probability – Computing the Correlation Coefficient of Two Random Variables


I am a bit confused on computing the $rho$ for two random variables. Let $f(X,Y) = 1$ with support given by $-x < y < x$ and $0 < x < 1$. I know that the definition of $rho$ is simply $rho = frac{cov(X,Y)}{sigma_Xsigma_Y}$. Hence, I have computed E(XY) =begin{equation} E(XY) =
int_{0}^{1} int_{-x}^{x} xy ,dy,dx
end{equation}

But clearly the above quantity is just $0$, and $E(Y) = 0$. Hence, by the definition of covariance $cov(X,Y) = E(XY) – E(X)E(Y)$, but both terms in the difference are zero, hence covariance is 0 and this implies $rho = 0$. Have I done this computation incorrectly? I feel like there is something not quite right, because there is a dependency in the support of the joint PDF, and hence it seems like there should be non-zero correlation between these random variables. Just looking for what I did wrong here if I did in fact do something wrong. Thanks for reading!