Help extending volumes of revolution to fourth dimension

In summary, the conversation is about extending the concept of volumes of revolution in calculus to revolving 3D functions over the x-axis into the fourth dimension. The individual has found a thread on Physics Forums discussing how to find the content of a 3-ball using this concept. They attempted to apply it to finding the content of the revolution about the x-axis of 4-x^2, but were unable to fully understand the method described in the thread. They mention the possibility of using surface area to describe the 3D solid of revolution, and ask for guidance on the topic. Several resources are provided for further explanation.
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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[itex]\pi[/itex]ssin(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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FAQ: Help extending volumes of revolution to fourth dimension

1. What is the fourth dimension?

The fourth dimension is a theoretical concept that exists beyond our three-dimensional world. It is often referred to as time, or the concept of time as a dimension.

2. How do you extend volumes of revolution to the fourth dimension?

Extending volumes of revolution to the fourth dimension involves using mathematical equations and geometric principles to calculate the volume of a solid object that has been rotated in the fourth dimension.

3. Why is extending volumes of revolution to the fourth dimension important?

Extending volumes of revolution to the fourth dimension allows us to better understand the properties and behaviors of higher-dimensional objects, which can have implications in fields such as physics, cosmology, and mathematics.

4. Can we visualize volumes of revolution in the fourth dimension?

It is difficult for humans to visualize anything beyond three dimensions, so visualizing volumes of revolution in the fourth dimension is not possible. However, we can use mathematical models and computer simulations to represent and understand these concepts.

5. What challenges do scientists face when studying volumes of revolution in the fourth dimension?

One of the main challenges scientists face is the limitations of our human perception and ability to visualize higher dimensions. Additionally, the mathematical and computational complexities involved in studying volumes of revolution in the fourth dimension can also pose challenges.

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