Integration doubt in notation.

Question

I come across two notations while solving integrals :
$1.\int_a^bf(x)dx$
$2.\int_{[a,b]}f(x)dx$.


For improper integrals,
$1.\int_{-\infty}^\infty f(x)dx$
$2.\int_{\mathbb{R}}f(x)dx$


Is there any difference, espectially for the improper integral case ?
I'm confused because we have the following identity :
$\int_a^bf(x)dx=-\int_b^af(x)dx$

But, in the other notation, there seems to be no differences between $[a,b]$ and $[b,a]$ ?

Answer 1

No. The first notation is more common for integrals over intervals, but the second type of notation is more convenient for more general domains (for example a complicated domain in $\mathbb R ^2$).