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

420 Chapter 5 Integrals
80. A model for the basal metabolism rate, in kcalyh, of a young
man is Rstd − 85 2 0.18 cossty12d, where t is the time in
hours measured from 5:00 am. What is the total basal metabolism
of this man, y 24
0
Rstd dt, over a 24-hour time period?
81. An oil storage tank ruptures at time t − 0 and oil leaks from
the tank at a rate of rstd − 100e 20.01t liters per minute. How
much oil leaks out during the first hour?
82. A bacteria population starts with 400 bacteria and grows at a
rate of rstd − s450.268de 1.12567t bacteria per hour. How many
bacteria will there be after three hours?
83. Breathing is cyclic and a full respiratory cycle from the
beginning of inhalation to the end of exhalation takes
about 5 s. The maximum rate of air flow into the lungs
is about 0.5 Lys. This explains, in part, why the function
f std − 1 2 sins2ty5d has often been used to model the rate of
air flow into the lungs. Use this model to find the volume of
inhaled air in the lungs at time t.
84. The rate of growth of a fish population was modeled by the
equation
Gstd −
60,000e20.6t
s1 1 5e 20.6t d 2
where t is measured in years and G in kilograms per year.
If the biomass was 25,000 kg in the year 2000, what is the
predicted biomass for the year 2020?
85. Dialysis treatment removes urea and other waste products
from a patient’s blood by diverting some of the bloodflow
externally through a machine called a dialyzer. The rate at
which urea is removed from the blood (in mgymin) is often
well described by the equation
ustd − r V C0e2rtyV
where r is the rate of flow of blood through the dialyzer (in
mLymin), V is the volume of the patient’s blood (in mL), and
C 0 is the amount of urea in the blood (in mg) at time t − 0.
Evaluate the integral y 30
0
ustd dt and interpret it.
86. Alabama Instruments Company has set up a production line
to manufacture a new calculator. The rate of production of
these calculators after t weeks is
dx
dt
− 5000S1 2 100
st 1 10d 2D calculatorsyweek
(Notice that production approaches 5000 per week as time
goes on, but the initial production is lower because of the
workers’ unfamiliarity with the new techniques.) Find the
number of calculators produced from the beginning of the
third week to the end of the fourth week.
87. If f is continuous and y 4
f sxd dx − 10, find y 2
f s2xd dx.
0
88. If f is continuous and y 9
f sxd dx − 4, find y 3
xf sx 2 d dx.
89. If f is continuous on R, prove that
y b
a
0
f s2xd dx − y 2a
f sxd dx
For the case where f sxd > 0 and 0 , a , b, draw a dia gram
to interpret this equation geometrically as an equality of
areas.
2b
90. If f is continuous on R, prove that
y b
a
f sx 1 cd dx − y b1c
f sxd dx
For the case where f sxd > 0, draw a diagram to interpret this
equation geometrically as an equality of areas.
a1c
91. If a and b are positive numbers, show that
y 1
0
x a s1 2 xd b dx − y 1
x b s1 2 xd a dx
92. If f is continuous on f0, g, use the substitution u − 2 x to
show that
y
xf ssin xd dx − 0 2 y f ssin xd dx
0
93. Use Exercise 92 to evaluate the integral
y
0
0
x sin x
1 1 cos 2 x dx
94. (a) If f is continuous, prove that
y y2
0
f scos xd dx − y y2
f ssin xd dx
(b) Use part (a) to evaluate y y2
cos 2 x dx and y y2
sin 2 x dx.
0
0
0
0
0
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