Solids of Revolution around y = x

[TheAbsoluTurk]

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?

 

[tiny-tim]

Hi TheAbsoluTurk!

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.

 

[TheAbsoluTurk]

Hi TheAbsoluTurk!

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.

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 ?

 

[tiny-tim]

Easier is to substitute x = (p+q)/2, y = (p-q)/2

 

[TheAbsoluTurk]

Easier is to substitute x = (p+q)/2, y = (p-q)/2

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

 

[tiny-tim]

uhh?

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

 

[TheAbsoluTurk]

uhh?

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?

 

[tiny-tim]

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?)

 

[TheAbsoluTurk]

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?

 

[TheAbsoluTurk]

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?

 

[tiny-tim]

(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 ​