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Let's consider $\int_a^b\int_c^d f(x,y)dxdy$ : do we know that $x$ will vary from $a$ to $b$ and $y$ from $c$ to $d$, or said similarly, is the ordering crucial in the definition ? Same question for the ordering of dx and dy ?

I'm not asking about Fubini theorem : I ask explictely if $\int_a^b\int_c^d f(x,y)dxdy$=$\int_c^d\int_a^b f(x,y)dxdy$ that is : could $x$ be either from $a$ to $b$ or from $c$ to $d$, without changing the result ?

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    $\begingroup$ You can search the Fubini's theorem. $\endgroup$ Oct 4, 2021 at 19:56
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    $\begingroup$ @Emilio Novati : I'm asking about the interpretation of $a$ and $b$ : are there necessary meaning the first variable. $\endgroup$ Oct 4, 2021 at 20:01
  • $\begingroup$ Have you tried any concrete examples, e.g. $$\int_0^1\int_0^2xy^2dxdy\mbox{ versus }\int_0^2\int_0^1xy^2dxdy?$$ $\endgroup$ Oct 4, 2021 at 20:03
  • $\begingroup$ first gives 4/3. Second gives 2/3 : so where is the rule that says that the first integral is meaning x ? How could we know ? $\endgroup$ Oct 4, 2021 at 20:06
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    $\begingroup$ This is the main reason why I prefer the 'forward-operating' notation for definite integrals as $\int_{a}^{b}dx\,f{\left(x\right)}$, and then double integrals $\int_{a}^{b}dx\int_{c}^{d}dy\,f{\left(x,y\right)}$. With this notation there is never any confusion or doubt over which $\int$ goes with which differential! $\endgroup$
    – David H
    Oct 4, 2021 at 23:21

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Usually $\int_a^b\int_c^d f(x,y)dxdy$ is interpreted as

$$\int_a^b\left(\int_c^d f(x,y)dx\right)dy$$ and this is, in general, different from $$\int_c^d\left(\int_a^b f(x,y)dx\right)dy$$ as you can see using a simple function as $f(x,y)=x$

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