James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 6.1 Areas Between Curves 431
We use the Midpoint Rule with n − 4 intervals, so that Dt − 4. The midpoints of
the intervals are t 1 − 2, t 2 − 6, t 3 − 10, and t 4 − 14. We estimate the distance between
the cars after 16 seconds as follows:
y 16
0
sv A 2 v B d dt < Dt f13 1 23 1 28 1 29g
− 4s93d − 372 ft n
Example 5 Figure 9 is an example of a pathogenesis curve for a measles infection.
It shows how the disease develops in an individual with no immunity after the measles
virus spreads to the bloodstream from the respiratory tract.
N
1500
FIGURE 9
Measles pathogenesis curve
Source: J. M. Heffernan et al., “An In-Host Model
of Acute Infection: Measles as a Case Study,”
Theoretical Population Biology
73 (2008): 134–47.
Number of infected cells per mL of blood plasma
1000
500
0
Pathogen
enters plasma
Symptoms
appear
Infectiousness
begins
Pathogen
is cleared
10–11 12 17–18 21
Infectiousness
ends
t
(days)
N
0
f
P¡
10 18
P
t
(days)
The patient becomes infectious to others once the concentration of infected cells
becomes great enough, and he or she remains infectious until the immune system
manages to prevent further transmission. However, symptoms don’t develop until the
“amount of infection” reaches a particular threshold. The amount of infection needed
to develop symptoms depends on both the concentration of infected cells and time,
and corresponds to the area under the pathogenesis curve until symptoms appear. (See
Exercise 5.1.19.)
(a) The pathogenesis curve in Figure 9 has been modeled by f std − 2tst 2 21dst 1 1d.
If infectiousness begins on day t 1 − 10 and ends on day t 2 − 18, what are the corresponding
concentration levels of infected cells?
(b) The level of infectiousness for an infected person is the area between N − f std and
the line through the points P 1 st 1 , f st 1 dd and P 2 st 2 , f st 2 dd, measured in (cellsymL) ? days.
(See Figure 10.) Compute the level of infectiousness for this particular patient.
FIGURE 10
Solution
(a) Infectiousness begins when the concentration reaches f s10d − 1210 cellsymL and
ends when the concentration reduces to f s18d − 1026 cellsymL.
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