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

In summary, the conversation is about calculating the volume of a solid of revolution when an area between a line and a parabola is rotated around the axis y=x. The problem was found to be interesting and original, and a previous thread was referenced for a similar problem. It was noted that the direction chosen for rotation is not particularly difficult, but the thickness of the volume element may need to be adjusted. It is also mentioned that rotating the coordinates can help solve the problem.
  • #1

PeroK

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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
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
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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  • #6
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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  • #7
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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What is the "Volume of Solid of Revolution (About the line y = x)"?

The Volume of Solid of Revolution (About the line y = x) is a mathematical concept that involves finding the volume of a 3-dimensional shape that is created by rotating a 2-dimensional shape around the line y = x.

How is the "Volume of Solid of Revolution (About the line y = x)" calculated?

The Volume of Solid of Revolution (About the line y = x) can be calculated using the method of cylindrical shells or the method of disks/washers. Both methods involve integrating a function that represents the cross-sectional area of the solid.

What is the difference between the method of cylindrical shells and the method of disks/washers?

The method of cylindrical shells involves integrating the circumference of the shell multiplied by its height, while the method of disks/washers involves integrating the area of a cross-section of the solid.

When do I use the method of cylindrical shells and when do I use the method of disks/washers?

The choice of method depends on the shape of the cross-section of the solid. If the cross-section is a circle, it is easier to use the method of disks/washers. If the cross-section is a rectangle, it is easier to use the method of cylindrical shells. However, both methods can be used for any shape of cross-section.

What are some real-life applications of the "Volume of Solid of Revolution (About the line y = x)"?

The concept of Volume of Solid of Revolution (About the line y = x) is used in engineering, architecture, and physics to calculate the volume of objects with rotational symmetry. For example, it can be used to calculate the volume of a water tank, a cylindrical chimney, or a spherical dome.

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