Chapter 5
1.
Find
‘( )Gx
for
2
( ) ( 6)
x
a
G x t dt=+
.
A)
26x+
B)
22
xa−
C)
2
x
D)
1
02t dt
2.
The rate of pollution pouring into a lake is measured every 10 days, with results in the
following table. About how many tons of pollution have entered the lake during the
first 40 days?
Time in days
0
10
20
30
40
Rate of pollution in tons/day
5
7
10
9
7
3.
Consider a sports car which accelerates from 0 ft/sec to 85 ft/sec in 5 seconds (85 ft/sec
= 58 mph). The car’s velocity is given in the following table. What is the lower bound
for the number of feet the car travels in 5 seconds?
t
0
1
2
3
4
5
()Vt
0
30
52
68
80
85
Chapter 5
4.
Consider a sports car which accelerates from 0 ft/sec to 85 ft/sec in 5 seconds (85 ft/sec
= 58 mph). The car’s velocity is given in the following table. Over which time
interval in the average acceleration the smallest?
t
0
1
2
3
4
5
()Vt
0
30
52
68
80
85
A)
The last
B)
The fourth
C)
The third
D)
The second
E)
The first
5.
The flow rate of water in a mountain stream due to spring runoff is given in the
following table. Give your best estimate (average of upper and lower limit) of how
many cubic meters of water flowed through from 6:00 pm to midnight.
Time
(hours since 6:00 pm)
0
1
2
3
4
5
6
Flow rate
(in cubic meters per hour)
300
360
410
455
490
520
545
Ans:
2657.5
change. difficulty: easy section: 5.1
Chapter 5
6.
The graph below shows the velocity of an object (in meters/second). Find an upper
bound for the number of meters traveled from t = 1 to t = 5 seconds.
Chapter 5
7.
The graph below shows the velocity of an object (in meters/second). At what time(s) is
the acceleration zero? Select all that apply.
A)
0
B)
2
C)
4
D)
6
E)
8
F)
10
Chapter 5
8.
The following figure shows the graph of the velocity, v, of an object (in meters/sec). If
the graph were shifted up 4 units, how would the distance traveled between t = 0 and t =
6 change?
A)
It would increase by 4 meters.
B)
It would decrease by 4 meters.
C)
It would increase by 24 meters.
D)
It would decrease by 24 meters.
E)
It would remain the same.
Ans: C Learning Objectives: Estimate the total change given information about a
rate of change. difficulty: easy section: 5.1
9.
A car is observed to have the following velocities at times t = 0, 2, 4, 6:
Time (sec)
0
2
4
6
Velocity (ft/sec)
0
21
40
65
Give a lower estimate for the number of feet the car traveled in first 6 seconds.
Ans:
122
Learning Objectives: Estimate the total change given information about a rate of
change. difficulty: easy section: 5.1
Chapter 5
10.
Two cars start at the same time and travel in the same direction along a straight road.
The following figure gives the velocity, v, of each car as a function of time, t. Which
car reached the larger maximum velocity?
A)
Car B
B)
Car A
rate of change. difficulty: easy section: 5.1
11.
A new factory worker is put to work assembling gadgets. He is fairly slow at first, but
over a period of days gets much faster. His employer constructs the following table to
evaluate his progress over the first 20 days he is on the job. Give a n upper estimate for
the number of gadgets the worker assembled the first 10 days on the job.
number of days on
the job
0
5
10
15
20
number of gadgets
assembled per day
8
15
20
23
24
Ans:
175
change. difficulty: medium section: 5.1
12.
Your velocity, in meters per second, is given by
2
( ) 5v t t=+
, where t is time in
seconds. Estimate your distance traveled over the first 6 seconds.
A)
97 meters
B)
103 meters
C)
109 meters
D)
115 meters
rate of change. difficulty: medium section: 5.1
Chapter 5
13.
The following figure shows the rate of change of enrollment for a community college.
Estimate the total change in enrollment over the first 6 years.
A)
1900 students
B)
2300 students
C)
2700 students
D)
2900 students
rate of change. difficulty: medium section: 5.1
14.
The balance in an investment account increases at a rate of
0.025
2000 t
Re=
dollars per
year, where t is time in years since 2000. Make a table of values for R and use it to
give a n upper estimate for the total change in the value of the account between 2000
and 2004.
A)
$8369
B)
$8444
C)
$8519
D)
$8594
rate of change. difficulty: medium section: 5.1
15.
At time t, in seconds, the velocity v, in miles per hour, of a car is given by
2
( ) 5 0.5v t t=+
for
08t
. Use
2t=
to estimate how many miles were traveled
during this time (average left- and right-hand sums).
Ans:
128
difficulty: medium section: 5.2
Chapter 5
16.
At time t, in seconds, the velocity v, in feet per second, of a car is given by
2
( ) 5 0.5v t t=+
for
06t
. A second car travels exactly 20 feet per second faster
than the first car. How much greater will the left- and right-hand estimates for the
distance traveled by the second car be that the left- and right-hand estimates for the
distance traveled by the first car?
Ans:
120 feet
Learning Objectives: Estimate a definite integral from a table, graph, or formula.
difficulty: medium section: 5.2
17.
At time t, in seconds, your velocity, v, in meters per second, is given by
2
( ) 6 2v t t=+
for
06t
. Use
2t=
to estimate how many meters you traveled during this time.
Ans:
188 meters
Learning Objectives: Estimate a definite integral from a table, graph, or formula.
difficulty: medium section: 5.2
18.
At time t, in seconds, your velocity, v, in meters per second, is given by
2
( ) 4 2v t t=+
for
06t
. Use
1t=
to estimate how many meters you traveled during this time.
Ans:
170 meters
Learning Objectives: Estimate a definite integral from a table, graph, or formula.
difficulty: medium section: 5.2
19.
Use a calculator to estimate
53
0( 1)x dx+
. Round your answer to 2 decimal places.
Ans:
161.25
Learning Objectives: Evaluate a definite integral using technology. difficulty: easy
section: 5.2
20.
Use a calculator to evaluate
6
1(2.5)tdt
. Round your answer to 2 decimal places.
Ans:
263.72
Learning Objectives: Evaluate a definite integral using technology. difficulty: easy
section: 5.2
21.
Use a calculator to evaluate
22
2ln( 1)t dt
−+
. Round your answer to 2 decimal places.
Ans:
2.87
Learning Objectives: Evaluate a definite integral using technology.
difficulty: medium section: 5.2
Chapter 5
22.
Use the following graph to estimate
70
10 ()f t dt
Ans:
13,200
difficulty: medium section: 5.2
using the left Riemann sum with three
terms.
23.
Which of the following is the best approximation for
20
5()f x dx
, where f is shown in
the following figure?
A)
350
B)
1000
C)
4150
D)
6025
Chapter 5
Page 10
24.
Given the following graph of f, which is the best approximation for
50
0()f x dx
?
A)
51,000
B)
78,750
C)
84,500
D)
100,000
25.
Consider the graph of the function
2
() x
f x e−
=
shown in the following figure.
A. Approximate
2
1.5
0
x
e dx
−
Part A:
Part B:
by using a right-hand sum with 3 subdivisions. Round to
2 decimal places.
B. Is your answer to part (A) a lower or an upper estimate?
formula. difficulty: medium section: 5.2
Chapter 5
Page 11
26.
You plan to approximate the definite integral
10 2
5(8 3)x dx+
by Riemann sums.
Which Riemann sum will be larger, the right or the left?
27.
Consider the function
( ) lnf x x=
, as shown in the following figure. Taking
0.5x=
,
find the upper sum estimate for
3
1ln x dx
. Round to 3 decimal places.
28.
A. Find a 4-term left Riemann sum approximation for the integral
34
30 31x dx+
.
Round to 1 decimal place.
B. Is your answer in part (A) an underestimate or an overestimate?
Chapter 5
29.
Using the following figure, calculate the value of the right-hand Riemann sum for the
function f on the interval
0 12t
using
3t=
.
30.
Use the following table to estimate
50
10 ()f x dx
using
10x=
.
x
0
10
20
30
40
50
()fx
30
35
45
50
70
85
31.
Find the area included between the curves
yx=
and
1
yx
=
, from x = 1 to x = 2.
Round to 2 decimal places.
Chapter 5
32.
In the following graph, does
3
0()f x dx
appear to be positive, negative, or zero?
Ans:
zero
33.
Use the following table to estimate the area between
()fx
and the x-axis on the
interval
66x−
.
x
-6
-4
-2
0
2
4
6
()fx
-3
-2
0
2
3
2
1
A)
3
B)
8
C)
22
D)
36
above the x-axis minus area below the x-axis. difficulty: medium section: 5.3
Chapter 5
34.
Your rich eccentric friend has hired you to cover his back yard with grass and patio
stone. If the southwest corner of his yard is taken as the origin, with the x-axis pointing
eastward and distances measured in feet, then the boundaries of the yard are the lines x
= 0, x = 100, y = 0, and f(x) = 110-0.5x. The border between grass and stone is
( ) 40 20cos(π / 40)g x x=+
, with grass covering all of the yard south of the curve. This
border also bounds one side of the pool, the other side of the pool being surrounded by
the curve
2
( ) 60 0.05( 40)h x x= − −
. All of the rest of the yard is to be covered by
stone. Estimate, to the nearest square yard, the area of the grass.
grass
pool
stone
f(x)
g(x)
h(x)
20
40
60
80
100
20 40 60 80 100
x
y
Chapter 5
35.
A car is moving along a straight road from A to B, starting from A at time t = 0. Below
is the velocity plotted against time. How many kilometers away from A is the car at
time t = 9?
36.
If the upper estimate of the area of a region bounded by the curve in the following
figure, the horizontal axis, and the vertical lines x = 3 and x = -3 is 15, what is the upper
estimate if the graph is shifted up 2 units?
Chapter 5
37.
Estimate the area above the curve
cosyx=
and below y = 4 for
0π/ 2x
. Round
to 2 decimal places.
Ans:
5.28
difficulty: medium section: 5.3
38.
Estimate the area of the region bounded by
cosyx=
,
yx=
,
π/ 2x=−
, and
0x=
.
Round to 2 decimal places.
Ans:
2.23
difficulty: easy section: 5.3
39.
Estimate the area of the region under the curve
35yx= − +
and above the x-axis for
3
05x
. Round to 2 decimal places.
Ans:
6.41
axis minus area below the x-axis. difficulty: medium section: 5.3
40.
Estimate the area of the region under the curve
sin( / 2)yx=
for
01x
. Round to
2 decimal places.
Ans:
0.24
axis minus area below the x-axis. difficulty: medium section: 5.3
41.
Suppose
( ) 3sin 6F x x x
= + +
. Find the total area bounded by
()Fx
,
0x=
,
πx=
,
and
0y=
. Round to 1 decimal place.
Ans:
29.8
axis minus area below the x-axis. difficulty: medium section: 5.3
42.
Use an integral to find the area under the graph of
3
75yx=−
for
0.89 10x
.
A)
17453.35
B)
17437.15
C)
17473.35
D)
17457.15
Chapter 5
43.
The upper half of the ellipse below has equation
2
42yx=−
. Find the area in the
first quadrant between the ellipse and the line
yx=
. Give your answer to two decimal
places.
A)
1.57
B)
2.57
C)
3.14
D)
1.14
curves. difficulty: medium section: 5.3
44.
What are the units of
()
b
af t dt
if t is measured in km/hr and f is in hours?
A)
km
B)
km/hr
C)
km/hr2
D)
km
hr
-5
-4
-3
-2
-1
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5
x
y
total change. difficulty: easy section: 5.4
Chapter 5
45.
A large ice cube is melting at a rate of
()r f t=
cm3 per minute, where t is time in
minutes. If
0
V
is the volume of the ice cube at time t = 0 minutes, which one of the
following expresses the volume of the ice cube after 4 hours?
A)
4
0
0
()V f t dt−
B)
240
0
0
()V f t dt−
C)
240
0
0
()V f t dt
D)
4
0
0
()V f t dt+
quantity given an initial value and a rate of change. difficulty: medium
section: 5.4
46.
A stamp collector has 3000 stamps in his collection on January 1, 2005, and is
collecting more stamps at a rate of
()r f t=
stamps per week, where t is time in weeks
since January 1, 2005. Which of the following expresses the number of stamps in his
collection at the end of 2005?
A)
52
0
3000 ( )f t dt+
B)
52
0
3000 ( )f t dt
C)
5
0
3000 ( )f t dt
D)
2005
0
3000 ( )f t dt+
quantity given an initial value and a rate of change. difficulty: medium
section: 5.4
47.
Sales of a new product are increasing at a rate of
300(1.04)t
units per month, where t is
time in months since the product was introduced. How many units were sold the first 3
months since the product was introduced? Set up an integral and use a calculator to
evaluate it to the nearest whole unit.
Ans:
955
Learning Objectives: Interpret the definite integral of rate of change as total change.
difficulty: medium section: 5.4
Chapter 5
48.
Equal numbers of two different species of ground squirrels are introduced into an area at
time t = 0, with t in years. They have the growth rates shown in the following figure.
Which species will have a larger population after 13 years?
A)
Species A
B)
Species B
total change. difficulty: easy section: 5.4