Line Integrals

Recall that an integral is a way to "sum up" changes in a function over some interval. We add the input by some small amount, multiply it by the output, and sum up all of these products. How would we do this if we were to integrate a function with multiple inputs?

One way to do this is called the line integral. Essentially, we draw a curve in the domain of the function, and we integrate the function over this curve.

Table of Contents

• Table of Contents
• Arc Lengths
• Arc Length of Any Parametric Curve
• Line Integrals of Scalar Fields
• Visualizing Scalar Field Line Integrals
• Example Problem: Arc Length of Different Functions
• Summary and Next Steps

Arc Lengths

The first step to understanding line integrals is to understand arc lengths. First, let's consider a circle with radius . The arc length of a circle is given by the .

The arc length, however, can also be used to describe the length of any curve, not just circles. We shall first consider the special case of a curve that is a function of .

Consider a curve (i.e. just ), and we want to find the arc length of the curve from to .

We can approximate the arc length by breaking the curve into small segments. Increment by some small amount , and calculate the distance between the two points.

Options
Segments

The distance between the two points is given by the Pythagorean theorem;

We can calculate the distance between the two points as . Given that this curve is essentially just a function of , treat as the independent variable.

We can find what by multiplying the derivative with the change in :

Substitute this back into the arc length:

The entire arc length can then be found by summing up all of these small segments:

As , this sum becomes an integral:

The general steps to find the arc length of a function are:

1. Find the derivative of the function.
2. Write in terms of the derivative and .
3. Use the Pythagorean theorem to find the arc length of the curve. Plug in the value of .
4. Integrate the expression to find the arc length.

Let's generalize this. Consider any , and we want to find the arc length of the curve from to . The arc length is given by:

Arc Length of Any Parametric Curve

Now that we have found the arc length of a curve that is a function of , let's consider any parametric curve. Consider a curve that is defined by for .

Start with the same expression for the arc length:

We wrote in terms of earlier, but since is no longer the independent variable, we need to write the whole expression in terms of :

Recall that a parametric curve can be written as a vector-valued function . Then, the arc length of the curve is:

This should make sense; an increment in changes the position of the point by . The arc length is then the magnitude of this change.

In order to define the boundary of the curve, we need to find the limits of . However, defining the boundaries also defines the variable of integration. Instead, we can write a in the integral boundaries, meaning:

"Integrate ... along the curve ."

encodes the curve and the limits of integration. Then, the arc length of the curve is:

Line Integrals of Scalar Fields

With the concept of arc lengths in mind, we can now define line integrals.

Consider a curve that is defined for . Recall from Equation that the arc length of the curve is:

Now, consider a scalar field . We can generalize the arc length integral and add the function to it:

This means, instead of adding the arc length, you first multiply the arc length by some function and then sum up all of these products.

Let's now consider writing this in terms of . Recall that is the increment in the arc length:

Substitute that into the line integral:

Next, the terms and are just coordinates of the curve . We can write them in terms of the position on the curve instead:

Finally, consider the bounds of the integral. We wrote as the integral boundaries, but now that we have written the integrand in terms of , we can write the limits of integration as the boundaries of itself. Since is defined for , we get the general form of the line integral:

Visualizing Scalar Field Line Integrals

Let's visualize a line integral of a scalar field. Recall that the integral, in a single-variable case, can be thought of as the area under the curve. For a function , you take increments of , multiply it by the output of the function, and sum up all of these products.

In the single-variable case, we can see it in action like this:

For a scalar field, we can visualize it in a similar way. Instead of incrementing along the -axis, we increment along the curve . Then, draw a line perpendicular to the curve at each point up to the function's surface, and draw a rectangle. If we do this over the entire curve, it will look like this:

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Segments

Example Problem: Arc Length of Different Functions

Find the arc length of the following functions:

1. from to (a semicircle)
2. from to

1. First, write in terms of the derivative and :

   The arc length is then:

   This integral is the arc length of a semicircle, which is . The appendix contains a very cool interpretation of this integral.

2. The derivative of is . The arc length is then:

   This is a non-elementary integral.

Summary and Next Steps

In this section, we introduced the first integral in multivariable calculus: the line integral. Specifically, we discussed line integrals of scalar fields.

Here are the key points to remember:

• The arc length of a curve can be found by integrating the magnitude of the derivative of the curve.

• The line integral of a scalar field is the sum of the products of the function and the arc length of the curve.

• The line integral has two forms: one in terms of and , and one in terms of :

In the next section, we will discuss line integrals of vector fields, and its connection to forces, work, and all that fun stuff.