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This is probably a really asinine question.

I was trying to identify an area of a ring, namely a really small ring such that its near enough the circumference of a circle. I thought I could approach it in two ways.

The first was to subtract a smaller circle of radius r

_{1}form a circle of larger radius r

_{2}:

[tex]\pi r_{2}^{2} - \pi r_{1}^{2} = \pi (r_{2}^{2} - r_{1}^{2})[/tex]

Using the limit r

_{1}tends to r

_{2}, this should be a ring with an area equal to that which I could obtain from using the circumference of a circle of radius r

_{2}multiplied by a tiny amount (essentially giving it some thickness:

This is only approximately a ring:

[tex]2\pi r_{2} (r_{2}-r_{1})[/tex]

But in the limit it would be a ring:

[tex]2\pi r_{2} \left dr[/tex]

Are the two equal? Conceptually, it seems they should be.

This was my reasoning:

[tex]\pi r_{2}^{2} - \pi r_{1}^{2} = \pi (r_{2}^{2} - r_{1}^{2}) = \pi (r_{2} + r_{1})(r_{2} - r_{1})[/tex]

In the limit:

[tex]r_{2} - r_{1}[/tex] Approaches 0, but is some infinitesimally small difference dr, which is explicitly non-zero.

The summation component:

[tex]r_{2} + r_{1}[/tex] Seems to approach 2.

So we have:

[tex]\lim \pi (r_{2} + r_{1})(r_{2} - r_{1}) = \pi(2r_{2}}{dr}[/tex]

Is it considered to equal 2 or is it explicitly not 2 as the difference component is explicitly not 0?

I am confused, is there an error in my reasoning?

Any help appreciated as this is very frustrating!

Thanks in advance,

Nobahar