Unknowledgable of single variabled integral

Question

Browsing Stack-Exchange and other sites, I have noticed this come up quite a few times, an integral with only one letter or number! This would be a great example:

$$\int_{a}$$

What in the world does this mean? Thanks for any help whatsoever.

Answer 1

It is a notation for the integration domain. The notation is used for measurable set, curve, surface, manifold, etc. Usually, if the exact content of the domain is not of interest or too complex to write out explicitly.

For example:

• If $S$ is measurable set of a measure space $(\Omega, \Sigma, \mu)$, then $\int_S f \mathrm d\mu$ is defined as $\int \mathbb 1_S f \mathrm d \mu$ for any measurable function $f:\Omega\to\mathbb R$, where $\mathbb 1_S$ is the indicator of $S$.

• If $\gamma:[a,b]\to\mathbb R^2$ is $C^1$ and $f:\mathbb R^2\to\mathbb R$, then $\int_\gamma f(x) \cdot \mathrm d x = \int_a^b f(\gamma(t))\cdot \gamma'(t) \mathrm d t$.