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In the previous section we looked at line integrals with
respect to arc length. In this section
we want to look at line integrals with respect to x and/or y.
As with the last section we will start with a
two-dimensional curve C with
parameterization,
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The line integral
of f with respect to x is,
The line integral
of f with respect to y is,
|
Note that the only notational difference between these two
and the line integral with respect to arc length (from the previous section) is
the differential. These have a dx or dy while the line integral with respect to arc length has a ds.
So when evaluating line integrals be careful to first note which
differential you’ve got so you don’t work the wrong kind of line integral.
These two integral often appear together and so we have the
following shorthand notation for these cases.
Let’s take a quick look at an example of this kind of line
integral.
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Example 1 Evaluate
where C is the line segment from to .
Solution
Here is the parameterization of the curve.
The line integral is,
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In the previous section we saw that changing the direction
of the curve for a line integral with respect to arc length doesn’t change the
value of the integral. Let’s see what
happens with line integrals with respect to x
and/or y.
|
Example 2 Evaluate
where C is the line segment from to .
Solution
So, we simply changed the direction of the curve. Here is the new parameterization.
The line integral in this case is,
|
So, switching the direction of the curve got us a different
value or at least the opposite sign of the value from the first example. In fact this will always happen with these
kinds of line integrals.
Fact
|
If C is any
curve then,
With the combined form of these two integrals we get,
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We can also do these integrals over three-dimensional curves
as well. In this case we will pick up a
third integral (with respect to z)
and the three integrals will be.
|
where the curve C is parameterized by
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As with the two-dimensional version these three will often
occur together so the shorthand we’ll be using here is,
Let’s work an example.