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In this section we are now going to introduce a new kind of
integral. However, before we do that it
is important to note that you will need to remember how to parameterize
equations, or put another way, you will need to be able to write down a set of
parametric equations for a given curve.
You should have seen some of this in your Calculus II course. If you need some review you should go back
and review some of the basics of
parametric equations and curves.
Here are some of the more basic curves that we’ll need to
know how to do as well as limits on the parameter if they are required.
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Curve
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Parametric Equations
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(Ellipse)
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Counter-Clockwise Clockwise
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(Circle)
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Counter-Clockwise Clockwise
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Line Segment From
to
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With the final one we gave both the vector form of the
equation as well as the parametric form and if we need the two-dimensional
version then we just drop the z
components. In fact, we will be using
the two-dimensional version of this in this section.
For the ellipse and the circle we’ve given two
parameterizations, one tracing out the curve clockwise and the other
counter-clockwise. As we’ll eventually
see the direction that the curve is traced out can, on occasion, change the
answer. Also, both of these “start” on
the positive x-axis at .
Now let’s move on to line integrals. In Calculus I we integrated ,
a function of a single variable, over an interval . In this case we were thinking of x as taking all the values in this
interval starting at a and ending at b.
With line integrals we will start with integrating the function ,
a function of two variables, and the values of x and y that we’re going
to use will be the points, ,
that lie on a curve C. Note that this is different from the double
integrals that we were working with in the previous chapter where the points
came out of some two-dimensional region.
Let’s start with the curve C that the points come from.
We will assume that the curve is smooth
(defined shortly) and is given by the parametric equations,
We will often want to write the parameterization of the
curve as a vector function. In this case
the curve is given by,
The curve is called smooth
if is continuous and for all t.
We use a ds here
to acknowledge the fact that we are moving along the curve, C, instead of the x-axis (denoted by dx) or
the y-axis (denoted by dy).
Because of the ds this is
sometimes called the line integral of f with respect to arc length.
We’ve seen the notation ds
before. If you recall from Calculus II
when we looked at the arc length of a
curve given by parametric equations we found it to be,
It is no coincidence that we use ds for both of these problems.
The ds is the same for both
the arc length integral and the notation for the line integral.
So, to compute a line integral we will convert everything
over to the parametric equations. The
line integral is then,
Don’t forget to plug the parametric equations into the
function as well.
If we use the vector form of the parameterization we can
simplify the notation up somewhat by noticing that,
where is the magnitude or norm of . Using this notation the line integral
becomes,
Note that as long as the parameterization of the curve C is traced out exactly once at t increases from a to b the value of the
line integral will be independent of the parameterization of the curve.
Let’s take a look at an example of a line integral.
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Example 1 Evaluate
where C
is the right half of the circle, . rotated in the counter clockwise direction.
Solution
We first need a parameterization of the circle. This is given by,
We now need a range of t’s
that will give the right half of the circle.
The following range of t’s
will do this.
Now, we need the derivatives of the parametric equations
and let’s compute ds.
The line integral is then,
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Next we need to talk about line integrals over piecewise smooth curves. A piecewise smooth curve is any curve that
can be written as the union of a finite number of smooth curves, ,…,
where the end point of is the starting point of . Below is an illustration of a piecewise
smooth curve.
Evaluation of line integrals over piecewise smooth curves is
a relatively simple thing to do. All we
do is evaluate the line integral over each of the pieces and then add them
up. The line integral for some function
over the above piecewise curve would be,
Let’s see an example of this.
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Example 2 Evaluate
where C
is the curve shown below.
Solution
So, first we need to parameterize each of the curves.
Now let’s do the line integral over each of these curves.
Finally, the line integral that we were asked to compute
is,
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Notice that we put direction arrows on the curve in the
above example. The direction of motion
along a curve may change the value of
the line integral as we will see in the next section. Also note that the curve can be thought of a
curve that takes us from the point to the point . Let’s first see what happens to the line
integral if we change the path between these two points.