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

536 Chapter 7 Techniques of Integration
66. Astronomers use a technique called stellar stereography
to determine the density of stars in a star cluster from the
observed (two-dimensional) density that can be analyzed from
a photograph. Suppose that in a spherical cluster of radius R
the density of stars depends only on the distance r from the
center of the cluster. If the perceived star density is given by
yssd, where s is the observed planar distance from the center of
the cluster, and xsrd is the actual density, it can be shown that
71. Determine how large the number a has to be so that
1
y`
dx , 0.001
a x 2 1 1
72. Estimate the numerical value of y`
sum of y 4 2 0 e2x dx and y`
Fssd − y`
f stde 2st dt
0
Gssd − sFssd 2 f s0d s . a
M − 2k y`
te kt dt
0
76. If y`
2`
show that
y a f sxd dx 1 y`
f sxd dx − y b f sxd dx 1 y`
2`
a
2`
b
77. Show that y`
x 2 0
e 2x 2 dx − 1 y` 2 2 0 e2x dx.
78. Show that y` 2 0 e2x dx − y 1 0
cNs1 2 e 2kt d
integrals as areas.
e 2t dt
0 k
1
y`
S 2
C dx
0 sx 2 1 4 x 1 2D
converges. Evaluate the integral for this value of C.
x
y`
S
0 x 2 1 1 2 C dx
3x 1 1D
converges. Evaluate the integral for this value of C.
ustd − r V C0e2rtyV
possible that y`
0
f sxd dx is convergent?
ustd and interpret it. 0
yssd − y R 2r
xsrd dr
s
sr 2 2 s 2
If the actual density of stars in a cluster is xsrd − 1 2 sR 2 rd2 ,
find the perceived density yssd.
67. A manufacturer of lightbulbs wants to produce bulbs that last
about 700 hours but, of course, some bulbs burn out faster than
others. Let Fstd be the fraction of the company’s bulbs that
burn out before t hours, so Fstd always lies between 0 and 1.
(a) Make a rough sketch of what you think the graph of F
might look like.
(b) What is the meaning of the derivative rstd − F9std?
(c) What is the value of y`
rstd dt? Why?
0
68. As we saw in Section 3.8, a radioactive substance decays
exponentially: The mass at time t is mstd − ms0de kt ,
where ms0d is the initial mass and k is a negative constant.
The mean life M of an atom in the substance is
For the radioactive carbon isotope, 14 C, used in radiocarbon
dating, the value of k is 20.000121. Find the mean life of a
14
C atom.
69. In a study of the spread of illicit drug use from an enthusiastic
user to a population of N users, the authors model the number
of expected new users by the equation
− y`
where c, k and are positive constants. Evaluate this integral
to express in terms of c, N, k, and .
Source: F. Hoppensteadt et al., “Threshold Analysis of a Drug Use Epidemic
Model,” Mathematical Biosciences 53 (1981): 79–87.
70. 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
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`
2 0 e2x dx by writing it as the
2
4 e2x dx. Approximate the first integral
by using Simpson’s Rule with n − 8 and show that the
second integral is smaller than y`
4 e24x dx, which is less than
0.0000001.
73. If f std is continuous for t > 0, the Laplace transform of f is
the function F defined by
and the domain of F is the set consisting of all numbers s for
which the integral converges. Find the Laplace transforms of
the following functions.
(a) f std − 1 (b) f std − e t (c) f std − t
74. Show that if 0 < f std < Me at for t > 0, where M and a are
constants, then the Laplace transform Fssd exists for s . a.
75. Suppose that 0 < f std < Me at and 0 < f 9std < Ke at for t > 0,
where f 9 is continuous. If the Laplace transform of f std is
Fssd and the Laplace transform of f 9std is Gssd, show that
f sxd dx is convergent and a and b are real numbers,
f sxd dx
s2ln y dy by interpreting the
79. Find the value of the constant C for which the integral
80. Find the value of the constant C for which the integral
81. Suppose f is continuous on f0, `d and lim x l` f sxd − 1. Is it
82. Show that if a . 21 and b . a 1 1, then the following
integral is convergent.
y`
0
x a
1 1 x b dx
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