Integration of Parametric Curves

Integration

Definition: Integration

Let $\gamma :\mathcal{D}\subseteq \mathbb{R}\to {\mathbb{R}}^n$ be a vector-valued function with component functions ${\gamma }_1,\cdots ,{\gamma }_n$.

We say that $\gamma$ is integrable on $D\subseteq \mathcal{D}$ if all of its component functions ${\gamma }_1,\cdots ,{\gamma }_n$ are integrable on $D$. The integral of $\gamma$ on $D$ is then defined as the following vector:

$${\int }_D\gamma \stackrel{\text{def}}{=}\left[\begin{array}{c}{\int }_D{\gamma }_1\\ ⋮\\ {\int }_D{\gamma }_n\end{array}\right]$$