Is the notation $\int_{\Bbb R}$ equivalent to $\int_{-\infty}^\infty$? The question is pretty straightforward. In class, I have seen $\int_{\Bbb R}$ and $\int_{-\infty}^\infty$ being used interchangeably. However $\int_{\Bbb R}$ contains no implicit sense of direction, so technically isn't $\int_\infty^{-\infty}$ also the same as $\int_{\Bbb R}$? 
Or $\int_{\infty}^{0}+\int_{-\infty}^{0}\equiv\int_{\Bbb R}$, provided all integrals exist? 
There could be several such possibilities. Is that why we have attached a conventional direction to $\int_{\Bbb R}$, i.e. from $-\infty$ to $\infty$?

As a Riemann integral, the notation $\int_\mathbb{R}$ necessarily implies a positive orientation, so the only interpretation is $\int_{-\infty}^\infty.$
This type of notation is more common in Riemann integrals when we talk about the multiple integrals $\iint_D f(x,y)\,dA$ and $\iiint_R f(x,y,z)\,dV$ where $D,R$ are domains of integration for the integrals.  In these cases, as in the one-dimensional case, the integral always is interpreted with a positive orientation unless otherwise specified.
