Study With Me: Line Integrals

Hey guys! I’m currently studying for the Mathematics Subject Test of the GRE, which I plan on taking in the fall. One of the ways I like to study is by explaining the material to someone else. I currently have weekends off from research, and since Saturdays are for the boys, it leaves Sundays for GRE preparation.

Because of this, every Sunday, I’ll explore a different undergraduate topic that could appear on the Mathematics Subject Test. This week: Line Integrals.

I’ll talk about the following:

- What is a line integral?
- How do you calculate a line integral?
- An Example

As a brief note, this post contains LaTeX code and will be much easier to read when viewed directly on my blog, where the code will compile!

**What is a line integral?**

**Let’s first recall what we already know about integrals. **We’re used to integrating functions of one variable over an interval [*a,b*]. We can think of this as integrating over the path on the *x*-axis from *a* to *b, *and the value of the integral as giving the area bounded by the curve *y=f(x)* over the path [*a,b*].

But! We can also integrate over paths that aren’t just straight lines along the *x*-axis. The resulting integral is called a **curve**, **contour**, or **path integral**. Most commonly, it is known as a *line integral. *

In this post, I’ll be talking about line integrals with respect to arc length.

Before we get into it, I’d like to start by defining what it means for a curve to be **smooth. **A curve, *C, *with parameterization **r**(*t*) = <*x(t), y(t)*> is **smooth** if the derivative **r**’(t) is *continuous *and *nonzero*. Additionally, we can say *C* is **piecewise smooth **if it is composed of a finite number of smooth curves joined at consecutive endpoints. Basically, this means I have a bunch of curves $C_1, C_2, …, C_n $ that are all individually smooth and Each $C_i$ has its endpoints connected to $C_{i-1}$ and $C_{i+1}$.

Back to line integrals. Suppose we have a function *f(x,y)* and a smooth curve, *C*, in the *x-y* plane. We want to think about breaking *C* into *n* tiny pieces of arc length $\Delta s_i$. For each of the tiny pieces of *C*, choose any point $P_i = (x_i, y_i)$ and then multiply $f(P_i) = f(x_i, y_i)$ by the length $\Delta s_i$. This process is fairly similar to how we define integration for the case where the path is a line on the *x*-axis. We want to sum up these multiplied terms for all *n* terms. If the value of that sum approaches a finite, limiting value as $n \rightarrow \infty$, then the result is the line integral of *f* along *C* with respect to arc length. Below is a comparison of the single variable case integrating over a path [*a,b*] on the *x*-axis (left) and the line integral with respect to arc length over the curve *C* (right).

Note the notation used for the line integral. If we’re integrating over a path *C*, we write *C* at the bottom of the integral.

**What does this mean geometrically?**

The value of this integral is the area of the region whose base is *C* and whose height above each (*x,y*) point is given by *f(x,y)*.

**How do we actually calculate the line integral?**

**First, parameterize C**

*.*That is, for a parameter

*t*, find the equations

*x=x(t)*and

*y=y(t)*for $a \leq t \leq b$. We consider

*C*to be

**directed**, which means we’re saying that we trace

*C*in a definite direction, which is called the

**positive**direction. Basically, we’re saying that

*t*runs from

*a*to

*b,*so

*A*= (

*x(a), y(a))*is the

**initial point**and

*B*=

*(x(b), y(b))*is the

**final point.**

Since we have $(ds)^2 = (dx)^2 + (dy)^2$ (think Pythagorean theorem), we can write:

$\frac{ds}{dt} = \pm \sqrt{( \frac{dx}{dt})^2+(\frac{dy}{dt})^2}$

which can then be rewritten as:

$\pm \sqrt{(x’(t))^2+(y’(t))^2}$.

We use the + sign if the parameter *t ***increases** in the positive direction on *C* and the - sign if *t* **decreases** in the positive direction on *C*.

So, we have:

$\int_C f \,ds = \int_{a}^{b} f(x(t), y(t))\frac{ds}{dt} \,dt$.

**An Example**

**Determine the value of the line integral of the function f(x,y) = x + y^2 over the quarter-circle x^2 + y^2 = 4 in the first quadrant, from (2,0) to (0,2).**

*Solution below.*