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).