From AMS Glossary
The integral of a function along a given curve.
Mathematically, if s is a linear coordinate along the curve, the line integral of a function f(s) between points A and B on the curve is
where f(s) is evaluated on the curve. The line integral of a vector field F along the same curve is
where dr is a vector element along the curve of magnitude ds. In general the integral will depend on the curve; in the special case where the field F is irrotational, it depends only on the points A and B and therefore vanishes for a closed curve. In meteorology, line integrals are of frequent occurrence, for example, in the circulation, and in the averaging of a quantity along a latitude circle.