Answer
(a) The integral of a rate of change is the net change:
$\int_{a}^{b}F'(x)~dx = F(b)- F(a)$
(b) $~~\int_{t_1}^{t_2}r(t)~dt~~$ represents the net change of the amount of water in the reservoir between the times $t_1$ and $t_2$.
Work Step by Step
(a) We can state the Net Change Theorem as follows:
The integral of a rate of change is the net change:
$\int_{a}^{b}F'(x)~dx = F(b)- F(a)$
(b) $~~r(t)~~$ is the rate at which water flows into the reservoir.
Therefore, $~~\int_{t_1}^{t_2}r(t)~dt~~$ represents the net change of the amount of water in the reservoir between the times $t_1$ and $t_2$.
