Expectations of Joint Distribution.

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How would you work out E[XY] where X,Y are dependent variables in a Joint Distribution.

I know there is a relationship with the conditional distributions but I can't understand the logic behind it, hence am hoping someone here can give me directions to work out this expectation.(I don't just want a formulae, I want the idea behind it)
 
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If you have the joint distribution, then E(XY) = integral of (xy multiplied by the joint density).
 

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