Sorry, this was incorrect. In this integral we're just calculating the geometrical question, what is is the 5d volume of the collection of 3 points whose circumcenter is on {(x, 0) : x in [0, 1]}. So the "overall" probability of the problem has nothing to do with it.
You've done
∫₀¹ ∫₀¹⁻ᶜ (2π r)³ dr dc
However,
- the `dr` integral is assuming that the radii are uniformly likely in [0, r]
- the `dc` integral is assuming that the centers are uniformly likely in [0, c]
You need to wait these integrals by the conditional probability distributions.