Question

If the birth rate of a population is b(t)= − 2200e

0.024t people per year and the death rate

is d(t)= − 1460e

0.018t people per year, find the area between these curves for 0 < t < 10.

What does this area represent? Also plot the curves using MATLAB.

Answer 1

The death curve is blue and birth curve is red.

From graph, we can see that b(t) is greater than d(t)

Therefore

 $\begin{gathered} \text { Area between the two curves }=\int_{0}^{10} b(t)-d(t) d t \\ =\int_{0}^{10} 2200 e^{0.024 t}-1460 e^{0.018 t} d t \\ =\left[\frac{2200 e^{0.024 t}}{0.024}-\frac{1460 e^{0.018 t}}{0.018}\right]_{0}^{10} \\ =\left[\frac{2200 e^{0.24}}{0.024}-\frac{1460 e^{0.18}}{0.018}\right]-\left[\frac{2200}{0.024}-\frac{1460}{0.018}\right] \approx 8868 \end{gathered}$

Since birth curve is greater than death curve we conclude that area between the two curves represents the increase in population between t=0 and t=10.