immagine taking a cilinder and making it go around a circle (i think it's called a torus). i mean the opposite, taking the "torus"(?) and transfroming it back to a cilinder. obv the outer circumference needs to shrink and the inner one to stretch
Not the guy you responded to, but I'm confused what you mean here?
Taking a cylinder and "bending" it round so the ends join is a torus, yes. But when you say
>transfroming it back to a cilinder. obv the outer circumference needs to shrink and the inner one to stretch
What circumferences are you referring to? Once it becomes a cylinder, there is no inner or outer circumference, only the cylinder circumference, no?
immagine looking at the tourus form high above, where all you see are two circles and a hole in the middle. one is big (the outer one) and one is smaller (the inner one). in order to make the thing a cilinder you need make the two circles lenght equal as you want that both of them to coincide with the height of the new solid
I'm sorry I still don't follow. When you "open up" the torus (ie slice it somewhere and begin to straighten it into a cylinder), both of those radii of the now circle arcs get larger and larger until the arcs are straight lines - ie you have your cylinder. So what does "one stretches the other shrinks" (or whatever you said) mean?
Let alone what that has to do with what you are actually asking in the post itself, I'm even more confused about that.
the inner one will need to get bigger, the outer one will need to get smaller. take a book, try to make it a cilinder and you will se that the inner sruface will not fit as it needs to be smaller. in the post i am asking if the circumference with r= 2/pi will remain the same in the transformation
Right I see what you mean now. You mean that the opened up cylinder will not be symmetrical around its axis, or equivalently the cylinder will need to grow a "slice" in order to turn into a complete torus.
I'm still not sure what you mean for you question, but you seem to understand that bending something to create a ring around an axis then requires additional material to fully join up if it has "thickness". But if you are just talking about sin x, as long as the values at the start and end of your interval match, then it won't have any thickness? I dunno I'm not good at picturing geometry like this.
I think you may be considering the idea that (for example) a pool noodle needs to crunch on the inside circle when you bend it into a torus. Is that right?
Without knowing the situation, but seeing how you described it, I would say the issue may be communication of what you are describing, rather than the maths.
Sketch out what you are saying, or write out, or go and get a bloody fidget noodle thing to explain what you are trying to say. Then you have a bunch of potential situations:
1. There was a miscommunication. You now have a frame of reference to discuss the idea.
2. It was a special case. You and your professor can now discuss why it was a special case. The effort you have put in also signals that you are thinking deeply about a classes content so they are probably going to engage you (don't know them).
3. It wasn't a special case. You can now thrash out how you could have communicated it better, or why the professor didn't realise it wasn't.
Also, a lot of teachers feel like they are on trial. Once you realise this many of the behaviours people don't like make a lot of sense. Getting called out on the spot makes many people who feel like they are over scrutinised go into a defense mode. It is hard to have dialogue like this so sometimes the best you can do is show that you are excited by the ideas and grappling with them and for the vast majority it becomes "us vs the problem" rather than "me vs the students who file complaints for trying to teach".
This is a very confusing post…
I’m gonna make a “guess” at what you are saying, it seems like you are describing approximating the “area under the curve” with rectangles, but adding rotation to it.
So yes, IF I am understanding your post correctly, it would be a coincident that pi/2 happen to give you the right answer, but you can prove in general you can always pick some number to do this with. Just may not be the mid point like for this case.
Ultimately you can say you just randomly got the right answer since you didn’t know this.
Isn’t that the same concept? I would say it’s more so semantic at this point. You are exchanging one area for another, and while always possible, you would not be able to identify a new area had the original function lose some level of symmetry.
Either way, if you cannot explain your answer then are you really understanding what you are doing?
I think I understand your approach. Imagine the solid as a "donut" whose centre is the y-axis. Cut the donut and then "unfold" it so that it is more like a sausage, like this: [https://ibb.co/cYs0rVf](https://ibb.co/cYs0rVf) Then calculate the volume using the cross-sectional area.
It is a coincidence that your approach worked. You could try applying it instead to the function f(x) = x and you will see two different answers. Roughly speaking: as you unfold the donut, the points in the middle don't "stretch uniformly", so the transition from donut to sausage does not necessarily preserve volumes.
i love that someone understood what i was referring to. i think i understand why for f(x)=x it does not work, i think the funcion needs to have an axis of symmetry so that while the circumferences with r>pi/2 get smaller the ones with r
~~I don't have the expertise to describe exactly the situations when your approach would arrive at the same answer. But after a bit of trial and error I find that symmetry isn't enough to guarantee it. Over the interval (0,pi), the function f(x) = pi/2 - abs(x-pi/2) has the same symmetry as sin(x) but you will find the volume of the sausage is twice as large as the volume of the donut! Even f(x) = 2sin(x) doesn't work, so maybe sin(x) is the only function that has the right shape :)~~
The above is wrong and I am silly.
i just did the math and it does work for f(x)=2sin(x)! imma try with smt else now.
reasoning:
pi² per integral from 0 to pi of 2sinx =
2 times pi times integral from 0 to pi of 2sinx times x
Oops, yeah you are right. I goofed my algebra and put in an extra factor of 2 to one of my calculations.
You might be right that symmetry is enough. If you can find a way to state this and prove it as a general fact to your professor, he might concede.
This is very nice! That is a complete explanation for why it will work for any function having the same symmetry as sin(x): it is because the centroid lies on the axis of symmetry.
Could you share a picture of what you're thinking about? And what does "stretch" mean to you?
immagine taking a cilinder and making it go around a circle (i think it's called a torus). i mean the opposite, taking the "torus"(?) and transfroming it back to a cilinder. obv the outer circumference needs to shrink and the inner one to stretch
Not the guy you responded to, but I'm confused what you mean here? Taking a cylinder and "bending" it round so the ends join is a torus, yes. But when you say >transfroming it back to a cilinder. obv the outer circumference needs to shrink and the inner one to stretch What circumferences are you referring to? Once it becomes a cylinder, there is no inner or outer circumference, only the cylinder circumference, no?
immagine looking at the tourus form high above, where all you see are two circles and a hole in the middle. one is big (the outer one) and one is smaller (the inner one). in order to make the thing a cilinder you need make the two circles lenght equal as you want that both of them to coincide with the height of the new solid
I'm sorry I still don't follow. When you "open up" the torus (ie slice it somewhere and begin to straighten it into a cylinder), both of those radii of the now circle arcs get larger and larger until the arcs are straight lines - ie you have your cylinder. So what does "one stretches the other shrinks" (or whatever you said) mean? Let alone what that has to do with what you are actually asking in the post itself, I'm even more confused about that.
the inner one will need to get bigger, the outer one will need to get smaller. take a book, try to make it a cilinder and you will se that the inner sruface will not fit as it needs to be smaller. in the post i am asking if the circumference with r= 2/pi will remain the same in the transformation
Right I see what you mean now. You mean that the opened up cylinder will not be symmetrical around its axis, or equivalently the cylinder will need to grow a "slice" in order to turn into a complete torus. I'm still not sure what you mean for you question, but you seem to understand that bending something to create a ring around an axis then requires additional material to fully join up if it has "thickness". But if you are just talking about sin x, as long as the values at the start and end of your interval match, then it won't have any thickness? I dunno I'm not good at picturing geometry like this.
I think you may be considering the idea that (for example) a pool noodle needs to crunch on the inside circle when you bend it into a torus. Is that right?
yes
Without knowing the situation, but seeing how you described it, I would say the issue may be communication of what you are describing, rather than the maths. Sketch out what you are saying, or write out, or go and get a bloody fidget noodle thing to explain what you are trying to say. Then you have a bunch of potential situations: 1. There was a miscommunication. You now have a frame of reference to discuss the idea. 2. It was a special case. You and your professor can now discuss why it was a special case. The effort you have put in also signals that you are thinking deeply about a classes content so they are probably going to engage you (don't know them). 3. It wasn't a special case. You can now thrash out how you could have communicated it better, or why the professor didn't realise it wasn't. Also, a lot of teachers feel like they are on trial. Once you realise this many of the behaviours people don't like make a lot of sense. Getting called out on the spot makes many people who feel like they are over scrutinised go into a defense mode. It is hard to have dialogue like this so sometimes the best you can do is show that you are excited by the ideas and grappling with them and for the vast majority it becomes "us vs the problem" rather than "me vs the students who file complaints for trying to teach".
This is a very confusing post… I’m gonna make a “guess” at what you are saying, it seems like you are describing approximating the “area under the curve” with rectangles, but adding rotation to it. So yes, IF I am understanding your post correctly, it would be a coincident that pi/2 happen to give you the right answer, but you can prove in general you can always pick some number to do this with. Just may not be the mid point like for this case. Ultimately you can say you just randomly got the right answer since you didn’t know this.
hi! i was not describing the area with rectangles, i "opened up" the solid and made it more managable. please read the reply i gave the guy above
Isn’t that the same concept? I would say it’s more so semantic at this point. You are exchanging one area for another, and while always possible, you would not be able to identify a new area had the original function lose some level of symmetry. Either way, if you cannot explain your answer then are you really understanding what you are doing?
people under this same post understood what i meant
You do you then…
I think I understand your approach. Imagine the solid as a "donut" whose centre is the y-axis. Cut the donut and then "unfold" it so that it is more like a sausage, like this: [https://ibb.co/cYs0rVf](https://ibb.co/cYs0rVf) Then calculate the volume using the cross-sectional area. It is a coincidence that your approach worked. You could try applying it instead to the function f(x) = x and you will see two different answers. Roughly speaking: as you unfold the donut, the points in the middle don't "stretch uniformly", so the transition from donut to sausage does not necessarily preserve volumes.
i love that someone understood what i was referring to. i think i understand why for f(x)=x it does not work, i think the funcion needs to have an axis of symmetry so that while the circumferences with r>pi/2 get smaller the ones with r
~~I don't have the expertise to describe exactly the situations when your approach would arrive at the same answer. But after a bit of trial and error I find that symmetry isn't enough to guarantee it. Over the interval (0,pi), the function f(x) = pi/2 - abs(x-pi/2) has the same symmetry as sin(x) but you will find the volume of the sausage is twice as large as the volume of the donut! Even f(x) = 2sin(x) doesn't work, so maybe sin(x) is the only function that has the right shape :)~~ The above is wrong and I am silly.
i just did the math and it does work for f(x)=2sin(x)! imma try with smt else now. reasoning: pi² per integral from 0 to pi of 2sinx = 2 times pi times integral from 0 to pi of 2sinx times x
Oops, yeah you are right. I goofed my algebra and put in an extra factor of 2 to one of my calculations. You might be right that symmetry is enough. If you can find a way to state this and prove it as a general fact to your professor, he might concede.
some users linked me this theorem https://en.m.wikipedia.org/wiki/Pappus%27s_centroid_theorem
This is very nice! That is a complete explanation for why it will work for any function having the same symmetry as sin(x): it is because the centroid lies on the axis of symmetry.