Integrating Vector-Valued Functions

A vector-valued function like v(t) = (x'(t), y'(t)) packages two separate rates of change into one arrow. Integrating it just means integrating each piece on its own: integrate x'(t) to get x(t), integrate y'(t) to get y(t), and glue the results back into a vector. Each integration leaves behind its own leftover constant, because x(t) and y(t) are independent functions that happen to be traveling together. Those two constants form a constant vector (C1, C2), not one shared number — there is no reason the horizontal drift and vertical drift should need the same offset.

That constant vector is exactly why velocity alone cannot tell you where something is. Knowing how fast and in what direction an object moves at every instant says nothing about its starting point — the same velocity function could describe a path starting at the origin or one starting a mile away. An initial position r(0) fixes C1 and C2 separately, one equation per component, and only then does integrating v(t) actually recover r(t).

This is the vector version of the same idea from single-variable integration: antiderivatives come with a +C until given a condition to pin them down. It also sets up displacement, since integrating v(t) from t = a to t = b gives the displacement vector r(b) - r(a) directly, without needing position at all. When a path returns to its starting point, that displacement vector is zero even though the object clearly moved — total distance traveled along the way can be large while net displacement vanishes.