# Tag Archives: scalar line integral

## Math2111: Chapter 3: Recommended reading: Line integrals and orientation

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 discuss in what sense the scalar line integrals do not depend on the orientation of the curve.

Let us start with integrals $\int_a^b f(t)\,\mathrm{d} t$ where $f$ is a Riemann integrable function. Let $a \textless b$ and let $P_n=\{x_0,\ldots, x_n\}$ be such that $a=x_0 \textless x_1 \textless x_2 \textless \cdots \textless x_{n-1} \textless x_n =b,$ let $x_{i-1} \le t_i \le x_{i}$ for $1\le i \le n$ and $\delta(P_n)=\max_{1\le i\le n} x_{i+1}-x_i.$ We can now form the Riemann sum

$\displaystyle \mathcal{S}(P_n) = \sum_{i=1}^n f(t_i) (x_i-x_{i-1}).$

By considering the limit $\lim \mathcal{S}(P_n)$ where $n\to\infty$ such that $\delta(P_n) \to 0,$ we obtain the Riemann integral $\int_a^b f(t)\,\mathrm{d} t.$ Continue reading