Solids of Revolution around y = x

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Discussion Overview

The discussion revolves around the concept of revolving a function around the line y = x, specifically exploring methods for calculating volumes of solids of revolution. Participants discuss various coordinate transformations and integration techniques applicable to this problem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant questions the feasibility of revolving a function around y = x and seeks methods that work for most functions.
  • Another participant suggests using new coordinates p = x + y and q = x - y to simplify the problem, indicating that the rotation can be treated as around the q axis.
  • A participant attempts to apply the transformation to the function y = x^2 and asks how to proceed with the integration after substitution.
  • There is a suggestion that substituting x = (p+q)/2 and y = (p-q)/2 may be easier for the calculations.
  • Participants express confusion about the meaning of r in the context of the volume formula and its relation to the distance from the axis of rotation.
  • There is a request for resources such as videos or articles that explain the process of revolving functions around y = x.
  • Clarifications are made regarding the steps to convert the problem into p and q coordinates and how to revert back to x and y after solving.

Areas of Agreement / Disagreement

Participants express differing opinions on the best method to approach the problem, with no consensus on a single technique or solution path. Some participants propose different coordinate transformations, while others seek clarification on the integration process.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in the transformations and integration, and there are uncertainties regarding the definitions and applications of the variables used.

TheAbsoluTurk
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Is it possible to revolve a function around y = x? If so how would you do it?

I suppose the main difficulty is in finding the radius for the area of a disk or cylinder. Is there any method that works will all or most functions?
 
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Hi TheAbsoluTurk! :smile:

Easiest way is to change to new coordinates p = x + y, q = x - y (or the same but divided by √2, if you prefer).

Then x = y is the q axis, so that's just a rotation about the q axis. :wink:
 
tiny-tim said:
Hi TheAbsoluTurk! :smile:

Easiest way is to change to new coordinates p = x + y, q = x - y (or the same but divided by √2, if you prefer).

Then x = y is the q axis, so that's just a rotation about the q axis. :wink:

Ok, let's say that I'm trying to rotate y = x^2 around y = x.

p = x + y

q = x - y

So do I have to insert (q + y) into x to make y = (q + y)^2 ?
 
Last edited:
Easier is to substitute x = (p+q)/2, y = (p-q)/2 :wink:
 
tiny-tim said:
Easier is to substitute x = (p+q)/2, y = (p-q)/2 :wink:

Do you know of any YouTube videos or articles on the internet which show how to do this?
 
uhh? :confused:

just do it … substitute those formulas into y = x2 !​
 
tiny-tim said:
uhh? :confused:

just do it … substitute those formulas into y = x2 !​

I understand that but I don't know what to do after that. Does r in ∏r^2 equal (p-q)/2? How do you integrate that?
 
TheAbsoluTurk said:
I understand that but I don't know what to do after that. Does r in ∏r^2 equal (p-q)/2? How do you integrate that?

no, the r is the distance from your axis

your axis (originally called x=y) is the q axis, so r is the distance from the q axis, which is p (or is it p/2?)
 
tiny-tim said:
no, the r is the distance from your axis

your axis (originally called x=y) is the q axis, so r is the distance from the q axis, which is p (or is it p/2?)

Let me get this straight:

What is the volume of y = x^2 rotated about y = x?

Define p = x +y

Define q = x - y

I don't understand why you chose to insert x = (p+q)/2 and y = (p-q)/2 ? How did you get these?
 
  • #10
TheAbsoluTurk said:
Let me get this straight:

What is the volume of y = x^2 rotated about y = x?

Define p = x +y

Define q = x - y

I don't understand why you chose to insert x = (p+q)/2 and y = (p-q)/2 ? How did you get these?

Ok, I understand how you got those expressions. But what's to do next? Do you solve for p?
 
  • #11
(just got up :zzz:)

first you convert everything into p and q

then you solve the problem, in p and q (you've said you know how to do this)

finally you convert your solution back to x and y :smile:​
 

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