The birth rate in a certain city is described by the following function $$ b(t)=4.48 -0.02 t^2 $$ The city's death rate is given by $$ d(t)=4+0.01 t^2 $$ Here, t is measured in years, and t = 0 corresponds to the start of the year 1990. The birth and death rates are measured in thousands of births or deaths per year. At the start of 1990, the population of the city is 300 thousand. Enter all the following answers correct to two decimal places.
Calculate the area between the curves $y = b(t)$ and $y = d(t)$ for t between 0 and 10.
First, you need to find the points of intersection of $b(t)$ and $d(t)$. For this end, solve the equation $$ 4.48 - 0.02 t^2 = 4 + 0.01 t^2, $$ and you will get $t_1 = -4$, $t_2 = 4$. Since you integrate from $0$ to $10$, your area will be defined as $$ S = \int_0^4 (b(t) - d(t)) dt + \int_4^{10} (d(t) - b(t)) dt, $$ or, equivalently, $$ S = \int_0^4 (0.48-0.03 t^2) dt + \int_4^{10} (-0.48+0.03 t^2) dt. $$ I hope, the further calculations will be not so hard for you.