 # Line Integral (Path Integral): Simple Definition, Examples

A line integral (also called a path integral) is the integral of a function taken over a line, or curve.

The integrated function might be a vector field or a scalar field; The value of the line integral itself is the sum of the values of the field at all points on the curve, weighted by a scalar function. That weight function is commonly the arc length of the curve, or—if you’re integrating over a vector field—the scalar product with a vector differential in the curve.

It’s this weighting which sets a line integral apart from the integrals studied early on in calculus—simpler integrals defined on intervals.

## Example of a Line Integral

A simple example of a line integral is finding the mass of a wire if the wire’s density varies along its path. If you were to divide the wire into x segments of roughly equal density (as shown above), you could sum all of the segment’s densities to find the total density using the following mass function: Where:

• dxi = length of each segment
• λi = linear density of each segment.

However, if those line segments approach a length of zero, you could integrate to find a more accurate number for density. ## Applications of Line Integrals in Physics

Many simple physical formulas can be written in terms of line integrals. For example, work is force times distance:
W = F · s.
If an object is moving through an electric or gravitational field, you can write it as: ## The Line Integral in a Scalar Field

The animation below shows how a line integral over some scalar field f can be thought as the area under the curve C along a surface z = f(x,y), where z is described by the field. If C is a smooth, piecewise curve, and f is a scalar field such that then Here r: [a,b]→ C is an arbitrary bijective parameterization of the cure C such that—and this is key—r(a) and r(b) give the endpoints of C. a is also defined to be less than b.

## References

Fleisch, D. (2008). A student’s Guide to Maxwell’s Equations. Cambridge University Press.
Sharma, A. Text Book of Vector Calculus.