Math2111: Chapter 3: Line integrals. Section 1: Scalar line integrals

In this blog entry you can find lecture notes for Math2111, several variable calculus. See also the table of contents for this course. This blog entry printed to pdf is available here.

We now discuss the concept of a scalar line integral. This is one of several ways in which integrals of functions of one variable can be generalized to functions of several variables. Here, instead of integrating over an interval ${}[a,b]$ on the ${}x$ axes, we now integrate over a curve $\boldsymbol{c}:[a,b]\to\mathbb{R}^3$. The integrand is then a function $f:D\subseteq\mathbb{R}^3\to\mathbb{R}$ such that $D\supseteq \{\boldsymbol{c}(t): t\in [a,b]\}$.

Riemann integrals of functions $f:[a,b]\to\mathbb{R}$ are defined as the limit of Riemann sums

$\displaystyle \sum_{n=1}^N f(z_n) [x_n-x_{n-1}],$

where $a=x_0\textless x_1 \textless \cdots \textless x_N=b$, where $x_{n-1}\le z_n \le x_n$ for $n =1,2,\ldots, N$ and where $\max_{1\le n\le N} |x_n-x_{n-1}|\to 0$. That is to say that we partition the interval ${}[a,b]$ and multiply the function value in a subinterval $[x_{n-1},x_n]$ with the length of this subinterval, namely $x_n-x_{n-1}$ and make the partition finer and finer.

Analogously we define scalar line integrals. Here, the domain is not an interval ${}[a,b]$, but a curve. Again, the idea is to partition the curve into subcurves $\mathcal{C}_1,\ldots,\mathcal{C}_N$ and compute the sum of some function value $f(\boldsymbol{p}_n)$, where $\boldsymbol{p}_n$ is a point on the subcurve $\mathcal{C}_n$, multiplied by the length of the subcurve $\mathcal{C}_n$. In other words,

$\displaystyle \sum_{n=1}^N f(\boldsymbol{p}_n) \mbox{ length of }\mathcal{C}_n \to \int_{\mathcal{C}} f \,\mathrm{d}s,$

where in the limit the length of the longest subcurve tends to ${}0$.

The following result shows how scalar line integrals can be computed.

Scalar line integral
Let $\boldsymbol{c}:[a,b]\to\mathbb{R}^3$ be a parameterisation of the curve $\mathcal{C}$. Assume that $\boldsymbol{c}$ is continuously differentiable and let $f:D\to\mathbb{R}$ be continuous such that $\{\boldsymbol{c}(t):t\in [a,b]\} \subseteq D\subseteq \mathbb{R}^3$. Then the scalar line integral is given by

$\displaystyle \int_{\mathcal{C}} f \,\mathrm{d} s= \int_a^b f(\boldsymbol{c}(t)) \|\boldsymbol{c}^\prime(t)\| \, \mathrm{d} t.$

The integral $\int_a^b f(\boldsymbol{c}(t)) \|\boldsymbol{c}^\prime(t)\| \, \mathrm{d} t$ does not depend on the parameterisation of the curve.

If $f(\boldsymbol{c}(t))=1$ for all $a\le t \le b$, then

$\displaystyle \int_{\mathcal{C}} \,\mathrm{d}s = \int_{a}^b \|\boldsymbol{c}^\prime(t)\|\,\mathrm{d}t$

is the length of the curve $\mathcal{C}$.

Example
Let the curve $\mathcal{C}$ be parameterised by $\boldsymbol{c}(t)=(\sin t, \cos t,t)$ where $0\le t\le 2\pi$ and let $f(x,y,z)=x+y+z$. Calculate $\int_{\mathcal{C}} f\,\mathrm{d}s$. $\Box$

Example
Let the curve $\mathcal{C}$ be parameterised by $\boldsymbol{c}(t)=(t,0)$ for $0 \le t \le \pi$ and let $f(x,y)= \mathrm{e}^y \sin x$. Calculate $\int_{\mathcal{C}} f\,\mathrm{d}s$ by interpreting the integral as an integral along the $x$-axis.

Question Using the geometrical interpretation of integrals $\int_p^q g(x)\,\mathrm{d} x$ as the area under the graph of ${}g$, how can you geometrically interpret the scalar line integral then?
$\Box$

Example
Let a curve $\mathcal{C}$ be parameterised by $\boldsymbol{c}:[0,1]\to\mathbb{R}^2$ given by $\boldsymbol{c}(t)=(t,t)$. Further let $f:\mathbb{R}^2 \to \mathbb{R}$ be given by $f(x,y)= \frac{x+y}{2}$. Calculate the line integral $\int_{\mathcal{C}} f \,\mathrm{d} s$ using a geometrical interpretation of scalar line integrals.
$\Box$

Question Let $f:[a,b]\to\mathbb{R}$ be a continuously differentiable function. Can you find a method to calculate the length of the graph of ${}f$?

Some properties of scalar line integrals
Let $\mathcal{C},\mathcal{C}_1,\mathcal{C}_2$ be curves and $f,g$ be functions defined on these curves. Let $\lambda\in\mathbb{R}$. Let $\mathcal{C}_1+\mathcal{C}_2$ denote the curve which consists of both, $\mathcal{C}_1$ and $\mathcal{C}_2$.

1. $\int_{\mathcal{C}} \lambda f\,\mathrm{d}s = \lambda\int_{\mathcal{C}} f\,\mathrm{d} s$
2. $\int_{\mathcal{C}_1+\mathcal{C}_2} f\,\mathrm{d} s= \int_{\mathcal{C}_1} f \,\mathrm{d}s+\int_{\mathcal{C}_2} f\,\mathrm{d}s$
3. $\int_{\mathcal{C}} (f+g)\,\mathrm{d} s= \int_{\mathcal{C}} f\,\mathrm{d}s + \int_{\mathcal{C}} f\,\mathrm{d}s$

Applications

Scalar line integrals can be used to calculate for instance the mass or center of mass of a wire. Let $\mathcal{C}$ be a curve in $\mathbb{R}^3$ and let ${}\delta=\delta(x,y,z)$ denote the density of the wire at a point $(x,y,z)$ on the curve.

1. Mass: $M = \int_{\mathcal{C}} \delta \, \mathrm{d} s,$ where $\delta(x,y,z)$ denotes the density.
2. First moments about coordinate planes: $M_{y,z} = \int_{\mathcal{C}} x \delta \, \mathrm{d} s,$ $M_{x,z} = \int_{\mathcal{C}} y \delta \, \mathrm{d} s$ and $M_{x,y} = \int_{\mathcal{C}} z \delta \, \mathrm{d} s.$
3. Center of mass: $\overline{x}=M_{y,z}/M,$ $\overline{y}=M_{x,z}/M$ and $\overline{z}=M_{x,y}/M.$