Line Integral

Definition (Line Integral) Let be a smooth curve, with parametric equations and for that lies within the domain of a function We say that is orientable if it is possible to describe direction along the curve for increasing Partition into sub-arcs, the of which has length Let be a point chosen arbitrarily from the sub-arc. Form the Riemann sum and let denote the largest sub-arc length in the partition. Then, if the limit   



exists, we call this limit the line integral of over and denote it by   



Additionally, if is a closed curve, then we denote the line integral by

Proposition (Line Integral) Suppose that the function is continuous at each point on a smooth curve ,  with parametric equations and for that lies within the domain of Then exists and   



and thus

    The definition of a line integral can be extended to curves that are piecewise smooth in the sense that they are the union of a finite number of smooth curves with only endpoints in common. In particular, if is comprised of a number of smooth sub-arcs then  

Proposition (Properties of Line Integrals) Let and be scalar functions defined on a piecewise smooth orientable curve Then, for any constants and

(i) Linearity rule:



(ii) Subdivision rule:  



where is the union of smooth orientable sub-arcs with only endpoints in common.

(iii) Opposite direction rule:

Example (Line Integral) Evaluate where is the circular helix given by the equations   

    Solution. We compute,

    Other line integrals are obtained by replacing by This is called the line integral of along with respect to    



and similarly with respect to and ,   



and   



When we want to distinguish the original line integral from these, we call it the line integral with respect to arc length. The following formulas say that the line integrals with respect to or can also be evaluated by expressing everything in terms of : and yielding:





and

Example (Line Integral Evaluation) Evaluate where consists of the line segment from to followed by the vertical line segment from to

    Solution. The curve is the union of the curves



and the curve   



Thus

Definition (Line Integral of Vector Field) Let  



be a vector field, and let be a piecewise smooth orientable curve with parametric representation   



Using we define the line integral of along by  



  

  

  
  

Proposition (Line Integral of Vector Field) Let be a smooth curve and let be a continuous function with domain containing the trace of Then the value of the integrals   



depends only on the initial point terminal point and the trace of That is, two different parametrizations having the same trace from to yield the same values for these integrals.

Example (Line Integral of Vector Field) Consider the smooth curves



and



Both and are smooth curves from to with the same trace which is the portion of the parabola for For we have    and therefore   

  

For we have    and therefore   

Proposition (Work as a Line Integral) Let be a continuous force field over a domain Then the work performed as an object moves along a smooth curve in is given by the integral where is the unit tangent at each point on and is the position vector of the object moving on

Example (Work as a Line Integral) Find the work done by the force field   



on an object moving along the curve defined parametrically by   



    Solution. We compute,
    


    

(from ) and



Thus,   

Proposition (Fundamental Theorem of Line Integrals) Let be a piecewise smooth curve that is parametrized by the vector function for and let be a vector field that is continuous on If is a scalar function such that then   



where and are the endpoints of

    Proof. Suppose and let be the composite function We have,

Example (Fundamental Theorem of Line Integrals) Find the work done by the gravitational field (Newton's law of gravitation)   



where and are the masses of two objects and is the gravitational constant, in moving a particle from the point to the point along a piecewise smooth curve

    Solution. We have where    Therefore,   


    

    

Definition (Independence of Path) The line integral is independent of path in a region if for any two points and in the line integral along every piecewise smooth curve in from to has the same value.

Example (Criterion for Conservative Vector Field) Show that the vector field   



is conservative and evaluate for any piecewise smooth path joining to

    Solution. First we determine We find,  
    






and so the vector field is conservative and we can find the scalar potential function. Now we set out to find with Since we know that



Then, we find and so comparing this with the given   we determine that and so So far we have,



Also since, and comparing this to the given we determine and so is a constant with respect to and Therefore, a scalar potential function is



(taking the constant to be zero). Finally,   

Proposition (Independence of Path) If is a continuous vector field on the open connected set then the following three conditions are either all true or all false:

    (i) is conservative on

    (ii) for every piecewise smooth closed curve in

    (iii) is independent of path within
    

Example (Independence of Path) Show that no work is performed when an object moves along a closed path in a connected domain where the force field is conservative.

    Solution. In such a force field where is a scalar potential of and because the path of motion is closed, it begins and ends at the same point Thus, the work is given by   

    Strategy for Line Integrals: Three ways for evaluating a given line integral , are

    (i)  Parametrize and use the parametrization to convert the line integral into a one-variable integral involving over an interval

    (ii)  Check to see if is conservative; if it is, then find a scalar potential function and use the fundamental theorem of line integrals.

    (iii)  If is conservative, find a convenient path with the same endpoints as and use the fact that since the line integral is independent of path.

Proposition (Line Integral for Area) If is a region bounded by a piecewise smooth simple closed curve oriented counterclockwise, then the area of is given by

Example (Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the circle

    Solution. We can parametrize the circle by and for Then the area is found by,
    




Example (Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the triangle with vertices

    Solution. We can parametrize the line segments by
    






Then the area is found by,
    









Example (Line Integral for Area) Use a line integral to find the area enclosed by the region defined by the curve for

    Solution. We have,
    






















by using the reduction formula,

• print this page
• pages linking here
• related pages

Cite this as:
Line Integral
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/line-integral.html