# 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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