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

[Zack K]

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.

 

[phinds]

Google Gabriel's Horn

 

[jedishrfu]

Wikipedia has it

https://en.m.wikipedia.org/wiki/Gabriel's_Horn

And from mathispower4u

 

[PeroK]

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.