One of the most important applications of line integrals is to the concept of work. I will specialize the discussion to the case of a smooth curve. However, the general case is obtained by writing dγ in place of γ^{′}
You have a force field field F which acts on an object as it moves along the curve C, in the direction determined by the given orientation of the curve which comes here from the given parametrization. From beginning physics or calculus, you defined work as the force in a given direction times the distance the object moves in that direction. Work is really a measure of the extent to which a force contributes to the motion but here the object moves over a curve so the direction of motion is constantly changing. In order to define what is meant by the work, consider the following picture.
In this picture, the work done by a constant force F on an object which moves from the point γ

where the wriggly equal sign indicates the two quantities are close. In the notation of Leibniz, one writes dt for h and

or in other words,

Defining the total work done by the force at t = 0, corresponding to the first endpoint of the curve, to equal zero, the work would satisfy the following initial value problem.

This motivates the following definition of work.
Definition 4.2.10 Let F

where the function γ is one of the allowed parameterizations of C in the given orientation of C. In other words, there is an interval
When you have a curve and γ :
Example 4.2.11 Say F ≡
From the above, it equals ∫ _{0}^{1}