# NLineIntegrate

NLineIntegrate[f,{x,y,}curve]

computes the numerical scalar line integral of the function f[x,y,] over the curve.

NLineIntegrate[{p,q,},{x,y,}curve]

computes the numerical vector line integral of the vector function {p[x,y,],q[x,y,],}.

# Details and Options

• Line integrals are also known as curve integrals and work integrals.
• Scalar line integrals integrate scalar functions along a curve. They typically compute things like length, mass and charge for a curve.
• Vector line integrals are used to compute the work done by a vector function along a curve in the direction of its tangent. Typical vector functions include a force field, electric field and fluid velocity field.
• The scalar line integral of the function f along a curve is given by:
• where is the measure of a parametric curve segment.
• The scalar line integral is independent of the parametrization and orientation of the curve. Any one-dimensional RegionQ object can be used as a curve.
• The vector line integral of the function F along a curve is given by:
• where is projection of the vector function onto the tangent direction so only the component in the tangent direction gets integrated.
• The vector line integral is independent of the parametrization of the curve, but it does depend on the orientation of the curve.
• The orientation for a curve is given by a tangent vector field over the curve.
• For a parametric curve ParametricRegion[{r1[u],,rn[u]},], the tangent vector field is taken to be ur[u].
• Special curves in with their assumed tangent orientations include:
•  Line[{p1,p2,…}] the orientation follows the points in the order they are given from p1 to p2 etc. HalfLine[{p1,p2}]HalfLine[p,v] the orientation is from p1 to p2 or in the v direction InfiniteLine[{p1,p2}]InfiniteLine[p,v] the orientation is from p1 to p2 or in the v direction Circle[p,r] the orientation is counterclockwise
• Special curves in with their assumed tangent orientations include:
•  Line[{p1,p2,…}] the orientation follows the points in the order they are given from p1 to p2 etc. HalfLine[{p1,p2}]HalfLine[p,v] the orientation is from p1 to p2 or in the v direction InfiniteLine[{p1,p2}]InfiniteLine[p,v] the orientation is from p1 to p2 or in the v direction
• Special curves in with their assumed tangent orientations include:
•  Line[{p1,p2,…}] the orientation follows the points in the order they are given HalfLine[{p1,p2}]HalfLine[p,v] the orientation is from p1 to p2 the orientation is given by v InfiniteLine[{p1,p2}]InfiniteLine[p,v] the orientation is from p1 to p2 the orientation is given by v
• The following options can be given:
•  AccuracyGoal Infinity digits of absolute accuracy sought MaxPoints Automatic maximum total number of sample points MaxRecursion Automatic maximum number of recursive subdivisions Method Automatic method to use MinRecursion 0 minimum number of recursive subdivisions PrecisionGoal Automatic digits of precision sought WorkingPrecision MachinePrecision the precision used in internal computations

# Examples

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## Basic Examples(6)

Line integral of a scalar field over a circle:

Line integral of a vector field over a line segment:

Line integral of a vector field over a space curve:

Line integral of a scalar field in two dimensions:

Curve over which to integrate:

A contour plot of and the curve:

The line integral:

Line integral of a vector field in two dimensions:

The vector field and the integration path:

The line integral:

Line integral of a vector field in three dimensions:

## Scope(33)

### Basic Uses(4)

Line integral of a scalar field:

Line integral of a vector field in three dimensions:

LineIntegrate works with many special curves:

Line integral over a parametric curve:

### Scalar Functions(11)

Line integral of a scalar field over a curve:

Contour plot of and the curve:

Line integral:

Line integral of a scalar field over an arc of a circle:

Line integral of a scalar field over a parametric curve:

Contour plot of the function and the curve:

Line integral of a scalar field over a circle:

Line integral of a scalar field over a space curve:

Line integral:

Line integral of a scalar field over the boundary of an annulus:

Contour plot of the function and the curve:

Line integral of a scalar field over a closed polygon:

Contour plot of the function and the curve:

Line integral of a scalar field over an elliptical path:

Contour plot of the function and the curve:

Line integral:

Line integral of a scalar field over a parametric curve:

Line integral:

Line integral of a scalar field over a circle:

Contour plot of and the curve:

Line integral:

Line integral of a scalar field over the boundary of a sector of a disk:

Contour plot of and the curve:

Line integral:

### Vector Functions(12)

Line integral of a vector field in three dimensions over a parametrized curve:

Visualization of the vector field and the curve:

Line integral:

Line integral of a vector field over a curve in two dimensions:

Line integral:

Line integral of a vector field over a circular arc:

Line integral of a vector field over a line segment:

Line integral of a vector field over a parametrized curve in three dimensions:

Line integral of a vector field over a curve:

Line integral of a vector field over an elliptical arc:

Line integral of a vector field over a parametric curve:

Line integral of a vector field over a parametric curve in three dimensions:

Line integral of a vector field over a parametrized curve:

Line integral of a vector field over an elliptical path:

Line integral of a vector field in higher dimensions:

### Special Curves(4)

Line integral over a circular arc:

Line integral of a vector field over the boundary of a circular sector of radius 1:

Line integral over a polygon:

Line integral over the boundary of an annulus:

### Parametric Curves(2)

Line integral of a vector field over a spiral in three dimensions:

Line integral of a scalar field over a parametric curve:

## Options(8)

### AccuracyGoal(1)

The option AccuracyGoal sets the number of digits of accuracy:

The result with default settings only sets a PrecisionGoal:

### MaxPoints(1)

The option MaxPoints stops the integration after a specified number of points has been evaluated:

With default options:

### MaxRecursion(1)

The option MaxRecursion specifies the maximum number of recursive steps:

Increasing the number of recursions:

The exact result is:

### Method(1)

The option Method can take the same values as in NIntegrate. For example:

With the default option:

Compare to the truncated exact result:

### MinRecursion(1)

The option MinRecursion forces a minimum number of subdivisions:

Compare to the exact result:

### PrecisionGoal(1)

The option PrecisionGoal sets the relative tolerance in the integration:

With default settings:

### WorkingPrecision(2)

If a WorkingPrecision is specified, the computation is done at that working precision:

The result has finite precision if the integrand has a finite precision:

## Applications(27)

### College Calculus(10)

Line integral of a function over a line segment:

Line integral of a vector field over a curve:

Mass of a thin circular wire of radius 1 with linear density :

Work done by the force field on a particle that moves along a line segment:

Line integral of a vector field along a path:

Line integral of a vector field along a curve:

Work done by the force as a particle moves along the curve :

Line integral of a vector field along the unit circle centered at the origin:

Line integral of a vector field along a circle of radius 2 centered at the origin:

Numerical value of the line integral of a vector field over a path:

### Lengths(3)

Circumference of a circle:

Perimeter of a cardioid using a line integral:

The length can also be calculated with RegionMeasure:

Perimeter of an astroid:

### Areas(5)

Area of an ellipse with semiaxes of length 2 and 3, calculated using a line integral:

Area of the right-hand loop of the lemniscate computed using a line integral:

Area of the epicycloid of parameters and :

Area of the cardioid using a line integral:

Area of an astroid using a line integral:

### Work by a Force(4)

Work done by a force force as an object is moved on a straight line:

Work done by the electric force as a charged particle of charge is moved from {1,1,0} to {2,2,0} in the electric field of a charged infinite wire of charge density :

Work done by an elastic force directed toward the origin as a quarter of an ellipse is traced:

Work of the electric force as a charge is moved along the axis from to infinity in the electric field of a charge :

### Centroids(2)

Mass of a closed semicircular wire of radius 2 and unit linear density:

coordinate of the center of mass:

coordinate of the center of mass:

Moments of inertia of a helix-shaped wire of unit linear density:

### Classical Theorems(3)

A vector field is conservative if its line integral depends only on the values at the endpoints, not on the path:

The field is the gradient of a scalar function :

All gradients of scalar fields are conservative. For example, the line integral of over the curve is:

This is the same as the difference of the values of at the endpoints of the curve:

Green's theorem. The line integral of the vector field over a closed curve is:

This can be related to a surface integral of over the region enclosed by the curve, where is defined as:

Stokes's theorem. The line integral of a vector field along a closed line in three dimensions is:

This is equal to the surface integral of the Curl of on any surface having the curve as its boundary:

The surface integral across a different surface with the same boundary is the same:

## Properties & Relations(5)

Apply N[LineIntegrate[]] to obtain a numerical solution if the symbolic calculation fails:

Find the center of mass of a triangular wire of unit linear density:

Find the total mass:

Find the component of the center of mass:

Find the component:

The center of mass can also be obtained using RegionCentroid:

Find the moment of inertia around the axis of a circular wire of unit linear density centered at the origin in the - plane:

The answer can also be computed with MomentOfInertia:

Find the length of an epicycloid:

The same answer can be obtained using ArcLength:

Find the area of an ellipse:

The result can be obtained using RegionMeasure:

## Neat Examples(2)

Length of a catenary:

Integral of a vector field over a Clelia curve:

Wolfram Research (2023), NLineIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/NLineIntegrate.html.

#### Text

Wolfram Research (2023), NLineIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/NLineIntegrate.html.

#### CMS

Wolfram Language. 2023. "NLineIntegrate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NLineIntegrate.html.

#### APA

Wolfram Language. (2023). NLineIntegrate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NLineIntegrate.html

#### BibTeX

@misc{reference.wolfram_2024_nlineintegrate, author="Wolfram Research", title="{NLineIntegrate}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/NLineIntegrate.html}", note=[Accessed: 23-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_nlineintegrate, organization={Wolfram Research}, title={NLineIntegrate}, year={2023}, url={https://reference.wolfram.com/language/ref/NLineIntegrate.html}, note=[Accessed: 23-July-2024 ]}