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Jun 13, 2014 at 4:43 comment added Jan Kyncl And for four discs, there are a continuum many configurations where the integral is $\pi +4$.
Jun 12, 2014 at 4:30 comment added Jan Kyncl Now for two such touching discs, the integral is $\pi + 2$, and for three discs like on the picture the integral is $\pi + 3$. Asymptotically, and approximately, it seems you want to pack the discs into another disc of smallest diameter (and thus perimeter).
Jun 12, 2014 at 1:50 vote accept Joseph O'Rourke
Jun 11, 2014 at 10:50 comment added Joseph O'Rourke Nice! So for one disk of diameter $1$, the integral is $\pi$.
Jun 11, 2014 at 5:16 history answered Jan Kyncl CC BY-SA 3.0