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

AI Thread Summary
The discussion focuses on calculating the volume of the solid of revolution formed by the area between the line y = x and the parabola y = x^2, rotated about the axis y = x, from x = 0 to x = 1. Participants note that while the problem may seem challenging at first, it is manageable with the right approach. A key point is that the thickness of the volume element should be based on arc length (ds) rather than the typical dx. The conversation emphasizes the importance of rotating coordinates to simplify the problem into standard form. Overall, the problem is deemed interesting and worth exploring despite previous discussions on similar topics.
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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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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?
 
jedishrfu said:
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.
 
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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.
 
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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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