Single Variable Calculus, Chapter 6, 6.1, Section 6.1, Problem 44

The equation $b(t) = 2200 + 52.3 t + 0.74t^2$ represents the birth rate of a population of people per year and the death rate $d(t) = 1460 + 28.8t$ people per year. Find the area between the curves for $0 \leq t \leq 10$. What does this area represents?







We can use a vertical strip to evaluate the area to be


$\begin{equation} \begin{aligned} A =& \int^{10}_0 [2200 + 52.3 t + 0.74t^2 - (1460 + 28.8t)] dt \\ \\ A =& \int^{10}_0 [740 + 23.5 t + 0.74 t^2] dt \\ \\ A =& \left[ 740 t + \frac{23.5 t^2}{2} + \frac{0.74 t^3}{3} \right]^{10}_0 \\ \\ A =& 8821.667 \text{ square units} \end{aligned} \end{equation}$


The area bounded by the curves represents the number of people living at interval $0 \leq t \leq 10$