Answer
An estimate for the total amount of infection needed to develop symptoms of measles is $~~7840$
Work Step by Step
The interval $[0,12]$ is divided into 6 subintervals.
$\Delta t = \frac{b-a}{n} = \frac{12-0}{6} = 2$
To find an estimate for the amount of infection, we can use the midpoint of each subinterval:
$t_1 = 1$
$t_2 = 3$
$t_3 = 5$
$t_4 = 7$
$t_5 = 9$
$t_6 = 11$
We can find an estimate for the amount of infection:
$\sum_{i=1}^{6} f(t_i)~\Delta t$
$=\sum_{i=1}^{6} -t_i(t_i-21)(t_i+1)~\Delta t$
$= (40+216+480+784+1080+1320)~(2)$
$= 7840$
An estimate for the total amount of infection needed to develop symptoms of measles is $~~7840$
