
# The Meaning of Definite Integrals of Vector-Valued Functions

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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).