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

chapter 7 Review 539
CAS
67. The speedometer reading (v) on a car was observed at
1-minute intervals and recorded in the chart. Use Simpson’s
Rule to estimate the distance traveled by the car.
t (min) v (miyh) t (min) v (miyh)
0 40 6 56
1 42 7 57
2 45 8 57
3 49 9 55
4 52 10 56
5 54
68. A population of honeybees increased at a rate of rstd bees
per week, where the graph of r is as shown. Use Simpson’s
Rule with six subintervals to estimate the increase in the
bee population during the first 24 weeks.
r
12000
8000
4000
0 4 8 12 16 20 24
t
(weeks)
69. (a) If f sxd − sinssin xd, use a graph to find an upper
bound for | f s4d sxd| .
(b) Use Simpson’s Rule with n − 10 to approximate
y f sxd dx and use part (a) to estimate the error.
0
(c) How large should n be to guarantee that the size of the
error in using S n is less than 0.00001?
70. Suppose you are asked to estimate the volume of a football.
You measure and find that a football is 28 cm long.
You use a piece of string and measure the circumference
at its widest point to be 53 cm. The circumference 7 cm
from each end is 45 cm. Use Simpson’s Rule to make
your estimate.
28 cm
71. Use the Comparison Theorem to determine whether the integral
is convergent or divergent.
2 1 sin x
(a) y`
dx
1
sx
(b) y`
1
1
s1 1 x 4 dx
72. Find the area of the region bounded by the hyperbola
y 2 2 x 2 − 1 and the line y − 3.
73. Find the area bounded by the curves y − cos x and y − cos 2 x
between x − 0 and x − .
74. Find the area of the region bounded by the curves
y − 1ys2 1 sx d, y − 1ys2 2 sx d, and x − 1.
75. The region under the curve y − cos 2 x, 0 < x < y2, is
rotated about the x-axis. Find the volume of the resulting
solid.
76. The region in Exercise 75 is rotated about the y-axis. Find the
volume of the resulting solid.
77. If f 9 is continuous on f0, `d and lim x l ` f sxd − 0, show that
y`
f 9sxd dx − 2f s0d
0
78. We can extend our definition of average value of a continuous
function to an infinite interval by defining the average value
of f on the interval fa, `d to be
lim
t l `
1
t 2 a yt f sxd dx
a
(a) Find the average value of y − tan 21 x on the interval f0, `d.
(b) If f sxd > 0 and y`
a
f sxd dx is divergent, show that the
average value of f on the interval fa, `d is lim x l ` f sxd, if
this limit exists.
(c) If y`
f sxd dx is convergent, what is the average value of f
a
on the interval fa, `d?
(d) Find the average value of y − sin x on the interval f0, `d.
79. Use the substitution u − 1yx to show that
y`
0
ln x
1 1 x 2 dx − 0
80. The magnitude of the repulsive force between two point
charges with the same sign, one of size 1 and the other of size
q, is
F −
q
4« 0r 2
where r is the distance between the charges and « 0 is a
constant. The potential V at a point P due to the charge q is
defined to be the work expended in bringing a unit charge to
P from infinity along the straight line that joins q and P. Find
a formula for V.
7et07rx70
10/14/09
MasterID: 00905
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