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The function $P=P(t)$ relates the population $P$ (in billions of persons) as a function of the time $t$ (in years). Suppose $\int_{0}^{10}P^\prime (t)\, dt=3.$ Since $P^{\prime} (t)$ is the rate of change of the population with respect to time, then the change in population from $t=0$ to $t=10$ is $3$ billion persons.
The function $v=v(t)$ is the speed $v$ (in meters per second) of an object at a time $t$ (in seconds). If $\int_{0}^{5}v(t) dt=8,$ then the object travels 8 m during the interval $0\leq t\leq 5.$ Do you see why? The speed $v=v(t)$ is the rate of change of distance $s$ with respect to time $t.$ That is, $v=\dfrac{ds}{dt}.$