LineIntegrate
LineIntegrate[f,{x,y,…}∈curve]
computes the scalar line integral of the function f[x,y,…] over the curve.
LineIntegrate[{p,q,…},{x,y,…}∈curve]
computes the 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 function 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 as it moves 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 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 orientation 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 orientation 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 orientation 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:
-
Assumptions $Assumptionsassumptions to make about parameters Direction Automaticorientation of the curve GenerateConditions Automaticwhether to generate answers that involve conditions on parameters WorkingPrecision Automaticthe precision used in internal computations - LineIntegrate uses a combination of symbolic and numerical methods when the input involves inexact quantities.
Examples
open allclose allBasic 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:
Line integral of a vector field in two dimensions:
The vector field and the integration path:
Line integral of a vector field in three dimensions:
Scope (32)
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 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 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 of a scalar field over a parametric curve:
Line integral of a scalar field over a circle:
Contour plot of and the curve:
Line integral of a scalar field over the boundary of a sector of a disk:
Contour plot of and the curve:
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 of a vector field over a curve in two dimensions:
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 the boundary of an annulus:
Parametric Curves (1)
Line integral of a vector field over a spiral in three dimensions:
Options (5)
Assumptions (1)
The option Assumptions can be used on parameters:
Direction (1)
The default Direction for closed circular paths is counterclockwise:
The clockwise direction can be chosen using the option Direction:
GenerateConditions (1)
LineIntegrate can work with symbolic parameters:
Suppress the conditions on the parameters:
WorkingPrecision (2)
If a WorkingPrecision is specified, a numerical result is given:
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 centered at the origin:
Numerical value of the line integral of a vector field over a path:
Lengths (3)
Perimeter of a cardioid using a line integral:
The length can also be calculated with RegionMeasure:
Areas (5)
Area of an ellipse with semiaxes and , 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 the gravitational force as an object is moved on a straight line:
Work done by the electric force as a 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 and linear density :
coordinate of the center of mass:
coordinate of the center of mass:
Moments of inertia of a helix-shaped wire of constant 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:
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:
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 component of the center of mass:
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:
The result can be obtained using RegionMeasure:
Neat Examples (2)
Text
Wolfram Research (2023), LineIntegrate, Wolfram Language function, https://reference.wolfram.com/language/ref/LineIntegrate.html.
CMS
Wolfram Language. 2023. "LineIntegrate." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LineIntegrate.html.
APA
Wolfram Language. (2023). LineIntegrate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LineIntegrate.html