The Meaning of Definite Integrals of Vector-Valued Functions

Now let's turn our attention to the meaning of a definite integral of a vector-valued function. The context in which this will make the most sense is where the function we integrate is a velocity function. That is,

$$\int_a^b \vecs v(t) \, \,dt$$

We know that the antiderivative of velocity is position and that this definite integral gives us the change in position over the time interval, $a \le t\le b$. In other words,

$$\int_a^b \vecs v(t) \, \,dt = \vecs r(b) - \vecs r(a)$$

Thus, the definite integral of velocity over a time interval $a \le t\le b$ gives us the displacement vector that indicates the change in position over this time interval.

In general, the definite integral

$\displaystyle \int_a^b \vecs r(t) \, \,dt = \vecs q(b) - \vecs q(a)$, where $\vecs q(t)$ is the antiderivative of $\vecs r(t)$,

gives us a change in the antiderivative of our vector-valued function over the given inteval $[a,b]$. This will always be a constant vector that would fit from tip to tip of the vectors given by the antiderivative function at $t = a$ and $t = b$, respectively (assuming the vectors were placed in standard position).