I think your reasoning checks out; if $x,y \in [0,1]$, then I think it does follow that $xy \in [0,1]$. I agree with your following inferences as well. Unless I've failed to consider something as well, I believe this is a correct argument. (Wolfram Alpha also evaluates the integral to be positive).
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I think your reasoning checks out; if $x,y \in [0,1]$, then I think it does follow that $xy \in [0,1]$. I agree with your following inferences as well. Unless I've also failed to consider something as well, something, I believe this is a correct argument. (Wolfram Alpha also evaluates the integral to be positive).