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.

Curve	Parametric Equations
(Ellipse)	Counter-Clockwise                 Clockwise
(Circle)	Counter-Clockwise                 Clockwise
Line Segment From   to

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 as 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.

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.

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.

So, the previous two examples seem to suggest that if we change the path between two points then the value of the line integral (with respect to arc length) will change.  While this will happen fairly regularly we can’t assume that it will always happen.  In a later section we will investigate this idea in more detail

Next, let’s see what happens if we change the direction of a path.

So, it looks like when we switch the direction of the curve the line integral (with respect to arc length) will not change.  This will always be true for these kinds of line integrals.  However, there are other kinds of line integrals in which this won’t be the case.  We will see more examples of this in the next couple of sections so don’t get it into your head that changing the direction will never change the value of the line integral.

Before working another example let’s formalize this idea up somewhat.  Let’s suppose that the curve C has the parameterization , .  Let’s also suppose that the initial point on the curve is A and the final point on the curve is B. The parameterization ,  will then determine an orientation for the curve where the positive direction is the direction that is traced out as t increases.  Finally, let  be the curve with the same points as C, however in this case the curve has B as the initial point and A as the final point, again t is increasing as we traverse this curve.  In other words, given a curve C, the curve  is the same curve as C except the direction has been reversed.

We then have the following fact about line integrals with respect to arc length.

Fact

So, for a line integral with respect to arc length we can change the direction of the curve and not change the value of the integral.  This is a useful fact to remember as some line integrals will be easier in one direction than the other.

Now, let’s work another example

To this point in this section we’ve only looked at line integrals over a two-dimensional curve.  However, there is no reason to restrict ourselves like that.  We can do line integrals over three-dimensional curves as well.

Let’s suppose that the three-dimensional curve C is given by the parameterization,

then the line integral is given by,

Note that often when dealing with three-dimensional space the parameterization will be given as a vector function.

Notice that we changed up the notation for the parameterization a little.  Since we rarely use the function names we simply kept the x, y, and z and added on the  part to denote that they may be functions of the parameter. 

Also notice that, as with two-dimensional curves, we have,

and the line integral can again be written as,

So, outside of the addition of a third parametric equation line integrals in three-dimensional space work the same as those in two-dimensional space.  Let’s work a quick example.

So, as we can see there really isn’t too much difference between two- and three-dimensional line integrals.