9. Line Integrals

We've seen that we can integrate a vector valued function by integrating each component.

But we can also integrate along a curve in space! Such an integral is called a line integral. Let's see the definition and then talk about it. For a curve $\large C$ described by $\large \vec{r}(t)$ with parameter domain $\large a\le t\le b$, the line integral of a function $\large f(x,y,z)$ is:

$$\large \int_C f(x,y,z)ds=\int_a^b f\left(\vec{r}(t)\right)|\vec{r'}(t)|dt=\int_a^b f\left(x(t),y(t),z(t)\right)|\vec{r'}(t)|dt$$

Okay, that looks daunting. Let's talk through it...on the left we have our notation for a line integral. The $\large C$ is the curve we are integrating over. The $\large ds$ is a notation to let us know we are doing a line integral on a curve (it's called an arc length element).

On the right, we see what we are actually dealing with. It's a regular, old-fashioned integral of a single independent variable $\large t$. We are working along the curve $\large C$, so we limit the $\large (x,y,z)$ values to values on the curve. To do this, we replace $\large (x,y,z)$ with their values along $\large \vec{r}(t)$. This is what is meant by $\large f\left(\vec{r}(t)\right)$ or $\large f\left(x(t),y(t),z(t)\right)$.

Let's just see how it's done on a simple example:

There are three facts about line integrals we should remember:

$\large \int_{C}1 ds = \int_a^b|\vec{r'}(t)|dt$ is the length of the curve $\large C$ from $\large a$ to $\large b$ (called the arc length of $\large C$).
The average value of the function $\large f$ on the curve $\large C$ is $\large \displaystyle \frac{\int_C f ds}{\int_C 1 ds}$ (the average is the integral divided by the length of the curve).
If $\large f$ represents a density (charge density, mass density, etc.), then $\large \int_a^b f(x,y,z)ds$ is the total amount (of charge, mass, etc.) on the curve $\large C$.

All similar integral facts are summarized on the Integral Fact Page.

Finally, the following video is optional. It is a 10 minute explanation of where line integrals come from via Riemann sums:

Tomorrow's class:

Learn about where this definition comes from.
Practice line integrals.

Section Outcomes:

I can:

Find the line integral of a function on a given curve.
Find the arc length of a curve, find the average value of a function on a curve, and find a total quantity by integrating a density along a curve.