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:

- dx
_{i}= length of each segment - λ
_{i}= 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.

Vector Calculus: Line Integrals. Retrieved from https://www.whitman.edu/mathematics/calculus_online/section16.02.html on March 8, 2019

Calculus Quest Study: Introduction to Line Integrals. Retrieved from https://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lint/lint.html on March 8, 2019

Vector Calculus: Fundamental Theorem of Line Integrals. Retrieved from https://www.whitman.edu/mathematics/calculus_online/section16.03.html on March 8, 2019.

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