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For $E[XY]$, if I understood correctly, there is a mistake. You are first choosing $X$ and than $Y=h(X)$, with $h$ deterministic. So you have to evaluate:
$E[XY]=\int f_X(x) xh(x)dx$
Instead it looks that you integrated over $y$. But consider when $h$ is invertible:
$\int f_X(x) xh(x)dx \ne \i...