# Is the ordering of integrals important in a 2D integral?

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 ?

• You can search the Fubini's theorem. Oct 4, 2021 at 19:56
• @Emilio Novati : I'm asking about the interpretation of $a$ and $b$ : are there necessary meaning the first variable. Oct 4, 2021 at 20:01
• Have you tried any concrete examples, e.g. $$\int_0^1\int_0^2xy^2dxdy\mbox{ versus }\int_0^2\int_0^1xy^2dxdy?$$ Oct 4, 2021 at 20:03
• 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 ? Oct 4, 2021 at 20:06
• 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! Oct 4, 2021 at 23:21

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 $$f(x,y)=x$$