Line Integrals — Mathematics Knowledge System

Overview

A line integral generalizes ordinary integration from a straight interval [a,b] to an arbitrary curve C in the plane or in space. A scalar line integral ∫_C f ds sums a scalar quantity (like mass density) along the curve's arc length; a vector line integral ∫_C F·dr sums the component of a vector field F along the curve's direction of travel, computing quantities like work done by a force along a path.

Intuition

For the scalar case, imagine a wire shaped like curve C with variable density f at each point; ∫_C f ds adds up mass along the wire's length. For the vector case, imagine pushing a particle along a path C through a force field F (like wind or gravity); ∫_C F·dr adds up the tiny amounts of work F·dr done at each infinitesimal step along the path — work being force times the component of displacement in the force's direction.

Interactive Graph

Formal Definition

Definition

Scalar and vector line integrals along a curve C parametrized by r(t), t∈[a,b]:

\int_C f\, ds = \int_a^b f(\mathbf r(t))\, |\mathbf r'(t)|\, dt

ds = |r'(t)| dt is the arc-length element

Scalar line integral

\int_C \mathbf F \cdot d\mathbf r = \int_a^b \mathbf F(\mathbf r(t)) \cdot \mathbf r'(t)\, dt

Vector line integral (work integral)

\int_C \mathbf F \cdot d\mathbf r = f(\mathbf r(b)) - f(\mathbf r(a))

Valid when F=∇f is conservative — the integral only depends on endpoints, not the path

Fundamental Theorem for Line Integrals

Notation

Notation	Meaning
$\int_C f\, ds$	Scalar line integral of f along curve C with respect to arc length
$\int_C \mathbf F \cdot d\mathbf r$	Vector line integral (work) of field F along curve C
$\oint_C$	Line integral around a closed curve C

Derivation

The vector line integral formula follows from the definition of work as force times displacement, applied incrementally: over a small parameter step dt, the particle moves by dr=r'(t)dt, and the incremental work is F·dr.

dW = \mathbf F(\mathbf r(t)) \cdot d\mathbf r = \mathbf F(\mathbf r(t)) \cdot \mathbf r'(t)\, dt

Incremental work over an infinitesimal displacement along the path

W = \int_a^b \mathbf F(\mathbf r(t)) \cdot \mathbf r'(t)\, dt

Summing (integrating) incremental work over the whole parameter range gives total work

Properties

Reversal of orientation

\int_{-C} \mathbf F \cdot d\mathbf r = -\int_C \mathbf F \cdot d\mathbf r

Condition: Reversing the direction of travel negates the vector line integral (scalar line integrals ∫f ds are unaffected)

Additivity over pieced curves

\int_{C_1 \cup C_2} \mathbf F \cdot d\mathbf r = \int_{C_1} \mathbf F\cdot d\mathbf r + \int_{C_2} \mathbf F \cdot d\mathbf r

Theorems

Theorem 1: Fundamental Theorem for Line Integrals

\text{If } \mathbf F=\nabla f \text{ and } C \text{ goes from } \mathbf a \text{ to } \mathbf b, \text{ then } \int_C \mathbf F\cdot d\mathbf r = f(\mathbf b)-f(\mathbf a)

Theorem 2: Path independence for conservative fields

\mathbf F \text{ conservative} \implies \int_C \mathbf F\cdot d\mathbf r \text{ depends only on the endpoints of } C, \text{ not the path taken}

Corollaries

Follows from Fundamental Theorem for Line Integrals

If F is conservative and C is a closed curve (a = b), then ∮_C F·dr = 0.

Applications

Work done by a force field on a particle moving along a path is exactly a vector line integral W=∫_C F·dr; conservative force fields (gravity, electrostatics) give path-independent work.

Worked Examples

Compute r'(t) = (1, 2t).
$\mathbf r'(t) = (1, 2t)$
Compute F(r(t)) = ⟨t², t⟩, then dot with r'(t).
$\mathbf F(\mathbf r(t))\cdot \mathbf r'(t) = t^2\cdot 1 + t\cdot 2t = t^2+2t^2 = 3t^2$
Integrate over t from 0 to 1.
$\int_0^1 3t^2\, dt = \left[t^3\right]_0^1 = 1$

Answer: 1

Practice Problems

Difficulty 6/10

Evaluate ∫_C F·dr where F=⟨2x,3y⟩ and C is r(t)=(cos t, sin t), t∈[0, π/2].

Difficulty 7/10

Show F=⟨2xy, x²⟩ is conservative, find its potential, and use it to evaluate ∫_C F·dr from (0,0) to (2,3).

Common Mistakes

Common Mistake

Assuming ∫_C F·dr is always path-independent.

Path independence only holds for CONSERVATIVE vector fields (F=∇f). For a general vector field, different paths between the same two points can give different line integral values.

Common Mistake

Forgetting to include |r'(t)| in a scalar line integral ∫f ds, treating it like a vector line integral.

Scalar line integrals require ds=|r'(t)|dt (arc-length element, always positive), while vector line integrals use dr=r'(t)dt directly (a vector, direction matters). Mixing these up gives wrong signs or magnitudes.

Quiz

A line integral ∫_C F·dr computes:

the area under a curvethe work done by a force field F along the path Cthe length of C onlythe volume of a region

For a CONSERVATIVE field F = ∇f, the line integral between two points:

depends on the path takendepends only on the endpoints (path-independent)is always zeroequals the arc length

Historical Background

Line integrals developed from 19th-century physics, particularly the concept of work done by a force along a path, formalized as vector calculus matured with Green, Stokes, and Maxwell's development of field theory in the mid-1800s. The distinction between path-independent (conservative) and path-dependent line integrals became central to understanding energy conservation in mechanics and electromagnetism.

1820s-1830s
George Green develops potential theory, laying groundwork for path-independent line integrals
George Green
1850s
William Thomson (Lord Kelvin) and Stokes formalize vector line integrals for physics applications
Lord Kelvin, George Gabriel Stokes

Summary

Line integrals generalize integration to curves: scalar ∫_C f ds sums a quantity over arc length; vector ∫_C F·dr sums a field's component along the direction of travel.
Vector line integrals compute work done by a force field along a path.
The Fundamental Theorem for Line Integrals: if F=∇f, then ∫_C F·dr = f(endpoint) − f(start), independent of the path.
Reversing a curve's orientation negates a vector line integral but leaves a scalar line integral unchanged.
Sets up the machinery (circulation, work) needed for Green's theorem and the broader Stokes/divergence theorems.

References

WebsiteWikipedia — Line integral