What does it mean to integrate a vector function? What is the meaning of a vector function and what is the geometric interpretation of integrating such a function?
 A: Integration of a vector function is an ambiguous term, it may mean a lot of completely different integrals.


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*integral of a vector function over an independent scalar parameter, $\int \vec{v}\,dt$ is simply 3 integrals for each component. Example: integration of velocity to get position in space.

*integral of a vector function over a path: $\int \vec{v}\cdot d\vec{r}$. This integrates the contribution of a vector field along a path, for instance, work as integral of force along a path, Faraday's law (induction) and Ampere's law, circulation in hydrodynamics and so on.

*integral of a vector function over a surface: $\iint \vec{v}\cdot d\vec{S}$. This tells you the flux of the vector field through a surface. For instance, actual flux of liquid for velocity field, magnetic flux (relevant for induction) and so on.
2 and 3 satisfy special relations (for potential field, (2) gives you zero for closed loop,  for sourceless field, (3) gives zero for closed surface,...) - check out Stokes and Gauss laws.
There are other less common options... but the essence is, you have to know what your vector is (is it simply a vector variable? is it defined everywhere in space - a field?), and what your expression means. Usually in physics, it's just one of the natural laws in general form when you get from local to global expression.
A: Vector function in 3d space is used to represent a function as a function of three variables.
Velocity is the integration of acceleration function which can be represented as a vector function in space. This integration can be done if the functions along x y and z co-ordinates are parameteized.
So integrating the vector function here gives you again a vector function which is velocity and you can interpret speed from this function.
