Net Change Theorem

The total change in a quantity over an interval equals the definite integral of its rate of change.

The Net Change Theorem says $\displaystyle\int_a^b f'(x)\,dx = f(b) - f(a)$ — integrating a rate of change over $[a,b]$ recovers the total change in the original quantity, an immediate consequence of FTC Part 2.

Examples

If $v(t)$ is velocity, $\int_0^5 v(t)\,dt$ gives the net displacement between $t=0$ and $t=5$ .