On single variable calculus integrals have a nice graphical interpretation as the area under the curve of a function $f(x)=y$ who's graph is built with 2 axis where the $x$ axis is the domain and the $y$ axis is the image of the function.
I would like to know if, Given some single parameter vector valued function, for example $f(t)=<x(t),y(t),z(t)>$, is there any graphical interpretation (or intuition) for $\int f(t)\,dt$?
Note: My question is limited to functions with domains in $\mathbb{R}^1$ but NOT limited to functions with images in $\mathbb{R}^3$
You can think of $\textbf{a}(t) = (a_{1}(t),a_{2}(t),a_{3}(t))$ as the acceleration of a moving particle within $\textbf{R}^{3}$. Consequently, its integral between the instants $t_{1}$ and $t_{2}$ gives the difference between its inicial and final velocities, respectively.
Similarly, you can think of $\textbf{v}(t) = (v_{1}(t),v_{2}(t),v_{3}(t))$ as the velocity of a moving particle within $\textbf{R}^{3}$. Consequently, its integral between the instants $t_{1}$ and $t_{2}$ gives the difference between its inicial and final position, respectively.
Another interesting example comes from physics as well. Let us consider the resultant force acting upon a particle is expressible as a function $\textbf{F}:\textbf{R}^{3}\rightarrow\textbf{R}^{3}$, where $\textbf{s}:\textbf{R}\rightarrow\textbf{R}^{3}$ is the function which expresses its position over time. Then the work corresponding to the time interval $t_{1}$ and $t_{2}$ is given by \begin{align*} W = \int_{t_{1}}^{t_{2}}\langle\textbf{F}(\textbf{s}(t)),\textbf{s}'(t)\rangle\mathrm{d}t \end{align*}