Re: Area of two intersecting circles
- To: mathgroup at smc.vnet.net
- Subject: [mg123379] Re: Area of two intersecting circles
- From: David Skulsky <edskulsky at gmail.com>
- Date: Mon, 5 Dec 2011 05:13:41 -0500 (EST)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <jbf968$l4j$1@smc.vnet.net>
- Reply-to: comp.soft-sys.math.mathematica at googlegroups.com
I don't know if a closed-form solution to this problem exists, but you can solve it numerically with the following expression:
\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2\ \[Pi]\)]\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(R\)]r\ \((Boole[
\*SuperscriptBox[\((r\ Cos[\[Theta]] - d)\), \(2\)] +
\*SuperscriptBox[\((r\ Sin[\[Theta]])\), \(2\)] <=
\*SuperscriptBox[\(r\), \(2\)]])\) \[DifferentialD]r \[DifferentialD]\
\[Theta]\)\)
where R is the radius of the circles and d is the distance between the centers of the circles.
A slightly simplified version of the same expression is
\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2\ \[Pi]\)]\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(R\)]\(r\ Boole[
\*SuperscriptBox[\(d\), \(2\)] <=
2\ d\ r\ Cos[\[Theta]]]\) \[DifferentialD]r \[DifferentialD]\
\[Theta]\)\)
David