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Why aren’t these additions of integrals and summations equal?

I am trying to expand all the terms in a sum in order to apply assignment rules later, but can't figure out how to do it. I've tried pretty much the commands I've seen around here. I have the following integral:

$$\iint(x^2+y^2) f(x,y)dxdy$$

and I would like to extend it to

$$\iint x^2 f(x,y)dxdy+\iint y^2 f(x,y)dxdy$$

in order to apply rules like $\iint x^2f(x,y)dxdy \to a$ and so on. I would like to do this in a generic way since the integrand would not always look like the one in the example here. Any idea?

  • $\begingroup$ Look at a related problem : mathematica.stackexchange.com/questions/8353/… $\endgroup$
    – Artes
    Nov 10, 2012 at 22:42
  • $\begingroup$ Please look at the link that Artes posted, which explains why such an expansion would be wrong in general and also a way to do it nevertheless. This is a duplicate of that question and will be closed as such @Artes, please vote to close instead of flagging, since you have the rep to do it :) $\endgroup$
    – rm -rf
    Nov 10, 2012 at 22:57
  • $\begingroup$ Thanks, I still can't make it work but I'll look through it a bit more and comment in said post if needed. Thanks again. $\endgroup$ Nov 10, 2012 at 23:08
  • $\begingroup$ Finally managed to make it work, I didn't know one had to be so careful defining the exact rules. $\endgroup$ Nov 10, 2012 at 23:45