Deck 8: Integration Techniques, L'Hôpital's Rule, and Improper Integrals

Question 1

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

A)49
B)13
C)
D)
E)diverges

Answer 1

diverges

Question 2

​Is the graph of concave up or concave down at ? Explain your answer. ​

A)​The graph of is concave up at
.
B)​The graph of is concave down at
.

Answer 2

​The graph of is concave down at
.

Question 3

​Which of the following integral expressions represents the area of the region bounded by the graphs of and the -axis? ​

A)​
B)​
C)​
D)​

Answer 3

​

Question 4

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

A)
B)
C)diverges
D)
E)

Question 5

Determine whether the improper integral diverges or converges. ​

A)
B)

Question 6

​Let ,where and is some finite positive constant,as pictured below. ​ ​
Let and .Which one of the following statements is true?
​

A)​
B)​
C)​
D)​No conclusion can be made about the relative values of and
.

Question 7

​Let be a differentiable function with the properties that and . ​

A)-3
B)​5
C)8
D)​∞

Question 8

​Let be the region bounded by the graphs of the function ,the -axis,and the lines and ,where .An expression for the volume of the solid generated by revolving about the -axis is ​

A)​
B)​
C)​
D)​

Question 9

​The area of the region bounded by the graphs of , , ,and the -axis is ​

A)​
B)​
C)​
D)​

Question 10

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

A)
B)
C)
D)
E)diverges

Question 11

​Knowing that ,explain what is meant by the expression .

Question 12

​Find an expression for ,in terms of only,that does not involve an integral.
​

Question 13

Let for and .
​
​Find an expression for in terms of only.

Question 14

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

A)
B)
C)
D)
E)diverges

Question 15

​The area of the Quadrant I region under the function is ​

A)​
B)​1
C)​
D)​divergent

Question 16

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

A)
B)
C)
D)
E)diverges

Question 17

​The base of a certain solid is the region bounded by the coordinate axes,the line ,and the graph of .For this solid,each cross section perpendicular to the -axis is a square whose side is across the base.The volume of this solid is ​

A)​10
B)​
C)​
D)​divergent

Question 18

​Assuming that ,find .

Question 19

​What is the area of the region in the first quadrant between the -axis, -axis, ,and ? ​ ​

A)​2
B)​3
C)​1
D)​∞

Question 20

​What is the area of the region in the first quadrant between the -axis,the graph ,and ? ​ ​

A)​1
B)​2
C)​0
D)​∞

Question 21

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

A)90
B)27
C)
D)9
E)diverges

Question 22

Evaluate the limit using L'Hopital's Rule if necessary.

A)0
B)
C)
D)1
E)does not exist

Question 23

Evaluate the using L'Hôpital's Rule if necessary. ​

A)
B)
C)
D)0
E)

Question 24

Determine whether the improper integral diverges or converges. ​

A)
B)

Question 25

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)
D)0
E)

Question 26

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

A)
B)
C)
D)
E)diverges

Question 27

Suppose the capitalized cost C is given by ,where is the original investment,t is the time in years,r is the annual interest rate compounded continuously,and is the annual cost of maintenance.Find the capitalized cost C of an asset forever if , ,and .Round your answer to the nearest dollar. ​

A)$1,031,414
B)$661,820
C)$1,031,443
D)$1,031,429
E)$687,820

Question 28

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)1
D)0
E)does not exist

Question 29

Evaluate the limit first by using techniques from Chapter 1 then by using L'Hopital's Rule. ​

A)
B)0
C)
D)
E)does not exist

Question 30

Suppose the capitalized cost C is given by ,where is the original investment,t is the time in years,r is the annual interest rate compounded continuously,and is the annual cost of maintenance.Find the capitalized cost C of an asset forever if , ,and .Round your answer to the nearest dollar. ​

A)676,400
B)1,463,400
C)1,497,141
D)676,420
E)1,463,000

Question 31

Evaluate the limit first by using techniques from Chapter 1 then by using L'Hopital's Rule. ​

A)
B)
C)
D)0
E)does not exist

Question 32

Evaluate the limit first by using techniques from Chapter 3 then by using L'Hopital's Rule. ​

A)
B)
C)
D)∞
E)does not exist

Question 33

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)0
C)
D)
E)does not exist

Question 34

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)
D)
E)0

Question 35

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

A)
B)
C)
D)
E)

Question 36

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

A)
B)

Question 37

Evaluate the using L'Hôpital's Rule if necessary. ​

A)
B)
C)
D)
E)0

Question 38

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)
D)0
E)

Question 39

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

A)
B)
C)
D)
E)diverges

Question 40

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

A)3
B)1
C)2
D)
E)diverges

Question 41

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 42

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)1
D)
E)

Question 43

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 44

Use integration tables to evaluate the integral . ​

A) ​
B) ​
C) ​
D) ​
E) ​

Question 45

Use integration tables to find the integral . ​

A)
B)
C)
D)
E)

Question 46

Evaluate the limit using L'Hopital's Rule if necessary.

A)
B)
C)∞
D)​- ∞
E)625

Question 47

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)1
B)
C)0
D)
E)

Question 48

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 49

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 50

Apply the Extended Mean Value Theorem to the functions and on the interval ,and find all values c in the interval such that . ​

A)
B)
C)
D)
E)

Question 51

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

A)
B)
C)
D)
E)

Question 52

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 5 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 53

Evaluate the integral . ​

A)
B)
C)
D)
E)

Question 54

Use integration tables to evaluate the integral . ​

A)
B)
C)
D)
E)

Question 55

Use integration tables to find . ​

A)
B)
C)
D)
E)

Question 56

Use integration tables to evaluate the integral . ​

A)
B)
C)
D)
E)

Question 57

A population is growing according to the logistic model ,where t is the time in days.Find the average population over the interval .Round your answer to one decimal place. ​

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

Question 58

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)
D)0
E)does not exist

Question 59

Use a table of integrals to find the integral. ​

A)
B)
C)
D)
E)

Question 60

Evaluate the limit ,using L'Hopital's Rule if necessary. ​

A)
B)
C)
D)
E)does not exist

Question 61

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)

Question 62

Evaluate the definite integral . ​

A)
B)
C)
D)
E)

Question 63

Find the integral using integration by parts. ​ ​

A)
B)
C)
D)
E)

Question 64

Evaluate the definite integral . ​

A)
B)
C)
D)
E)

Question 65

Use partial fractions to find . ​

A)
B)
C)
D)
E)

Question 66

Use substitution and partial fractions to find the indefinite integral. . ​

A)
B)
C)
D)
E)

Question 67

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

A)
B)
C)
D)
E)

Question 68

Use partial fractions to find . ​

A)
B)
C)
D)
E)

Question 69

Use substitution to find the integral . ​

A)
B)
C)
D)
E)

Question 70

Use partial fractions to find the integral . ​

A)
B)
C)
D)
E)

Question 71

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

A)
B)
C)
D)
E)

Question 72

Use partial fractions to find the integral . ​

A)
B)
C)
D)
E)

Question 73

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)

Question 74

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

A)
B)
C)
D)
E)

Question 75

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)

Question 76

Use partial fractions to find . ​

A)
B)
C)
D)
E)

Question 77

Use partial fractions to find the integral . ​

A)
B)
C)
D)
E)

Question 78

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)

Question 79

Find the area of the region bounded by the graphs of ,y = 0,x = 0,and x = 1. ​

A)
B)
C)
D)
E)

Question 80

Find the indefinite integral by making the substitution . ​ ​

A)
B)
C)
D)
E)