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Today in class we were introducing complex line integrals. And that got me thinking, I don't know of a good interpretation for integrals of functions from $\mathbb{R}$ to $\mathbb{R}^2$ or $\mathbb{R}^3$. By a good interpretation I mean some (possibly geometric) way of intuitively understanding what such an integral means and why it's calculated like it is.

It's easy to understand why, if $\mathbf{r}(t)$ describes the position of a particle at time $t$, then $\int_a^b \mathbf{r}'(t)\ dt = \mathbf{r}(b)-\mathbf{r}(a)$, by imagining lots of small tangents to the curve $\mathbf{r}(t)$ being added up. It's harder to picture what $\int_a^b \mathbf{r}(t)\ dt$ means without thinking of $\mathbf{r}(t)$ as a derivative. Does anyone know of an intuitive interpretation for this kind of integrals?

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