Deck 8: Integration Techniques and Improper Integrals

Question 1

Use integration tables to find . ​

A)
B)
C)
D)
E)

Answer 1

E

Question 2

Find the indefinite integral.

A)
B)
C)
D)
E)

Answer 2

D

Question 3

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. ​
X
0
100
200
300
400
500
600
700
800
900
1000
Y
125
125
120
112
90
90
95
80
75
35
0
​ ​

A) 88,450
B) 88,250
C) 87,950
D) 88,950
E) 88,650

Answer 3

A

Question 4

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 5

Find the indefinite integral by making the substitution .

A)
B)
C)
D)
E)

Question 6

Find the definite integral.

A)
B)
C)
D)
E)

Question 7

Use the error formula to estimate the error in approximating the integral with using Simpson's Rule. Round your answer to six decimal places. ​

A) 0.003341
B) 0.023613
C) 0.001312
D) 0.010495
E) 0.031484

Question 8

Use Simpson's Rule with to approximate using the equation . Round your answer to five decimal places. ​

A) 3.14159
B) 3.13645
C) 3.14673
D) 3.14381
E) 3.13937

Question 9

Use integration tables to find the integral . ​

A)
B)
C)
D)
E)

Question 10

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 11

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 16 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. ​ ​

A) The Trapezoidal rule gives 5.008090 and Simpson's rule gives 5.258495.
B) The Trapezoidal rule gives -5.008090 and Simpson's rule gives -5.008382.
C) The Trapezoidal rule gives 5.008090 and Simpson's rule gives 5.008382.
D) The Trapezoidal rule gives 5.007217 and Simpson's rule gives 5.258495.
E) The Trapezoidal rule gives 5.007217 and Simpson's rule gives 5.008382.

Question 12

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 10 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. ​ ​

A) The Trapezoidal rule gives 13.7000 and Simpson's rule gives 14.3588.
B) The Trapezoidal rule gives 13.7000 and Simpson's rule gives 13.6667.
C) The Trapezoidal rule gives -13.6750 and Simpson's rule gives -13.6667.
D) The Trapezoidal rule gives 13.6750 and Simpson's rule gives 13.6667.
E) The Trapezoidal rule gives 13.6750 and Simpson's rule gives 14.3588.

Question 13

Find the indefinite integral by making the substitution .

A)
B)
C)
D)
E)

Question 14

Suppose a damping force affects the vibration of a spring so that the displacement of the string is given by . Find the average value of y on the interval from to . Round your answer to three decimal places.

A) 1.000
B) 0.286
C) 0.255
D) 0.318
E) 0.800

Question 15

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 16

Solve .

A)
B)
C)
D)
E)

Question 17

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 18

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A)
B)
C)
D)
E) diverges

Question 19

Find the definite integral.

A)
B)
C)
D)
E)

Question 20

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A)
B)
C)
D)
E) diverges

Question 21

Estimate the error in using (a) the Trapezoidal Rule and (b) Simpson's Rule with when approximating the following integral. ​ ​

A) The error for the Trapezoidal Rule is 0.0255 and for Simpson's Rule it is 0.0013.
B) The error for the Trapezoidal Rule is 0.0000 and for Simpson's Rule it is 0.0200.
C) The error for the Trapezoidal Rule is 0.1020 and for Simpson's Rule it is 0.0200.
D) The error for the Trapezoidal Rule is 0.1020 and for Simpson's Rule it is 0.0000.
E) The error for the Trapezoidal Rule is 0.0000 and for Simpson's Rule it is 0.0000.

Question 22

Use substitution to find the integral .

A)
B)
C)
D)
E)

Question 23

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 24

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 20 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. ​ ​

A) The Trapezoidal rule gives 92.289382 and Simpson's rule gives 96.886125.
B) The Trapezoidal rule gives 92.272500 and Simpson's rule gives 92.266873.
C) The Trapezoidal rule gives 92.289382 and Simpson's rule gives 92.266873.
D) The Trapezoidal rule gives 92.272500 and Simpson's rule gives 96.886125.
E) The Trapezoidal rule gives -92.272500 and Simpson's rule gives -92.266873.

Question 25

Use integration tables to evaluate the integral . ​

A)
B)
C)
D)
E)

Question 26

Find the definite integral.

A)
B)
C)
D)
E)

Question 27

Use integration tables to evaluate the integral . ​

A)
B)
C)
D)
E)

Question 28

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 29

Find .

A)
B)
C)
D)
E)

Question 30

Use partial fractions to find .

A)
B)
C)
D)
E)

Question 31

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 32

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. ​

A) 208
B) 189
C) 196
D) 201
E) 188

Question 33

Find the indefinite integral by making the substitution .

A)
B)
C)
D)
E)

Question 34

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 35

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A) 8
B) 6
C) 2
D)
E) diverges

Question 36

Write the form of the partial fraction decomposition for the following rational expression.

A)
B)
C)
D)
E)

Question 37

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A)
B)

Question 38

Evaluate the integral . ​

A)
B)
C)
D)
E)

Question 39

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)

Question 40

Use partial fractions to find .

A) (1/ )
B)
C) (1/
D)
E) artcat( )

Question 41

Find the indefinite integral by making the substitution .

A)
B)
C)
D)
E)

Question 42

Find the indefinite integral .

A)
B)
C)
D)
E)

Question 43

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A)
B)
C)
D)
E) diverges

Question 44

Evaluate the definite integral .

A)
B)
C)
D)
E)

Question 45

Solve the differential equation .

A)
B)
C)
D)
E)

Question 46

Find .

A)
B)
C)
D)
E)

Question 47

Use Simpson's Rule to approximate the value of the definite integral with . Round your answer to four decimal places. ​

A) 16.9283
B) 22.3017
C) 21.1805
D) 24.5441
E) 18.5139

Question 48

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 49

Solve the differential equation .

A)
B)
C)
D)
E)

Question 50

Determine whether the improper integral diverges or converges.

A)
B)

Question 51

Evaluate . Round your answer to three decimal places.

A)
B)
C)
D)
E)

Question 52

Use the Trapezoid Rule to approximate the value of the definite integral with . Round your answer to four decimal places.

A) 11.0000
B) 8.5000
C) 10.5000
D) 9.0000
E) 44.0000

Question 53

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 54

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 55

Use integration tables to evaluate the integral . ​

A)
B)
C)
D)
E)

Question 56

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 57

Use partial fractions to find the integral .

A)
B)
C)
D)
E)

Question 58

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 59

A hydraulic cylinder on an industrial machine pushes a steel block a distance of x feet where the variable force required is pounds. Find the work done in pushing the block the full 4 feet through the machine. Round your answer to three decimal places. ​

A) ft. lbs
B) ft. lbs
C) ft. lbs
D) ft. lbs
E) ft. lbs

Question 60

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)

Question 61

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 62

Use a table of integrals to find the integral. ​

A)
B)
C)
D)
E)

Question 63

Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 16 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral. ​ ​

A) The Trapezoidal rule gives 1.4019 and Simpson's rule gives 1.4022.
B) The Trapezoidal rule gives 1.4010 and Simpson's rule gives 1.4720.
C) The Trapezoidal rule gives 1.4019 and Simpson's rule gives 1.4720.
D) The Trapezoidal rule gives 1.4010 and Simpson's rule gives 1.4022.
E) The Trapezoidal rule gives -1.4019 and Simpson's rule gives -1.4022.

Question 64

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 65

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 66

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 67

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 68

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

A)
B)
C)
D)
E) diverges

Question 69

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ​

A)
B)
C)
D)
E) diverges

Question 70

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 71

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 72

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ​

A)
B)
C)
D)
E) diverges

Question 73

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 74

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 75

Find the area between the x-axis and the graph of the function .

A)
B)
C)
D)
E)

Question 76

Use a table of integrals with forms involving the trigonometric functions to find the integral. ​

A)
B)
C)
D)
E)

Question 77

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 78

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 79

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 80

Suppose a model for the ability M of a child to memorize, measured on a scale from 0 to10, is given by where t is the child's age in years. Find the average value of this model between the child's first and second birthdays. Round your answer to three decimal places.

A) 2.209
B) 0.892
C) 1.734
D) 5.059
E) 3.750