Have a cone and divide it into infinately small slices

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In summary, the conversation revolves around the concept of dividing a cone into infinitely small slices and whether or not the cone would become a cylinder. The first person argues that as the slices become infinitely small, the areas on both sides of the slice would become equal, making the cone a cylinder. The second person disagrees, stating that the slices would never actually be completed and the assumption of equal areas is not based in reality. They also discuss the difference between infinitely small and zero.
  • #1
Lorentz
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If I have a cone and divide it into infinately small slices. Wouldn't both sides of one slice have the same area and wouldn't the next slice (and so on) have the same area as the slice before. So wouldn't your cone actually be a cylinder?

My answer is no, because the reasoning is wrong. If I had infinately small slices I would never complete the cone/cylinder in the first place. And the assumption of both sides having the same area is an assumption to be able to integrate, but is not reality. If we're talking about the perfect cone then both sides of the slices should have different areas even if the slices were infinately small.

What's your view on this?
 
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  • #2
Lorentz said:
If I have a cone and divide it into infinately small slices. Wouldn't both sides of one slice have the same area and wouldn't the next slice (and so on) have the same area as the slice before. So wouldn't your cone actually be a cylinder?
It would something more like, as the number of slices approaches infinity, the width of each slice approaches zero, and the areas on the two faces of the slice approach each other.

If I had infinately small slices I would never complete the cone/cylinder in the first place.
Here's something to think about. You should know that a line is made up of infinite points. Each point has zero size. So, given an infinite number of zero-sized points put together, how is it that you get a line with non-zero-size?

And the assumption of both sides having the same area is an assumption to be able to integrate, but is not reality.
Well, be careful here, because you can't really make any arguments "from reality" when dealing with math. Math is a useful tool in modelling reality, but that doesn't mean that it is based on reality (it is based on its own axioms, some of which seem rather unnatural), nor does it mean that it is an accurate tool in modelling reality.
If we're talking about the perfect cone then both sides of the slices should have different areas even if the slices were infinately small.
What does it mean to be infinitely small? Can something be smaller than infinitely small? What would the difference in area be between the two faces?
 
  • #3
AKG said:
Can something be smaller than infinitely small? What would the difference in area be between the two faces?

erm... the difference would be infinitely small? But that would still be a difference which makes it possible to glue the slices together and get the cone back again rather then a cylinder. If the difference would be zero we would get the cilinder.
 
  • #4
This question just popped into my mind: Is there a difference between infinitely small and zero?
 
  • #5
Lorentz said:
This question just popped into my mind: Is there a difference between infinitely small and zero?

Yes, IF you are working in "non-standard analysis" and, by "infinitely small", you mean "infinitesmal". Otherwise "infinitely small" is just a (misleading) shorthand for "in the limit".
 

1. What is the concept of dividing a cone into infinitely small slices?

The concept of dividing a cone into infinitely small slices is based on the mathematical concept of limits. It involves breaking down a cone into smaller and smaller pieces, with the goal of reaching an infinitely small size.

2. How is this concept related to calculus?

This concept is closely related to calculus, specifically the concept of integration. By dividing the cone into infinitely small slices, we can calculate the volume of the cone by summing up all the slices using integration.

3. Can a cone truly be divided into infinitely small slices?

Technically, no. In reality, there will always be a smallest possible size for the slices. However, in mathematical calculations, we can approach infinitely small slices by making them smaller and smaller. This is known as taking the limit.

4. What are the practical applications of this concept?

The concept of dividing a cone into infinitely small slices has many practical applications in fields such as physics, engineering, and economics. It allows us to calculate areas and volumes of irregular shapes and to model real-world phenomena with more accuracy.

5. Is this concept limited to cones or can it be applied to other shapes as well?

This concept can be applied to any shape, not just cones. It is a fundamental concept in calculus and is used to solve problems involving any curved or irregular shape.

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