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

[PeroK]

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 $y = x$.

 

[jedishrfu]

There was an earlier thread on this type of problem here:

https://www.physicsforums.com/threads/solids-of-revolution-around-y-x.669647/

It showed up in the related threads area of this page.

 

[PeroK]

There was an earlier thread on this type of problem here:

https://www.physicsforums.com/threads/solids-of-revolution-around-y-x.669647/

It showed up in the related threads area of this page.

Okay, but still worth doing, IMO.

 

[jedishrfu]

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?

 

[PeroK]

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.

 

[Mark44]

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.

 

[ergospherical]

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.