@sinexitoalmiedo Жыл бұрын
@@LearnPlaySolve yep, i see that, i realy was confusing, haha, thank you, you are the best
@the_eternal_student10 ай бұрын
The part where you describe plugging the equations for the top and bottom half of the circle was confusing; it was not clear what "it" you were referring to when you said "under it".
@LearnPlaySolve10 ай бұрын
Sorry about that. I should have explained better. The top equation represents the top half of the circle. So using it in the equation means you are taking all of the area under the the top half of the circle, all the way down to the x-axis, and revolving that area around the x-axis to end up with a certain volume. Likewise, if you do the same thing with the bottom half of the circle, and then subtract that from the previous answer, you will be left with the inside of the circle rotated about the x-axis. I hope that makes more sense.
@George6r42 жыл бұрын
If we rotated a triangle around a point to make triangular looking donut, would the volume than be V=(base*height)/2 * 2*pi*R?
@LearnPlaySolve2 жыл бұрын
That’s a very good question, and the answer is yes, but only under very specific conditions. If you had an equilateral triangle and it was oriented just the right way, and the radius went up to the exact center of the triangle, then yes it would work. But such restrictive conditions in my opinion make it an unreliable formula for anything other than a circle.
@George6r42 жыл бұрын
@@LearnPlaySolve funnily enough, I’ve actually tested it using spaceclaim (3D modelling software with a volume function) and the equation works for any triangle and I think any shape. You do need additional information as you suggested. The surface area of the shape is one, but the most important information is the centre of mass of the shape. So for a triangle the centre of mass is H/3, so if your triangle is pointing out of the donut it’s R+H/3 in if the triangle is pointing in its R+2H/3. Where R is the radius of the donut in your equation. Thanks for your response.
@LearnPlaySolve2 жыл бұрын
Very interesting. Thank you for sharing it. I tried with an isosceles right triangle oriented with the hypotenuse down. It works, but if I rotate it so one of the legs is down, my radius has to change to accommodate the new location of the triangle’s center.
@vinuthomas71932 жыл бұрын
As diagrammed, isn't x = r cos theta? The answer would still be the same, of course.
@vinuthomas71932 жыл бұрын
When factoring out the 1/2, shouldn't the 1 change to 1/2?
@LearnPlaySolve2 жыл бұрын
That's a great point! Although we can substitute anything for x that helps us evaluate the integral, it would have been less confusing to use r cos theta. Thanks for the input.
@circleoffifth90482 жыл бұрын
Just cut it and make it a cylinder Volume =base*altitude =πr2* 2π*(Ro-R1)/2
@Bonzeaux_Bleuxgrene Жыл бұрын
Read the pinned comment
@veronicalin24926 ай бұрын
why x=rsin theata not rcos?
@LearnPlaySolve6 ай бұрын
Either one would work. I just like to end up with a positive derivative.
@rbettsx2 жыл бұрын
Or.. just cut through the donut, bend it straight, and take the volume of the cylinder?
@LearnPlaySolve2 жыл бұрын
Yes... this demonstrates why that works. Thank you.
@mujtabaandrabi54102 жыл бұрын
gurvir singh lol, your 7months l8. But cheers m8
@aliberro2 жыл бұрын
The way I did it was by thinking of it as a flexible tube, cutting it in one place then finding the area of this cylinder, which has a length of 2piR and a cross-sectional area of pir^2, thus giving 2R(pi r)^2
@Bonzeaux_Bleuxgrene Жыл бұрын
Read the pinned comment
@marktolands96922 жыл бұрын
its just a cylinder idk what the big deal is lmfao
@circleoffifth90482 жыл бұрын
I was also thinking the same...😂😂
@marktolands96922 жыл бұрын
@@circleoffifth9048 hes out here acting like he solved E =MC^2 and its just a tube fr lmfao