Intuitive understanding of integral of vector valued functions

Solution 1:

You're essentially integrating distance, which gives you a quantity that has units of (distance)$\times$(time), say "meter-seconds" (NOT meters-per-second).

Think of an integral of a function $r(t)$ as the average value $r_{av}$ of that function on an interval $[a,b]$:

$$r_{av}:=\frac{1}{b-a}\int_a^br(t)dt.$$

Then if your $r(t)$ represents displacement, your integral is essentially the average displacement (which becomes a vector in higher dimensions), times the amount of time spent moving. Or, $r_{av}$ is exactly the average displacement over a time period $[a,b]$.

As far as physical units of meter-seconds, I haven't seen this used anywhere in physics as a fundamental quantity like velocity, force, momentum, etc.

Solution 2:

If you want an intuitive understanding of an integral $\int_a^b{\bf r}(t)\>dt$ you first have to give some intuitive sense to products of the form $${\bf r}\>\Delta t\ ,\tag{1}$$ where $t$ is a one-dimensional variable (not necessarily time), $\Delta t$ an interval-length in this variable, and ${\bf r}$ is a constant vector.

Given such a physical interpretation of $(1)$ we then can envisage situations of the following kind: We are given a vector-valued function of this variable $t$ (not necessarily a parametrical representation of a curve) $$t\mapsto{\bf r}(t)\qquad(a\leq t\leq b)\ ,\tag{2}$$ and we want to know the total impact in the sense of $(1)$ this variable vector has along the interval $[a,b]$. This total impact is called the integral of ${\bf r}(\cdot)$ over $[a,b]$, and is denoted by $$\int_{[a,b]}{\bf r}(t)\>{\rm d}t\ .$$ The integral should have the following properties: It is linear in ${\bf r}(\cdot)$, additive with respect to the concatenation of intervals, and for a constant ${\bf r}(\cdot)$ one has $$\int_{[a,b]}{\bf r}\>{\rm d}t={\bf r}\cdot(b-a)\ .$$ Using these axioms it is easy to see that for continuous functions $(2)$ the integral $\int_{[a,b]}{\bf r}(t)\>{\rm d}t$ has to be a limit of Riemann sums: $$\int_{[a,b]}{\bf r}(t)\>{\rm d}t=\lim_\ldots\>\sum_{k=1}^N {\bf r}(\tau_k)\>(t_k-t_{k-1})\ ,$$ where one has partitions $$a=t_0<t_1<\ldots<t_N=b$$ and sampling points $\tau_k\in[t_{k-1},t_k]$ in mind.

Note that these explanations did not mention the FTC, and they make sense as well for multivariate functions defined on domains $B\subset {\mathbb R}^d$. The FTC and Fubini's theorem come into play when we actually want to compute such limits of Riemann sums in a systematic way.