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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:

  1. What is a line integral?
  2. How do you calculate a line integral?
  3. 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.

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John Whitney. Permutations. 1966.

In Permutations, each point moves at a different speed and moves in a direction independent according to natural laws’ quite as valid as those of Pythagoras, while moving in their circular field. Their action produces a phenomenon more or less equivalent to the musical harmonies. When the points reach certain relationships (harmonic) numerical to other parameters of the equation, they form elementary figures. -John Whitney