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!