5.5 Integration by Substitution
332
____ 18. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 19. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 20. Find the following indefinite integral.
a.
b.
c.
d.
e.
333 Chapter 5: Integration
____ 21. Find an equation for the function whose derivative is and whose
graph passes through the point .
a.
b.
c.
d.
e.
____ 22.
Use a graphing utility to check your answer.
a.
b.
c.
d.
e.
Evaluate the following definite integral.
5.5 Integration by Substitution
334
____ 23. Evaluate the definite integral.
a.
b.
c.
d.
e.
____ 24. Evaluate the definite integral.
a.
b.
c.
d.
e.
335 Chapter 5: Integration
____ 25. Evaluate the following definite integral.
Use a graphing utility to check your answer.
a.
b.
c.
d.
e.
____ 26. Find the area of the shaded region.
a.
b.
c.
d.
e.
5.5 Integration by Substitution
336
____ 27.
The rate of disbursement
the square of
Time t is measured in days
and
is the amount that remains to
be disbursed. Find the amount that remains to be disbursed after 30 days. Assume that all the money
will be disbursed in 100 days.
$686,000
$50,700
$101,400
$33,800
$1,029,000
____
28.
The rate of depreciation
of a machine is inversely proportional to the square
of
where V is the value of the machine t years after it was purchased. The initial value of the
machine was
and its value decreased
in the first year. Estimate its value after 6
years. Round your answer to the nearest integer.
$275,000
$475,000
$550,000
$350,000
$500,000
____ 29. The sales S (in thousands of units) of a seasonal product are given by the model where t is
the time in months, with corresponding to January. Find the
average sales for the first quarter . Round your answer to three decimal places.
90.916 thousand units
92.084 thousand units
101.436 thousand units
129.632 thousand units
72.486 thousand units
____ 30. The oscillating current in an electrical circuit is where I is
measured in amperes and t is measured in seconds. Find the average current for the time interval
. Round your answer to three decimal places.
1.637 amps
0.637 amps
1.273 amps
2.546 amps
2.273 amps
337
Chapter 5: Integration
____
31.
Suppose that the probability that a person will remember between 100a% and 100b%
of material learned in an experiment is
where x represents the proportion
remembered. Determine from the figure below, the probability that a randomly chosen individual
will recall between 50% and 75% of the material? Express your answer as a percent rounded to three
decimal places.
19.900%
12.500%
11.000%
35.300%
106.600%
5.5 Integration by Substitution
338
5.5 Integration by
Substitution Answer Section
339
Chapter 5: Integration
22.
23.
28.
29.
5.6 Numerical Integration
340
5.6 Numerical Integration
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use the Trapezoid Rule to approximate the value of the definite integral wth
. Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 2. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite
integral using subintervals. Round your answer to six decimal places and compare the result
with the exact value of the definite integral.
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives 1.366795 and Simpson’s rule gives .
The Trapezoidal rule gives 1.366795 and Simpson’s rule gives .
____ 3. Use Simpson’s Rule to approximate the value of the definite integral with
. Round your answer to four decimal places.
a.
b.
c.
d.
e.
341
Chapter 5: Integration
____
4.
Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the
definite integral using
subintervals. Round your answer to six decimal places and compare the
result with the exact value of the definite integral.
The Trapezoidal rule gives 16.3400 and Simpson’s rule gives .
The Trapezoidal rule gives 16.3400 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
____ 5. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite
integral using subintervals. Round your answer to six decimal places and compare the result
with the exact value of the definite integral.
The Trapezoidal rule gives 43.377044 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives 43.377044 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
____ 6. Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite
integral using subintervals. Round your answer to six decimal places and compare the result
with the exact value of the definite integral.
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives 1.3973 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives 1.3973 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
5.6 Numerical Integration
342
____ 7.
Apply the Trapezoidal Rule and Simpson’s Rule to approximate the value of the
definite integral using subintervals. Round your answer to six decimal places and compare the
result with the exact value of the definite integral.
The Trapezoidal rule gives 0.7100 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives 0.7100 and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
The Trapezoidal rule gives and Simpson’s rule gives .
____ 8. Use the Trapezoidal Rule to approximate the value of the definite integral
with . Round your answer to four decimal places.
–8.0654
–4.0327
–2.0164
–1.0082
–0.5041
____ 9. Use Simpson’s Rule to approximate the value of the definite integral
with . Round your answer to four decimal places.
2.9784
3.0627
8.0100
10.6800
2.6700
____ 10. Use the error formula to estimate the error in approximating the integral
with using Trapezoidal Rule.
3
1
6
0
9
343 Chapter 5: Integration
____ 11. Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson’s Rule with
when approximating the following integral.
The error for the Trapezoidal Rule is 0.0051 and for Simpson’s Rule it is 0.0013.
The error for the Trapezoidal Rule is 0.0000 and for Simpson’s Rule it is 0.0200.
The error for the Trapezoidal Rule is 0.0204 and for Simpson’s Rule it is 0.0200.
The error for the Trapezoidal Rule is 0.0204 and for Simpson’s Rule it is 0.0000.
The error for the Trapezoidal Rule is 0.0000 and for Simpson’s Rule it is 0.0000.
____ 12. Use the error formula to estimate the error in approximating the integral
with using Simpson’s Rule. Round your answer to six decimal places.
0.007084
0.141676
0.004723
0.003936
0.002530
____ 13.
Find the smallest n such that the error estimate from the error formula in the
approximation of the definite integral
is less than 0.00001 using the Trapezoidal Rule.
49
26
73
30
15
____ 14.
Find the smallest n such that the error estimate in the approximation of the definite
integral
is less than 0.00001 using Simpson’s Rule.
a.
10
b.
16
c.
6
d.
8
e.
13
5.6 Numerical Integration
344
____ 15.
Find the smallest n such that the error estimate in the approximation of the definite
integral
is less than 0.00001 using Simpson’s Rule.
a.
48
b.
8
c.
11
d.
14
e.
19
____ 16. Find the smallest n such that the error estimate in the approximation of the definite
integral is less than 0.00001 using the Trapezoidal Rule. Use a graphing utility to
estimate the maximum of the absolute value of the second derivative.
208
189
196
201
188
____ 17.
The table lists several measurements gathered in an experiment to approximate an
unknown continuous function
. Approximate the integral
using the Trapezoidal
Rule. Round your answer to three decimal places.
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
4.12
4.27
4.62
6.37
6.82
7.47
7.72
8.67
8.92
a.
13.115
b.
12.000
c.
7.373
d.
104.920
e.
24.600
345
Chapter 5: Integration
____
18.
The table lists several measurements gathered in an experiment to approximate an
unknown continuous function
. Approximate the integral
using the Simpson’s Rule.
Round your answer to three decimal places.
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
4.72
4.77
4.82
6.17
6.52
7.17
7.32
7.97
8.02
12.197
8.518
15.975
12.865
11.628
____ 19. Use Simpson’s Rule with to approximate using the equation
. Round your answer to five decimal places.
3.14159
3.13595
3.14723
3.14381
3.13937
5.6 Numerical Integration
346
____ 20. Use the Trapezoidal Rule to estimate the number of square meters of land in a lot where x
and y are measured in meters as shown in the figure below. The land is bounded by a stream and
two straight roads that meet at right angles.
0
100
200
300
400
500
600
700
800
900
1000
110
110
105
97
75
75
80
65
60
20
0
74,200
74,000
73,700
74,700
74,400
____ 21.
Use a computer algebra system and Simpson’s Rule with
to approximate t in
the integral equation
. Round your answer to three decimal places.
a.
3.501
b.
3.581
c.
3.901
d.
3.529
e.
3.171
347 Chapter 5: Integration
5.6 Numerical Integration
Answer Section
5.6 Numerical Integration
348
349 Chapter 5: Integration
5.7 The Natural Logarithmic Function: Integration
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Find the indefinite integral of
.
a.
b.
c.
d.
e.
____
2.
Find the indefinite integral.
a.
b.
c.
d.
e.
____ 3. Find the indefinite integral .
a.
b.
c.
d.
e.
5.7 The natural Lodarithmic Function: Integration
350
____ 4. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 5. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 6. Find .
a.
b.
c.
d.
e.