# Is line integration a generalization of the definite integral in $\mathbb{R}$?

Recently I've been writing integrals in the following way, for example

$$\int\limits_{[0,1]} {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt}$$

$$\int\limits_0^1 {{t^{y - 1}}{{\left( {1 - t} \right)}^{x - 1}}dt}$$

or

$$\int\limits_{\mathbb{R}} {\frac{1}{{1 + {t^2}}}dt}$$

$$\int\limits_{ - \infty }^\infty {\frac{1}{{1 + {t^2}}}dt}$$

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?

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A line integral is ultimately a Riemann integral if that is what you're asking. FWIW, I prefer the standard notation over yours. – ShawnD Feb 14 '12 at 17:09
If you've taken a course or read a book on multivariable calculus, then line integrals are definitely a generalization. While @Shawn is welcome to his opinion, I rather like your notation +1. – Joe Johnson 126 Feb 14 '12 at 17:14
I like the notation $\int_{I} f(x)~dx$ fine, but writing out intervals makes my LaTeX look cluttered. – ShawnD Feb 14 '12 at 17:51
@Shawn: Something I had not thought about. – Joe Johnson 126 Feb 14 '12 at 18:08
If you write $\int_X$ and $X$ is just a set, then what you have must be some kind of Lebesgue integral, i.e. unoriented. Philosophically, if one wants to be able to distinguish between $\int_a^b$ and $\int_b^a$ in this notation, one must take $X$ be something with a chosen orientation – in particular, it can't be a set. But it could be a oriented manifold, or it could be a suitable homology class... – Zhen Lin Feb 14 '12 at 19:21

Regarding the notation, your choice is unusual for the basic calculus integral, but is definitely better, in the sense that when you move to Lebesgue integration, you can integrate over sets which are not defined by two endpoints, and you write $$\int_X\, f$$ Since in Lebesgue integration $X$ may be even non-numerical, in that context the classical notation makes no sense.