Line Integrals

In this topic:

    (1) Define a line integral on a piecewise smooth curve.
    
    (2) State a proposition on how to evaluate a line integral given parametric equation for the curve.
    (3) State three basic properties of line integrals.
    
    (4) Evaluate where is the circular helix given by the equations   
    
    (5) Evaluate where consists of the line segment from to followed by the vertical line segment from to
    

    Line integrals have a great many applications in the sciences, but were originally constructed mainly to compute work and mass in physics. Generally, if direction can be described along a curve using a parametric representation (called an orientable curve), then as a generalization of the Riemann integral it is possible to define a line integral where Riemann sums are constructed along the piecewise-smooth orientable curve.

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

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