Solids of Revolution around y = x

  1. 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?
     
  2. jcsd
  3. tiny-tim

    tiny-tim 26,054
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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:
     
  4. 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: Feb 5, 2013
  5. tiny-tim

    tiny-tim 26,054
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    Easier is to substitute x = (p+q)/2, y = (p-q)/2 :wink:
     
  6. Do you know of any YouTube videos or articles on the internet which show how to do this?
     
  7. tiny-tim

    tiny-tim 26,054
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    uhh? :confused:

    just do it … substitute those formulas into y = x2 !​
     
  8. 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?
     
  9. tiny-tim

    tiny-tim 26,054
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    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?)
     
  10. 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?
     
  11. Ok, I understand how you got those expressions. But what's to do next? Do you solve for p?
     
  12. tiny-tim

    tiny-tim 26,054
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    (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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