# Line integral

A line integral is an integral in which a function is evaluated along a curve, rather than along an interval of the x-axis. The curve may lie in a plane or in space, and the integrand may be a scalar function or a vector field. Line integrals generalize the single-variable integral \( \int_a^b f(x)\,dx \), whose domain of integration can be viewed as a line segment on the x-axis, to integration over arbitrary curves<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>.

| Key fact | Detail |
|---|---|
| Two types | Scalar line integrals integrate a scalar function over a curve; vector line integrals integrate a vector field along an oriented curve<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup> |
| Scalar formula | For a regular parametrization \( \gamma(t) \), \( t \in [a,b] \): \( \int_C f\,ds = \int_a^b f(\gamma(t))\,\|\gamma'(t)\|\,dt \)<sup>[2](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-line-integrals-4.html)</sup> |
| Vector formula | \( \int_C \mathbf{F} \cdot \mathbf{T}\,ds = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt \)<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup> |
| Existence | If \( f \) is continuous on a smooth curve \( C \), the scalar line integral \( \int_C f\,ds \) always exists<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup> |
| Orientation | Scalar line integrals do not depend on orientation or parametrization; reversing an oriented curve changes the sign of a vector line integral<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup> |
| Path independence | A line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) is independent of path when its value depends only on the endpoints; on an open connected region this is equivalent to \( \mathbf{F} \) being conservative<sup>[3](https://sites.science.oregonstate.edu/math/BridgeBook/book/sguide/path.html)</sup><sup> • </sup><sup>[4](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)</sup> |
| Physics uses | Work done by a force along a path, mass of a wire of varying density, and the circulation laws of electromagnetism (Ampère's law, Faraday's law)<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup><sup> • </sup><sup>[5](https://physicslibrary.org/encyclopedia/LineIntegral.html)</sup> |

## Definition of the scalar line integral

Let \( C \) be a smooth curve in a plane or in space, and let \( f \) be a function whose domain includes \( C \). The curve is chopped into small pieces; for each piece a point \( P \) is chosen, the value \( f(P) \) is multiplied by the arc length \( \Delta s \) of the piece, and the products \( f(P)\,\Delta s \) are added over all pieces. The scalar line integral is the limit of these sums as the arc lengths of the pieces shrink to zero<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>.

Formally, if \( C \) is parameterized by \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \) for \( a \le t \le b \), the parameter interval is partitioned into \( n \) subintervals, giving curve pieces of lengths \( \Delta s_1, \ldots, \Delta n \) with sample points \( P_i^* \), and

\[ \int_C f(x,y,z)\,ds = \lim_{n \to \infty} \sum_{i=1}^{n} f(P_i^*)\,\Delta s_i \]

provided the limit exists. Unlike the single-variable integral, the pieces of the curve need not have equal length; this difference has no effect in the limit<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>.

For computation, the arc-length definition is converted to an ordinary integral in the parameter. If \( s(t) \) measures distance along the curve from the start, then \( s'(t) = \|\gamma'(t)\| \), and the substitution rule gives

\[ \int_C f\,ds = \int_a^b f(\gamma(t))\,\|\gamma'(t)\|\,dt. \]

This value is independent of the particular regular parametrization chosen, and the integral over a piecewise smooth curve, one built from finitely many smooth arcs joined end to end, is defined as the sum of the integrals over the smooth parts<sup>[2](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-line-integrals-4.html)</sup>.

## Geometric and physical meaning

When \( f(x,y) \ge 0 \) on a planar curve \( C \), the scalar line integral has a direct geometric reading: project the curve up to the graph of \( f \), and drop a "sheet" from that lifted curve down to the plane. The area of this sheet is \( \int_C f\,ds \)<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>.

Two physical interpretations follow the same pattern. If a fence is built along a boundary curve \( C \) with height \( h(x,y) \) varying from point to point, the amount of fencing, the surface area of the fence, is \( \int_C h\,ds \). If \( C \) is a bent wire whose mass per length is \( \rho(x,y,z) \), the total mass of the wire is \( \int_C \rho\,ds \)<sup>[2](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-line-integrals-4.html)</sup>. More generally, in physics a scalar line integral sums a distributed quantity along a wire, trajectory, streamline, orbit, or field line<sup>[5](https://physicslibrary.org/encyclopedia/LineIntegral.html)</sup>.

## Vector line integrals and work

The second type of line integral integrates a vector field \( \mathbf{F} \) along an oriented curve. Its motivating application is work. If a particle moves along \( C \) through a force field, the work done on a small piece of the path is approximately \( \mathbf{F}(P_i^*) \cdot (\Delta s_i\,\mathbf{T}(P_i^*)) \), where \( \mathbf{T} \) is the unit tangent vector. Summing and taking the limit defines the vector line integral<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>:

\[ \int_C \mathbf{F} \cdot \mathbf{T}\,ds = \lim_{n \to \infty} \sum_{i=1}^{n} \mathbf{F}(P_i^*) \cdot \mathbf{T}(P_i^*)\,\Delta s_i. \]

Because \( \mathbf{T} = \mathbf{r}'(t) / \|\mathbf{r}'(t)\| \) and \( ds = \|\mathbf{r}'(t)\|\,dt \), the computation reduces to an ordinary integral<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>:

\[ \int_C \mathbf{F} \cdot \mathbf{T}\,ds = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt. \]

In the common shorthand \( d\mathbf{r} = \mathbf{r}'(t)\,dt \), the same integral is written \( \int_C \mathbf{F} \cdot d\mathbf{r} \); to evaluate it, the vector field must be composed with the parametrization before integrating in \( t \) over \( [a,b] \)<sup>[6](https://sites.ualberta.ca/~vbouchar/MATH315/section_line_integrals.html)</sup>. These are the "work integrals" of standard vector calculus textbooks<sup>[6](https://sites.ualberta.ca/~vbouchar/MATH315/section_line_integrals.html)</sup>.

## Orientation and parametrization

The two types of integral behave differently under reversal of the path. A scalar line integral does not change if the direction of travel along the curve is reversed, because \( ds \) is a nonnegative length element; the area of the sheet above a closed planar curve is the same whether the circle is traversed clockwise or counterclockwise<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup><sup> • </sup><sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch15.S3.html)</sup>. A vector line integral does depend on orientation: reversing the path changes the sign, so if \( C \) is an oriented curve, \( -C \) denotes the same curve with opposite orientation and \( \int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r} \)<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>. The work interpretation makes this concrete: hiking up a mountain, gravity does negative work; walking down the same path, it does positive work<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup>.

Neither type depends on the choice of parametrization, as long as the curve is traversed exactly once and, for vector integrals, the orientation is preserved<sup>[1](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)</sup><sup> • </sup><sup>[2](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-line-integrals-4.html)</sup>.

## Path independence and conservative fields

A line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) is said to be independent of path if its value depends only on the endpoints of \( C \), not on the particular curve connecting them<sup>[3](https://sites.science.oregonstate.edu/math/BridgeBook/book/sguide/path.html)</sup>. The fundamental theorem for line integrals connects this property to potential fields: if \( \mathbf{F} \) is a continuous vector field on an open connected region and \( \int_C \mathbf{F} \cdot d\mathbf{r} \) is independent of path for any path in the region, then \( \mathbf{F} \) is a [conservative vector field](https://www.edgepedia.org/conservative-vector-field) there, meaning it is the gradient of a potential function<sup>[4](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)</sup>.

Electrostatic fields have this property: the line integral of \( \mathbf{E} \) around any closed path is zero. Static magnetic fields do not. A conservative vector field is one whose line integral between two endpoints is the same regardless of the path chosen, and magnetic fields lack this property; instead, their circulation around a closed loop is tied to the electric current through the loop<sup>[8](https://openstax.org/books/university-physics-volume-2/pages/12-5-amperes-law)</sup>.

## Line integrals in electromagnetism

Two of Maxwell's equations are stated as line integrals around closed loops, denoted with a circle on the integral sign. Ampère's law relates the circulation of the magnetic field to the current passing through the loop:

\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}. \]

In the form used for [magnetostatics](https://www.edgepedia.org/magnetostatics), the circulation of \( \mathbf{B} \) around any closed curve equals the current through the loop divided by \( \varepsilon_0 c^2 \)<sup>[8](https://openstax.org/books/university-physics-volume-2/pages/12-5-amperes-law)</sup><sup> • </sup><sup>[9](https://www.feynmanlectures.caltech.edu/II_13.html)</sup>. Faraday's law gives the corresponding statement for the electric field: the circulation of \( \mathbf{E} \) around a closed loop equals the rate of change of the magnetic flux through the enclosed area, which is why the line integral of \( \mathbf{E} \), zero for electrostatic fields, is no longer zero when fields change in time<sup>[9](https://www.feynmanlectures.caltech.edu/II_13.html)</sup>.

These closed-loop integrals are evaluated most easily when symmetry makes the field constant or zero along the chosen path: for an infinitely long straight wire, the field is constant on a circle centered on the wire, so \( \oint \mathbf{B} \cdot d\mathbf{l} \) reduces to \( B \) times the circumference, which can be solved for \( B \)<sup>[8](https://openstax.org/books/university-physics-volume-2/pages/12-5-amperes-law)</sup>.

## References

1. [6.2 Line Integrals, Calculus Volume 3, OpenStax](https://openstax.org/books/calculus-volume-3/pages/6-2-line-integrals)
2. [Integrals of a Scalar Function, Vector Calculus, University of Sydney](https://www.maths.usyd.edu.au/u/daners/publ/vector-calculus/chapter-line-integrals-4.html)
3. [Path Independence, The Geometry of Vector Calculus, Oregon State University](https://sites.science.oregonstate.edu/math/BridgeBook/book/sguide/path.html)
4. [Calculus III, Fundamental Theorem for Line Integrals, Paul's Online Notes](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)
5. [Line Integral, Physics Library Encyclopedia](https://physicslibrary.org/encyclopedia/LineIntegral.html)
6. [Line Integrals, MATH 315, University of Alberta](https://sites.ualberta.ca/~vbouchar/MATH315/section_line_integrals.html)
7. [Line Integrals over Vector Fields, APEX Calculus LT](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch15.S3.html)
8. [12.5 Ampère's Law, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/12-5-amperes-law)
9. [The Feynman Lectures on Physics Vol. II Ch. 13: Magnetostatics](https://www.feynmanlectures.caltech.edu/II_13.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Integrals and integration theory*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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