If I have a vector function from $\mathbb{R}$ to $\mathbb{R}^n$, I understand that I can use component-wise integration to recover a velocity function from an acceleration function, a position function from a velocity function, and so on.
Say now though that I have some vector function $\mathbf{r}$ that is unaffiliated with any sort of application. Is there a geometric meaning of $\int\mathbf{r}$?
We know that $\mathbf{r}^\prime$ will tell us about the direction of a curve, and if $f$ is a scalar function, then $\int f$ tells us about the area under $f$, but what does $\int\mathbf{r}$ tell us?