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

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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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BvU
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where did I go wrong
The initial 4 is not right: the area of a disc is ##\pi r^2##
 
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@BvU ... How stupid am I ... Thanks ...
 

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