# Line Integral

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## line integral

[′līn ¦int·ə·grəl]**x**=

**x**(

*t*), and a vector function

**V**defined on this curve, the line integral of

**V**along the curve is the integral over

*t*of the scalar product of

**V**[

**x**(

*t*)] and

*d*

**x**/

*dt*; this is written ∫

**V**·

*d*

**x**.

*x*=

*x*(

*t*),

*y*=

*y*(

*t*), and a scalar function ƒ depending on

*x*and

*y*, the line integral of ƒ along the curve is the integral over

*t*of ƒ[

*x*(

*t*),

*y*(

*t*)] · √(

*dx*/

*dt*)

^{2}+ (

*dy*/

*dt*)

^{2}); this is written ∫ ƒ

*ds*, where

*ds*= √(

*dx*)

^{2}+ (

*dy*)

^{2}) is an infinitesimal element of length along the curve.

*z*=

*z*(

*t*), and a function ƒ depending on

*z*, the line integral of ƒ along the curve is the integral over

*t*of ƒ[

*z*(

*t*)] (

*dz*/

*dt*); this is written ∫ ƒ

*dz*.

## Line Integral

an integral taken along some curve in the plane or in space. We distinguish line integrals of the first kind and line integrals of the second kind. A line integral of the first kind arises, for example, in problems involving the calculation of the mass of a curve of variable density and is denoted by

∫_{C}^{f (P)ds}

where *C* is the given curve, *ds* is the differential of its arc, and *f(P)* is the function of a point on the curve and is the limit of the corresponding integral sums. In the case of a plane curve *C* given by the equation *y* = *y(x),* a line integral of the first kind reduces to an ordinary integral. Specifically,

A line integral of the second kind arises, for example, in connection with the work of a force field. In the case of a plane curve *C* the integral takes the form

∫_{C}*P(x,y)dx* + *Q(x,y)dy*

and is also the limit of the corresponding integral sums. A line integral of the second kind can be expressed as an ordinary integral. Specifically,

where*x = x(t)* and *y = y(t)* for α ≤ *t* ≤ β, is the parametric equation of the curve *C*. Its connection with a line integral of the first kind is given by the equality

∫_{C}*P(x,y)dx* + *Q* (*x,y*)*dy* = ∫_{C} {*P* cosα + *Q* sin α} *ds*

Here, α is the angle between the *Ox* axis and the tangent to the curve pointing in the direction of the increasing arc length.

A line integral of the second kind in space is defined similarly. *(See*VECTOR CALCULUS for a treatment of line integrals of the second kind from the standpoint of vectors.)

Suppose *D* is some region and *C* is its boundary. Under certain conditions, the line integral along the curve *C* and the double integral over the region *D* (*see*) are connected by the relation

Similarly, the line integral and the surface integral are connected by the relation

Line integrals are of great importance in the theory of functions of a complex variable. They are widely used in various branches of mechanics, physics, and engineering.