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What is the average distance of the points within a circle of radius $r$ from a point a distance $d$ from the centre of the circle (with $d>r$, though a general solution without this constraint would be nice)?

The question arose as an operational research simplification of a real problem in telecoms networks and is easy to approximate for any particular case. But several of my colleagues thought that such a simple problem should have a simple formula as the solution, but our combined brains never found one despite titanic effort. It looks like it might involve calculus.

I'm interested in both how to approach the problem, but also a final algebraic solution that could be used in a spreadsheet (that is, I don't want to have to integrate anything).

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    Perhaps not simple; but, taking the center of the circle $C$ (with interior) to be the origin and the point to be $(a,b)$ it's ${1\over \pi r^2}\int\kern-2pt\int_C \sqrt{(x-a)^2 +(y-b)^2 }\,dx\,dy$.2012-01-11
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    @DavidMitra I may need to clarify what I mean by formula. This is useful but only a halfway house if you need to use it in a spreadsheet. Is there a simple algebraic solution?2012-01-11
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    Perhaps it will simply things a bit to note that WLOG we can take the point to be $(d,0)$.2012-01-11
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    You would think there is a simple expression for the *perimiter* of an ellipse, but there isn't one. Again it falls into elliptic integrals category as is your question.2012-01-12

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