In multiple definite integrals, if the region is the same, does the order of the differentials matter?
I think the question is clear, but as an example in 2 dimensions, is it true that
$$\int_X\int_X f(x,y)dx\ dy=\int_X\int_X f(x,y)dy\ dx$$
for all valid regions $X$?
Let $f:[0,1]^2\to\mathbb R$ be defined by $$ f(x,y) = \frac{x^2-y^2}{(x^2+y^2)^2}\cdot\mathsf 1_{(x,y)\ne(0,0)}. $$ Then \begin{align} \int_0^1\int_0^1 f(x,y)\ \mathsf dx\ \mathsf dy &= \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2}\ \mathsf dx\ \mathsf dy\\ &= \int_0^1 -\left(\frac1{1+y^2}\right)\ \mathsf dy\\ &= -\frac\pi4 \end{align} while $$ \begin{align} \int_0^1\int_0^1 f(x,y)\ \mathsf dy\ \mathsf dx &= \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2}\ \mathsf dy\ \mathsf dx\\ &= \int_0^1 \left(\frac1{1+x^2}\right)\ \mathsf dx\\ &= \frac\pi4. \end{align} $$