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Lizzie Wagner
Line Integrals
Line integrals can be calculated by two methods: with respect to arclength and with
respect to coordinate variables.
When integrating through arclength, we know a property
along
the line, while when we integrate with coordinate variables, we know the vector field
around
the line.
Although this may sound like a subtle difference now, but their applications are different
and how they are calculated also contrast.
The uses for calculating through arclength include
mass of a thin object, like a wire, and require knowing a specific characteristic at every point on
the line, for example density.
When calculating mass, it is not essential to know which
direction the particle is moving along the curve.
A real life example of this is when someone
needs a certain length of string, but must make sure it is not too heavy to use in their design.
Arclength is acceptable to use in this sort of computation because there is no such thing as
negative mass or negative density.
However, this is the place where the two forms of line
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 Winter '08
 Boeri
 Integrals, Multivariable Calculus

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