Net Change Theorem

The net change theorem states that the definite integral of a rate of change gives the signed change of the underlying quantity. If is continuous on and its derivative on extends continuously to , then

This is the second fundamental theorem of calculus with the integrand interpreted as a rate (Strang and Herman 2016).

For motion on a line, integrating velocity gives displacement. Integrating speed, the absolute value of velocity, gives total distance traveled. For instance, a particle with velocity for has zero displacement but travels a total distance of 2.