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.
Line Segment From
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.
The line integral of
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,
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.
Example 1 Evaluate
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,
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,