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

Section 5.1 Areas and Distances 377
18. The velocity graph of a car accelerating from rest to a speed
of 120 kmyh over a period of 30 seconds is shown. Estimate
the distance traveled during this period.
√
(km/h)
80
40
0 10 20 30 t
(seconds)
19. In someone infected with measles, the virus level N (measured
in number of infected cells per mL of blood plasma)
reaches a peak density at about t − 12 days (when a rash
appears) and then decreases fairly rapidly as a result of
immune response. The area under the graph of Nstd from
t − 0 to t − 12 (as shown in the figure) is equal to the total
amount of infection needed to develop symptoms (measured
in density of infected cells 3 time). The function N has been
modeled by the function
f std − 2tst 2 21dst 1 1d
Use this model with six subintervals and their midpoints
to estimate the total amount of infection needed to develop
symptoms of measles.
N
1000
0
N=f(t)
12
t (days)
Source: J. M. Heffernan et al., “An In-Host Model of Acute Infection: Measles
as a Case Study,” Theoretical Population Biology 73 (2006): 134– 47.
20. The table shows the number of people per day who died
from SARS in Singapore at two-week intervals beginning on
March 1, 2003.
Date Deaths per day Date Deaths per day
March 1 0.0079 April 26 0.5620
March 15 0.0638 May 10 0.4630
March 29 0.1944 May 24 0.2897
April 12 0.4435
(a) By using an argument similar to that in Example 4,
estimate the number of people who died of SARS in
21
Singapore between March 1 and May 24, 2003, using
both left endpoints and right endpoints.
(b) How would you interpret the number of SARS deaths
as an area under a curve?
Source: A. Gumel et al., “Modelling Strategies for Controlling SARS
Outbreaks,” Proceedings of the Royal Society of London: Series B 271
(2004): 2223–32.
21–23 Use Definition 2 to find an expression for the area
under the graph of f as a limit. Do not evaluate the limit.
21. f sxd − 2x
x 2 1 1 , 1 < x < 3
22. f sxd − x 2 1 s1 1 2x , 4 < x < 7
23. f sxd − ssin x , 0 < x <
24–25 Determine a region whose area is equal to the given
limit. Do not evaluate the limit.
24. lim
n l ` o n 3
1
i−1 nÎ1 3i
n
25. lim
n l ` o n
i−1
4n
tan
i
4n
26. (a) Use Definition 2 to find an expression for the area
under the curve y − x 3 from 0 to 1 as a limit.
(b) The following formula for the sum of the cubes of
the first n integers is proved in Appendix E. Use it to
evaluate the limit in part (a).
2
nsn 1 1d
1 3 1 2 3 1 3 3 1 ∙ ∙ ∙ 1 n −F G
3
2
27. Let A be the area under the graph of an increasing continuous
function f from a to b, and let L n and R n be the
approximations to A with n subintervals using left and
right endpoints, respectively.
(a) How are A, L n, and R n related?
(b) Show that
R n 2 L n − b 2 a f f sbd 2 f sadg
n
Then draw a diagram to illustrate this equation by
showing that the n rectangles representing R n 2 L n
can be reassem bled to form a single rectangle whose
area is the right side of the equation.
(c) Deduce that
R n 2 A , b 2 a
n
f f sbd 2 f sadg
28. If A is the area under the curve y − e x from 1 to 3,
use Exercise 27 to find a value of n such that
R n 2 A , 0.0001.
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