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Double integral on one function of x and another of y

  1. May 1, 2012 #1
    Hi there. I think I have proved on little theorem on double integrals, showed below. Are my arguments 'correct' (I mean, rigorous enough)?

    Let [itex]f[/itex] be a function of [itex]x, f(x)[/itex], [itex]g[/itex] be a function depending only on [itex]y, g(y)[/itex], and last, let [itex]A[/itex] be the set determined by [itex]a≤x≤b[/itex] and [itex]c≤y≤d[/itex]. By Fubini's theorem, [itex]\int\int_{A}f(x)g(y)dxdy=\int^{d}_{c}[\int^{b}_{a}f(x)g(y)dx]dy.[/itex] Knowing [itex]g[/itex] is a function only dependent on [itex]y[/itex], it is a constant when integrating over x, hence we have [itex]\int^{d}_{c}[g(y)\int^{b}_{a}f(x)dx]dy.[/itex] Still, as [itex]f[/itex] depends only on [itex]x[/itex], so does any integral over [itex]f[/itex], then finally [itex]\int^{d}_{c}g(y)dy \int^{b}_{a}f(x)dx.[/itex]

    Many thanks.
  2. jcsd
  3. May 1, 2012 #2

    Kudos if you reached this result by yourself, but in fact it follows in a pretty simple way from well known properties of integrals.

  4. May 1, 2012 #3
    Yes it does follows easily, nice to know it is correct. It can simplify many problems as I was testing just now. Thanks!
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