Recently I've been writing integrals in the following way, for example
$\int\limits_{[0,1]} {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt} $
instead of
$\int\limits_0^1 {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt} $
or
$\int\limits_{\mathbb{R}} {\frac{1}{{1 + {t^2}}}dt} $
instead of
$\int\limits_{ - \infty }^\infty {\frac{1}{{1 + {t^2}}}dt} $
I did this because I thought the new notation highlights the fact that we're integrating over a line interval and not only in the extremes of the interval, so as no to "degrade" the definite integral to
$\int\limits_a^b {f\left( t \right)dt} = F\left( b \right) - F\left( a \right)$
Although I know virtually nothing about it, I remembered that in complex integration you integrate over a line, a curve in $\mathbb{R}^2$ as opposed to integrating in $\mathbb{R}$ (an interval). It also rang a bell that integrating over $(a,b)$ is the opposite as integrating over $(b,a)$ (i.e. taking the inverse "path" over the line) and I'm guessing this also happens in complex integration, i.e, the path you take changes the value of the integral.
So that's my doubt, is complex integration a generalization of the common integration in the real domain?