Volume of Solid of Revolution (About the line y = x)

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TL;DR Summary
Volume of Solid of revolution about unusual axis
I found this problem, which I thought was interesting and somewhat original:

Calculate the volume of the solid of revolution of the area between the line ##y = x## and the parabola ##y = x^2## from ##x = 0## to ##x = 1## when rotated about the axis ##y = x##.
 

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  • #4
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I agree just wanted to be helpful here. When I first looked at it, I didn’t know how to tackle it until I saw the older post.

How intractable is it if an arbitrary direction is chosen?
 
  • #5
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I agree just wanted to be helpful here. When I first looked at it, I didn’t know how to tackle it until I saw the older post.

How intractable is it if an arbitrary direction is chosen?
It's not particularly hard.
 
  • #6
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I think the only tricky part is that the thickness of the typical volume element needs to be ds rather than dx; i.e., an increment of arc length along the parabola.
 
  • #7
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Not even that, because the secret to any problem involving a rotated geometric figure is to rotate the co-ordinates ##\mathbf{x}' = \mathsf{R}\mathbf{x}## so that you have it in standard form.
 
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