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There are many different ways to integrate in three dimensions. We can integrate
over a volume, over a surface, or along a path. In every case the integral is
defined as the limit of a sum, as in ordinary one dimensional integration. The
sum in general is obtained by dividing the specified region surface or path
into pieces (whose sizes then approach 0) and summing the volume area or length
of the piece times the integrand (exactly as Riemann sums and limits of them
are defined on the path consisting of the real line).
In the case of paths or surfaces we can take into account the orientation of
the piece or not, as we wish. We can integrate vectors or numerical functions.
Since the definitions are in every case the same, rather than repeating ourselves
in each case, we can give one general definition, but make it in terms of a
"measure" which describes what kind of entity we are considering.
Among the possible measures the following are very important.
1. volume measure: to each piece we associate its volume multiplied by the integrand. This is usually denoted as
This measure allows you to relate density f of whatever with the amount of it in R.
2. flux measure: to each piece of a surface we associate the component of the vector integrand that is normal to that piece of the surface multiplied by its area. We usually denote this by
This is used to compute flow through a surface.
3.circulation measure: to each piece of a path we associate the component of its vector integrand in the direction tangent to the path times its length. Notation:
This measure describes work done to or by an object in its motion along a path.
4. area measure: to each piece of a surface we associate its area times the (scalar) integrand. Notation:
You use this measure with f = 1 to find the area of a surface.
5. length measure: to each piece of a curve we associate its length times the integrand. Notation:
You use this measure with f = 1 to compute arc length.
Any assignment of weight to pieces of whatever kind of point set can be a measure if it is ''finitely additive": the sum of the measures assigned to two disjoint sets must be the measure of the combined set.