Vector Integration - Intuition

Question

I understand that an integral of a scalar valued function can be visualized as "signed area under the curve". But what about integration of a vector valued function by its parameter? Is there a visually intuitive geometric explanation for what $\int_a^b \mathbf{r}(t)\, \mathrm{d}t$ represents if $\mathbf{r}(t)$ is a function from, say, $\mathbb{R}$ to $\mathbb{R}^3$?

Answer 1

One way is to think of $\int r(t) dt$ as the "antideriviative", so $r(t)$ is the vector rate of change of $\int r(t) dt$. In space, $r(t)$ would be the "velocity" of $\int r(t) dt$, or its directed rate of change.

In this illustration, we approximate the antiderivative curve $\int r(t)dt$ by breaking the interval $[a,b]$ into time slices and geometrically adding $r(t)$ to determine the trajectory. We then get the definite integral by taking the total vector displacement from $t=a$ to $t=b$.