Got it. Think of your object with density $\rho(\vec{x})$ moving rigidly with constant velocity $\vec{v}$. In this case, the $\vec{v}$ can be pulled out of the equation and the integral is just as you wrote. If the velocity is not constant within the volume, then $\vec{v}$ is an average velocity.

Do you know calculus? You seem hung up on the integral and not the physics.