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We can try to extend the idea of breaking up a curve into small pieces and
summing the difference between the endpoints of each piece times some function
to define integration along a curve C in Euclidean space. If f is a scalar field,
that is a function defined on points in our space, we can sum There are two important examples in which such integrals occur. First, suppose we integrate a unit tangent vector along C, which for
small intervals will be a unit vector in the direction of Second, in physics, the work done by a force F in pushing an object along a path C is given by an integral of the form In general, we define such integrals by breaking the curve up
into ever smaller pieces, taking the dot product indicated within each piece,
and summing over the pieces. So far we have generalized the notion of area or rather our approach to defining it to define integration in the complex plane, or along a path. The same idea can be applied to integration over an area in two dimensions, a surface in three dimensions, a volume in three dimensions, or what you will in higher dimensions: break what you are interested in into pieces whose diameters go to zero, and sum the integrand over these pieces. When integrating over a surface you have to take into consideration that surfaces have directions like curves do, and so we must make our definitions accordingly, as we just did for curves.
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represent subdivision points on the path, and again r' is a point in the
i-th subinterval. This can be defined, but here the difference in r's will
be a vector, and the sum will be a vector field. 
over
the intervals on the curve C that are defined by the endpoints 
.
This can be defined exactly as area is in the definite integral. The standard
notation for it is
The result is then the integral of 
over the curve C. Since this just sums the length |dl| of each of the subintervals
of the curve, its sum over the entire curve will be the length of C.
