Help extending volumes of revolution to fourth dimension

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SUMMARY

This discussion focuses on extending the concept of volumes of revolution from calculus into the fourth dimension, specifically revolving 3D functions over the x-axis. The user references a Physics Forums thread that explains how to find the content of a 3-ball and attempts to apply this method to the function 4 - x². They express difficulty in adapting the method for solids of revolution beyond spheres and seek guidance on calculating the surface area and circumference of these 3D solids. The user also shares resources for further understanding integrals related to this topic.

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  • Understanding of volumes of revolution in calculus
  • Familiarity with surfaces of revolution
  • Basic knowledge of 3D geometry and integrals
  • Ability to interpret mathematical resources such as PDFs and presentations
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  • Research methods for calculating volumes of revolution in higher dimensions
  • Study the concept of 3-balls and their properties in geometry
  • Learn about surface area calculations for solids of revolution
  • Explore advanced calculus topics, including surface integrals and their applications
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Students of calculus, particularly those interested in advanced geometric concepts, as well as educators and mathematicians exploring higher-dimensional mathematics.

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I am currently learning about volumes of revolution in calulus, and have looked ahead to surfaces of revolution as well. I want to try and extend this concept to revolving 3d functions over the x-axis into the fourth dimension. I found this thread:
https://www.physicsforums.com/showthread.php?t=64536
and saw how to find the content of a 3-ball (5th post).

I attemted to use this idea to find the content of the revolution about the x-axis of the revolution about the x-axis of 4-x2 but I couldn't quite puzzle through the method described in the above thread for anything other than spheres. The closest I came was the surface area of the solid of revolution, and realizing that I need to describe the circumference of the 3d solid of revolution with respect to the radius and/or height (for a sphere of radius 1 it would be 2\pissin(r))

Can anyone point me in the right direction, or does this seem way too far past a Calculus DC student in high school? (and if so I'm still really interested in how this would work)
 
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