A random question comes to mind, about the infinitesimal area of rings

In summary, the conversation is about calculating the area of a thin ring and using the formula ##2\pi rdr##. The question is whether the result from this formula is the same as using the usual way of calculating the area of a ring. The conversation then discusses an equation that leads to the conclusion that the second term can be ignored. However, there is a discrepancy of 4 times more than the original formula, and the person is seeking an explanation for this.
  • #1
Laudator
18
1
I know the area of a thin ring of radius ##r## can be expressed as ##2\pi rdr##, however, I wonder if I use the usual way of calculating area of a ring, can I reach the same conclusion? I got this:
$$4\pi(r+dr)^2-4\pi r^2=4\pi r^2+8\pi rdr+4\pi (dr)^2-4\pi r^2=8\pi rdr+4\pi (dr)^2$$And now I'm stuck. I think the second term can be ignored because it's so small, but how to deal with the fact that it's 4 times more than ##2\pi rdr##? Can someone explain this to me or tell me why I shouldn't think this way, where did I go wrong?
 
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  • #2
Laudator said:
where did I go wrong
The initial 4 is not right: the area of a disc is ##\pi r^2##
 
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Likes Laudator
  • #3
@BvU ... How stupid am I ... Thanks ...
 

1. What is meant by the infinitesimal area of rings?

The infinitesimal area of rings refers to the concept of dividing a ring into infinitely small sections in order to calculate its total area. It is often used in calculus and geometry to find the area of curved shapes.

2. How is the infinitesimal area of rings calculated?

The infinitesimal area of rings is calculated using the formula A = πr², where A is the area of the ring and r is the radius. This formula assumes that the ring is made up of infinitely small sections, allowing for a more accurate calculation of its area.

3. What is the significance of the infinitesimal area of rings in mathematics?

The concept of infinitesimal area of rings is important in mathematics as it allows for the calculation of the area of curved shapes, which are often encountered in real-world scenarios. It also plays a crucial role in the development of calculus and other branches of mathematics.

4. Can the infinitesimal area of rings be applied to other shapes besides rings?

Yes, the concept of infinitesimal area can be applied to any shape that can be divided into infinitely small sections. This includes circles, ellipses, and other curved shapes.

5. Are there any limitations to using the concept of infinitesimal area of rings?

One limitation of using the infinitesimal area of rings is that it assumes the ring is made up of infinitely small sections, which may not always be the case in real-world scenarios. Additionally, it may not be applicable to irregularly shaped rings or rings with varying thickness.

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