Prove: Volume of a Torus (Shell Method)

[@okoch2967]

your video goes crazy fast, but it totally just saved my life. thank you so much!

[@sergiokorochinsky49]

may I introduce the "physics" method? Cut the torus and unroll it to make a cylinder. By symmetry, you stretch the inner side as much as you compress the outer side, so the volume is still the same. The base of the cylinder has an area of pi*r^2 and the height is 2*pi*R, so the volume is 2*pi^2*r^2*R. tah dah! :-)

[@claudemiropacheco]

I think I like more your method :)....But it's fun to see this guy solving those problems. He is like a freaking machine spitting numbers and formulas

[@RAJSINGH-of9iy]

Yeah i also thought about that, when i see the videos of it by washer method. But i have a doubt. Let i cut it and make it a cylinder such that no volume changes. Then area of base=pi*r2. But there are two heights. Onr is 2*pi*R and another is 2*pi*r. As both are diff, then it can't be a cylinder. I hope u understand what m trying 2 say. Coz this type of shape can not be made by cylinder. Its like outer line must have greater length than inner. However,That is also equal to the same result by cancelling and adding of some volume. Like rubber band ,when we cut it, two lenghts are not same.

[@yudong8820]

nice proof! and we can also proof this by using the Theorem of Pappus, the volume is the product of area pi*r^2 and the distance travelled by the centroid 2*pi*R.

[@truonghung101]

This video is toooo good, that I can understand immediately after 1 play. Thanks so mucchhhhh

[@btmillion2813]

Thank you for such a coherent explanation.With this talent at instruction, you could make explaining something like Physics C intuitively rewarding.

[@LetsSolveMathProblems]

Thank you so much, BTMillion! I have personally taken AP Physics C (both Mechanics and E/M), and I would love to make some videos about it in the future; physics is such a fascinating subject built on the music of mathematics.

[@DrQuatsch]

is it me or is your definition of lower case r in the beginning different from what you later on use? you defined r as the radius of the hole, but later on you use the radius of the circle of the torus.

[@LetsSolveMathProblems]

Oh, wow! You are absolutely right! I did not see the mistake at all until you pointed it out. The lowercase r should always be the radius of the circle rotated to make the torus, not the radius of the hole. I sincerely apologize for the lack of consistency. Thank you so much for finding the error! =)

[@PhongNguyen-fs1gp]

I love the way u taught, u are legend

[@Jasmine-fj6vu]

MY MIND IS BLOWN you are amazing thank you i needed this

[@marie-antoinettemanzan430]

Did I just subscribe to somebody else besides TOCT? Amazing work. This would be fun if I wasn't in a time crunch doing all my assignments, but thank you so much!

[@puppylove33806]

I love fast teachers! Idk why but it makes way more sense than the slower ones haha

[@arturoreina4292]

Excellent content, keep it up man!

[@samuelyeo5450]

Hi sorry could you explain 10:42? How come you can change the definite integral into an equation for a circle? I understand that the function within the integral is that of a semicircle but I don't understand the implication? Thank you for the video by the way. EDit: Oh I understand now. Good video!

[@willowz1378]

great video! tysm for your explanation

[@javieraranazelaya2248]

If you are integrating and odd function but the bounds aren't equal and opposite (for example b, -a. where b and a are not equal), would it still cancel? or does the condition for it to cancel require the bounds to be equal and opposite (a, -a).

[@javieraranazelaya2248]

oh nvm you explained it in the video

[@mtdrei]

goated video thank you

[@GoldenMada1]

Thanks so much mate

[@LetsSolveMathProblems]

Glad it helped. =)

[@itstimi-XD]

Thx so much😭❣️

[@tyd2649]

some to calculate the v of torus, we only have three methods in total? is this right? And love your ascent, so lovely^

[@puppa8619]

and they are 1) Pappus's Centroid Theorem 2) Washer Method 3) Cylindrical Shell Method right?

[@zenicv]

great video (though I needed to watch at 0.75x speed ;-) )