Why can an infinite area have a finite volume or SA?

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Discussion Overview

The discussion centers around the concept of how an infinite area can coexist with a finite volume or surface area, particularly in the context of calculus and geometric shapes. Participants explore examples and thought experiments related to this phenomenon.

Discussion Character

  • Exploratory, Conceptual clarification, Homework-related

Main Points Raised

  • One participant mentions a calculus midterm and expresses interest in understanding the relationship between infinite area and finite volume, referencing the "famous painters example" as unsatisfactory.
  • Another participant suggests looking up "Gabriel's Horn" as a relevant example to explore this concept further.
  • A third participant provides a link to Wikipedia for additional information on Gabriel's Horn.
  • One participant proposes a thought experiment involving a cube, suggesting that it can be viewed as a stack of an infinite number of square surfaces, implying that the cube contains an infinite surface area while maintaining a finite volume.
  • This participant also describes a method of generating an infinite surface area by removing an infinite sequence of slices from the cube, indicating that the volume remains unchanged despite the infinite surface area created.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are multiple viewpoints and examples presented without resolution of the underlying questions.

Contextual Notes

The discussion includes assumptions about geometric properties and the implications of infinite sequences, but these assumptions are not fully explored or resolved.

Who May Find This Useful

Students preparing for calculus exams, individuals interested in mathematical concepts related to infinity, and those exploring geometric properties in higher mathematics.

Zack K
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I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it.

I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.
 
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Google Gabriel's Horn
 
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Zack K said:
I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it.

I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.

If you think of a cube, you can imagine it as a stack of an infinite number of square surfaces. So, there already is an infinite surface in there, so to speak. A simple way to generate an infinite surface area is simply to remove an infinite sequence of slices: if the cube is 1 unit high, you could remove the slices at ##z = 1/2, 1/3, 1/4 \dots##.

This would leave a shape with actually the same volume as before, but an infinite surface area.
 

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