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Laurent One has $\displaystyle {\rm Corr}(X,Y)=\frac{{\rm Cov}(X,Y)}{\sqrt{{\rm Var}(X){\rm Var}(Y)}}=\frac{E[XY]-E[X]E[Y]}{\sqrt{(E[X^2]-E[X]^2)(E[Y^2]-E[Y]^2)}}$, right? So what you need to compute is $\displaystyle E[XY]$, $\displaystyle E[X]$, $\displaystyle E[Y]$, $\displaystyle E[X^2]$ and $\displaystyle E[Y^2]$. In fact, the pdf of $\displaystyle (X,Y)$ is symmetric in $\displaystyle x,y$, so that $\displaystyle X$ and $\displaystyle Y$ have same distribution and the formula reduces to: $\displaystyle {\rm Corr}(X,Y)=\frac{E[XY]-E[X]^2}{E[X^2]-E[X]^2}$.
To compute $\displaystyle E[XY]$, just integrate $\displaystyle xy$ times the pdf of $\displaystyle (X,Y)$ (over the square $\displaystyle 0\leq x,y\leq 1$).
You can do the same with the other ones (integrating $\displaystyle x$ and $\displaystyle x^2$), but you may find it quicker to first determine the pdf of $\displaystyle X$. For that, you just have to integrate the pdf of $\displaystyle (X,Y)$ with respect to the variable $\displaystyle y$, keeping $\displaystyle x$ fixed.