travelintegral

Travel integral is a term used in mathematical modeling to describe the accumulation of a quantity along a travel path. It behaves similarly to a line or path integral in calculus but is often used in the context of transportation, navigation, or movement analysis to capture quantities that are transported or accumulated as an object travels.

For a parametric path r(t), t in [a,b], and a scalar field f defined along the path, the travel integral of f along the path is ∫_a^b f(r(t)) |r'(t)| dt, equivalently ∫_C f ds, where ds is the differential of arc length. If the integrand depends on direction or velocity, one may consider ∫_a^b g(r(t), r'(t), t) dt. The travel integral can also represent time along a path if the speed is known: T = ∫ ds / v(s).

As an application, travel integrals are used in transportation planning to quantify total exposure to a pollutant along a route, total energy or fuel consumption given a per-distance rate, or cumulative scenic value by weighting each segment accordingly.

The term is not standard in all texts; in many areas a path or line integral is used, with f representing density and ds or F·dr representing work. The travel integral is thus a flexible framing for accumulating movement-related quantities and may be given different names in applied fields.