Expected value of distance

Calculus Level 5

Let \(R\) be the region on the Cartesian plane bounded by \(0 \le x \le 1\) and \(0 \le y \le 1\). Let \(P\) be a point chosen at random in \(R\) with a uniform probability distribution. Let \(v\) be the expected value of the distance between \(P\) and the origin.

If \(v = \dfrac{\sqrt{a} + \ln\left(\sqrt{b} + c\right)}{d}\), where \(a,b,c,d \in \mathbb{N}\) and \(a,b\) are squarefree, find the value of \(a+b+c+d\).

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