8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals

In motion problems, derivatives connect position to velocity and velocity to acceleration, and integrals reverse those relationships. If v(t)=s'(t), then the change in position over a time interval is ${\int_a^b v(t)\,dt}$. If a(t)=v'(t), then the change in velocity is ${\int_a^b a(t)\,dt}$. This means integrals let you move backward from a rate function to the quantity being changed. So if you know acceleration, you can recover velocity changes, and if you know velocity, you can recover position changes.

You should also distinguish between displacement and distance traveled. Displacement is the net change in position and is found by integrating velocity directly. Distance traveled is the total amount of motion and is found by integrating v(t). If velocity changes sign, then displacement and distance are not the same. This is one of the most important applications of integrals in the course because it ties together Units 2, 4, 6, and 8 in one framework.