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 ?

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 $f(x,y)=x$
