Say we have some vector-valued function that, when feed a t-value, gives a particles position. So, this vector function measures the distance from the origin to a point on the curve that the vector function traces. Now, if we were to integrate this function from a point "a" to a point "b", where a > b, would this act correspond to me adding up all the vectors that define the curve between "a" and "b"? And would this tell me how far the particle has traveled relative to the origin?
If this isn't the case, how should I be viewing the act of integrating a vector function?