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

410 Chapter 5 Integrals
53. If oil leaks from a tank at a rate of rstd gallons per minute at
time t, what does y 120
0
rstd dt represent?
54. A honeybee population starts with 100 bees and increases
at a rate of n9std bees per week. What does 100 1 y 15
0
n9std dt
represent?
55. In Section 4.7 we defined the marginal revenue function R9sxd
as the derivative of the revenue function Rsxd, where x is the
number of units sold. What does y 5000
1000
R9sxd dx represent?
56. If f sxd is the slope of a trail at a distance of x miles from the
start of the trail, what does y 5 3
f sxd dx represent?
57. If x is measured in meters and f sxd is measured in newtons,
what are the units for y 100
0
f sxd dx?
58. If the units for x are feet and the units for asxd are pounds per
foot, what are the units for daydx? What units does y 8 2
asxd dx
have?
59–60 The velocity function (in meters per second) is given for a
particle moving along a line. Find (a) the displacement and (b) the
distance traveled by the particle during the given time interval.
59. vstd − 3t 2 5, 0 < t < 3
60. vstd − t 2 2 2t 2 3, 2 < t < 4
66. Suppose that a volcano is erupting and readings of the rate
rstd at which solid materials are spewed into the atmosphere
are given in the table. The time t is measured in seconds and
the units for rstd are tonnes (metric tons) per second.
t 0 1 2 3 4 5 6
rstd 2 10 24 36 46 54 60
(a) Give upper and lower estimates for the total quantity
Qs6d of erupted materials after six seconds.
(b) Use the Midpoint Rule to estimate Qs6d.
67. The marginal cost of manufacturing x yards of a certain fabric
is
C9sxd − 3 2 0.01x 1 0.000006x 2
(in dollars per yard). Find the increase in cost if the production
level is raised from 2000 yards to 4000 yards.
68. Water flows into and out of a storage tank. A graph of the rate
of change rstd of the volume of water in the tank, in liters per
day, is shown. If the amount of water in the tank at time t − 0
is 25,000 L, use the Midpoint Rule to estimate the amount of
water in the tank four days later.
61–62 The acceleration function (in mys 2 ) and the initial velocity
are given for a particle moving along a line. Find (a) the velocity
at time t and (b) the distance traveled during the given time
interval.
61. astd − t 1 4, vs0d − 5, 0 < t < 10
62. astd − 2t 1 3, vs0d − 24, 0 < t < 3
r
2000
1000
0 1 2 3 4
t
63. The linear density of a rod of length 4 m is given by
sxd − 9 1 2sx measured in kilograms per meter, where x
is measured in meters from one end of the rod. Find the total
mass of the rod.
64. Water flows from the bottom of a storage tank at a rate of
rstd − 200 2 4t liters per minute, where 0 < t < 50. Find
the amount of water that flows from the tank during the first
10 minutes.
65. The velocity of a car was read from its speedometer at
10-second intervals and recorded in the table. Use the Midpoint
Rule to estimate the distance traveled by the car.
t (s) v smiyhd t (s) v smiyhd
0 0 60 56
10 38 70 53
20 52 80 50
30 58 90 47
40 55 100 45
50 51
_1000
69. The graph of the acceleration astd of a car measured in ftys 2 is
shown. Use the Midpoint Rule to estimate the increase in the
velocity of the car during the six-second time interval.
a
12
8
4
0
2 4 6
t (seconds)
70. Lake Lanier in Georgia, USA, is a reservoir created by
Buford Dam on the Chattahoochee River. The table shows
the rate of inflow of water, in cubic feet per second, as
measured every morning at 7:30 am by the US Army Corps
of Engineers. Use the Midpoint Rule to estimate the amount
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