The line integral is a concept in calculus that allows you to find the area under a curve, given a curve and an area unit. A line integral is given a path (the curve) and an area unit (typically, this is assumed to be one unit of length, but it can be any unit as long as it is consistent).
There are two main ways to evaluate a line integral. The first way is to use the Fundamental Theorem of Calculus (FTC). The FTC states that if f(x) is continuous on [a, b] then:
$$\displaystyle \int_{a}^{b} f(x) \; dx = \frac{b} {a} \; f(a) – \frac{b – a}{b}f(b)\,,$$
where the integration is taken in the opposite order of the domain. This allows you to swap the domain and range of integration, then reverse the order of integration. This trick also applies to other cases of the FTC, like when there is only one variable.
Example of line integral
A common physics example of the line integral is finding the length of a wire. Given a wire of a given length and a compass, you can find the total length of the wire by drawing a circle around the wire and measuring the diameter.
The difference is that you are drawing a circle on a flat surface, which is your curve C in this case. You then find the width of the wire and measure its length, subtracting its ends to get its length along curve C.
This case also shows how you can integrate across multiple dimensions. A three-dimensional wire would have more volume, so drawing it in two dimensions makes it look thinner. Finding its volume would just be a matter of multiplying its length by its height and depth!
Line integrals are not limited to two- or three-dimensional cases, however. Any number of dimensions can be analyzed using this method.
Understanding the curve
The first step in evaluating the line integral is to understand the given curve. What is the curve? How many times does it go around?
The curve in this problem is a circle with radius r and a central angle of θ radians. r is the distance along the curve and θ is how much the curve goes around.
There are many ways to understand this problem, but one way that will be helpful in other problems is to break down the circle into small segments and count them.
There are n of these segments, so n = πr, where n is the number of segments and πr is the length of each segment. Now we can write our equation for C: C = (nπr) × (T3 − T).
Computing the curve
After you have the curve, or given curve, you need to compute the curve. This can be done in a few ways, depending on what program you are using.
Some software requires you to draw the entire curve and then breaks it down into smaller segments that can be computed separately. Others require you to input points and it draws the curve between those points.
Whatever software you are using, make sure it is drawing the correct shape by checking its dimensions and comparing them to the given information.
Some curves cannot be broken down into straight line integrals, so make sure your software is able to compute the integral for the given curve!
Curves such as circles or spirals can be more difficult to compute than straight lines or curves.
Applying the line integral
Now that we have the basic understanding of line integrals, let’s apply it to some sample problems.
Example: Find the line integral of f(x) = 3x from x = a to x = b, where a and b are constants.
Solution: We know that the line integral is the limit of the sum of products, so we need to find the sum of all f(x) values between a and b. We can do this by finding all points (a, b) such that a ≤ x ≤ b and then calculating f(x) for each point.
We can also do it another way by finding all points (a, b) such that a ≤ x ≤ b and then calculating the average value of f(x) for all points in between. Then, we would calculate how many of these average values exist between a and b, and divide that number by two to find the total number of average values.
Example using vector fields
Another interesting application of the line integral is in evaluating vector fields. A vector field is a set of vectors that define a flow, or how a quantity (such as velocity or energy) moves from point to point.
A common example of a vector field is the wind. The wind can be described as flowing from west to east, and it increases in speed as it gets closer to the equator.
We can evaluate the length of a path traveled by the wind using line integrals. First, we must divide the path into small segments, and then we must calculate the average velocity of the wind across each segment. These two values can be substituted for t and v in the line integral.
This can be difficult to understand, so let’s look at an example.
Examples of line integrals
Line integrals are commonly given in physics and engineering courses. Some examples of line integrals include calculating the distance traveled due to a constant velocity, calculating the total distance traveled due to a constant acceleration, and calculating the average value of some property along a given curve.
The first example can be thought of as finding the displacement vector, = , where is the initial position and is the final position.
The second example can be thought of as finding the average acceleration, , where is the initial position and is the final position.
The third example can be thought of as finding some property , where is on some curve .
All of these can be simplified by thinking about what it would mean to have no change in position, acceleration, or property.
Applications of line integrals
Line integrals have many applications. Some of the most common applications include velocity, displacement, and area.
As mentioned before, the line integral can be used to find velocity. Given a curve C and a constant T, the difference in magnitude of the velocity of an object at two points on the curve is given by the length of the line integral between those two points.
Displacement can be found by taking the absolute value of the line integral over a given curve. Area can be found using the equation for area, A=L*W, where L is the length and W is the width.
Another common application is finding force via line integrals over distance. Given a constant T and a curve C, if an object moves along C over time T, then its force (F) is given by F=Lint(T)=int(T)*C=int(T)*A=int(T)*displacement*time=int(T)Atimewhere A = displacement.
The curl of a vector field and line integrals
In this section, you will learn how to evaluate line integrals where the path of integration is a curve defined by a vector field.
First, you need to find the vector field for the given curve. You can do this by finding the gradient of the curve and representing that as a set of vectors.
Then, you need to find an equivalent Cartesian coordinate system in which your curve is a straight line. You need to do this so that you can integrate the curve along its length, or its orientation, using basic derivatives.
Finally, you must calculate the integral of the given vector field over the length of your curve in order to find the line integral on your given curve.
You must be careful to not double count any part of the line integral due to winding issues.
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