Deck 6: Integration

Question 1

Find . Round your answer to the nearest integer.

A)
B)
C)
D)
E)

Answer 1

Question 2

Suppose an investment is expected to generate income at the rate of dollars/year for the next 6 years. Find the present value of this investment if the prevailing interest rate is 8% / year compounded continuously. Round your answer to the nearest dollar.

A) $950,930
B) $953,042
C) $951,802
D) $957,375

Answer 2

$953,042

Question 3

A certain country's income distribution is described by the function Compute and .

A)
B)
C)
D)
E)

Answer 3

Question 4

Aiso deposits $110/month in a savings account paying 8% / year compounded continuously. Estimate the amount that will be in his account after 15 years.

A) $37,048
B) $36,139
C) $38,282
D) $42,603

Question 5

The management of the Titan Tire Company has determined that the quantity demanded x of their Super Titan tires/week is related to the unit price p by the relation where p is measured in dollars and x is measured in units of a thousand. Titan will make x units of the tires available in the market if the unit price is dollars. Determine the consumers' surplus and the producers' surplus when the market unit price is set at the equilibrium price. Round your answers to the nearest dollar.

A) CS = $337,012 , PS = $174,988
B) CS = $343,476 , PS = $168,524
C) CS = $341,333 , PS = $170,667
D) CS = $342,567 , PS = $169,433

Question 6

Estimate the present value of an annuity if payments are $500 monthly for 19 years and the account earns interest at the rate of 8% / year compounded continuously. Round your answer to the nearest dollar.

A) $56,454
B) $57,363
C) $62,918
D) $58,597

Question 7

A state lottery commission pays the winner of the "Million Dollar" lottery 10 annual installments of $40,000 each. If the prevailing interest rate is 5% / year compounded continuously, find the present value of the winning ticket. Round your answer to the nearest dollar.

A) $316,918
B) $310,454
C) $316,009
D) $314,775

Question 8

The demand function for a certain make of replacement cartridges for a water purifier is given by where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $5/cartridge. Round your answer to the nearest dollar.

A) $16,333
B) $16,800
C) $16,667
D) $16,886

Question 9

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of college teachers was described by the function and that of lawyers by the function Compute the coefficient of inequality for each Lorentz curve. Round your answer to four decimal places. Which profession has a more equitable income distribution?

A) ,
, college teachers
B) ,
, college teachers
C) ,
, college teachers
D) ,
, college teachers

Question 10

Sinclair wishes to supplement his retirement income by $200 per month for the next 12 years. He plans to obtain a reverse annuity mortgage (RAM) on his home to meet this need. Estimate the amount of the mortgage he will require if the prevailing interest rate is 12% per year compounded continuously.

A) $14,068
B) $15,261
C) $15,451
D) $29,552
E) $12,541

Question 11

Find the amount of an annuity if $300 per month is paid into it for a period of 17 years, earning interest at the rate of 9% per year compounded continuously.

A) $144,727
B) $118,909
C) $245,790
D) $179,614
E) $179,511

Question 12

A firm purchased a 16-year franchise for a computer outlet store that is expected to generate income at the rate of dollars per year. If the prevailing interest rate is 10% per year compounded continuously, find the present value of the franchise.

A) $1,114,979
B) $2,250,532
C) $4,660,006
D) $3,192,414
E) $1,756,632

Question 13

A certain country's income distribution is described by the function Sketch the Lorentz curve for this function. Compute and and interpret your results.

A)
;
The lowest 10 percent of the people receive 1.5 percent of the total income.
The lowest 30 percent of the people receive 10.1 percent of the total income.
B)
;
The lowest 10 percent of the people receive 1.5 percent of the total income.
The lowest 30 percent of the people receive 10.1 percent of the total income.
C)
;
The lowest 30 percent of the people receive 1.5 percent of the total income.
The lowest 10 percent of the people receive 10.1 percent of the total income.

Question 14

The demand function for a certain brand of compact disc is given by The supply function for the compact discs of is given by where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the producers' surplus if the wholesale market price is set at the equilibrium price. Round your answer to the nearest dollar.

A) $11,894
B) $11,667
C) $11,792
D) $11,346

Question 15

The Armstrongs wish to establish a custodial account to finance their children's education. If they deposit $600 monthly for 10 years in a savings account paying 6% / year compounded continuously, how much will their savings account be worth at the end of this period? Round your answer to the nearest dollar.

A) $99,877
B) $98,654
C) $100,786
D) $94,327

Question 16

It is known that the quantity demanded of a certain make of portable hair dryer is x hundred units/week and the corresponding wholesale unit price is dollars. Determine the consumers' surplus if the wholesale market price is set at $20/unit. Round your answer to the nearest dollar.

A) $5,093
B) $5,217
C) $4,975
D) $4,661

Question 17

The quantity demanded (in units of a hundred) of the miniature cameras per week is related to the unit price (in dollars) by and the quantity (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price (in dollars) by If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus. Please round the answers to the nearest dollar.

A) The consumers' surplus: $26,704, the producers' surplus: $142,718
B) The consumers' surplus: $142,718, the producers' surplus: $26,704
C) The consumers' surplus: $18,333, the producers' surplus: $20,000
D) The consumers' surplus: $20,000, the producers' surplus: $18,333
E) The consumers' surplus: $44,040, the producers' surplus: $21,478

Question 18

The demand function for a certain brand of compact disc is given by where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the wholesale market price is set at $2/disc. Round your answer to the nearest dollar.

A) $16,346
B) $16,790
C) $16,880
D) $16,667

Question 19

Estimate the present value of an annuity if payments are $1,500 monthly for 12 years and the account earns interest at the rate of 9% per year compounded continuously.

A) $105,563
B) $86,116
C) $129,803
D) $132,081
E) $104,177

Question 20

Find . Round your answer to the nearest integer.

A) 866,981
B) 912,104
C) 854,636
D) 851,747

Question 21

Estimate the present value of an annuity if payments are $700 monthly for 19 years and the account earns interest at the rate of 5% / year compounded continuously. Round your answer to the nearest dollar.
$__________

Question 22

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of stockbrokers was described by the function and that of high school teachers by the function Compute the coefficient of inequality for each Lorentz curve. Round your answers to the nearest hundredth. Which profession has a more equitable income distribution?

A) ,
, stockbrokers
B) ,
, teachers
C) ,
, stockbrokers
D) ,
, stockbrokers

Question 23

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of stockbrokers was described by the function and that of high school teachers by the function Compute the coefficient of inequality for each Lorentz curve. Round your answers to the nearest hundredth.
The coefficient of inequality for stockbrokers is __________.
The coefficient of inequality for teachers is __________.
Which profession has a more equitable income distribution?

Question 24

Suppose an investment is expected to generate income at the rate of dollars/year for the next 5 years. Find the present value of this investment if the prevailing interest rate is 0.05% / year compounded continuously. Round your answer to the nearest dollar.
$__________

Question 25

Find the amount of an annuity if $210 per month is paid into it for a period of 24 years, earning interest at the rate of 9% per year compounded continuously. Please round the answer to the nearest dollar.
$__________

Question 26

The demand function for a certain make of replacement cartridges for a water purifier is given by where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $1/cartridge. Round your answer to the nearest dollar.
CS = $__________

Question 27

Aiso deposits $140 / month in a savings account paying 8% / year compounded continuously. Estimate the amount that will be in his account after 15 years.
$__________

Question 28

Verify that .

Question 29

The demand function for a certain brand of compact disc is given by The supply function for the compact discs is given by where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the producers' surplus if the wholesale market price is set at the equilibrium price. Round your answer to the nearest dollar.
$__________

Question 30

A state lottery commission pays the winner of the "Million Dollar" lottery 10 annual installments of $50,000 each. If the prevailing interest rate is 5% / year compounded continuously, find the present value of the winning ticket. Round your answer to the nearest dollar.
$__________

Question 31

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of college teachers was described by the function and that of lawyers by the function Compute the coefficient of inequality for each Lorentz curve. Round your answer to four decimal places.
The coefficient of inequality for college teachers is __________.
The coefficient of inequality for lawyers is __________.
Which profession has a more equitable income distribution?

Question 32

The quantity demanded (in units of a hundred) of the miniature cameras per week is related to the unit price (in dollars) by and the quantity (in units of a hundred) that the supplier is willing to make available in the market is related to the unit price (in dollars) by If the market price is set at the equilibrium price, find the consumers' surplus and the producers' surplus. Please round the answers to the nearest dollar.
The consumers' surplus: $__________
The producers' surplus: $__________

Question 33

The management of the Titan Tire Company has determined that the quantity demanded x of their Super Titan tires/week is related to the unit price p by the relation where p is measured in dollars and x is measured in units of a thousand. Titan will make x units of the tires available in the market if the unit price is dollars. Determine the consumers' surplus and the producers' surplus when the market unit price is set at the equilibrium price. Round your answers to the nearest dollars.
CS = $__________
PS = $__________

Question 34

The Armstrongs wish to establish a custodial account to finance their children's education. If they deposit $600 monthly for 8 years in a savings account paying 7% / year compounded continuously, how much will their savings account be worth at the end of this period? Round your answer to the nearest dollar.
$__________

Question 35

Sinclair wishes to supplement his retirement income by $450 per month for the next 15 years. He plans to obtain a reverse annuity mortgage (RAM) on his home to meet this need. Estimate the amount of the mortgage he will require if the prevailing interest rate is 11% per year compounded continuously. Please round the answer to the nearest dollar.
$__________

Question 36

The demand function for a certain brand of compact disc is given by where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the wholesale market price is set at $4/disc. Round your answer to the nearest dollar.
$__________

Question 37

Estimate the present value of an annuity if payments are $1,400 monthly for 14 years and the account earns interest at the rate of 10% per year compounded continuously. Please round the answer to the nearest cent dollar.
$__________

Question 38

It is known that the quantity demanded of a certain make of portable hair dryer is x hundred units/week and the corresponding wholesale unit price is dollars. Determine the consumers' surplus if the wholesale market price is set at $10/unit. Round your answer to the nearest dollar.
$__________

Question 39

A firm purchased a 11-year franchise for a computer outlet store that is expected to generate income at the rate of dollars per year. If the prevailing interest rate is 9% per year compounded continuously, find the present value of the franchise. Please round the answer to the nearest dollar.
$__________

Question 40

Find the area of the shaded region.

A) A = 23.6 sq units.
B) A = 19.9 sq units.
C) A = 18.1 sq units.
D) A = 21.3 sq units.

Question 41

Sketch the region completely enclosed by the graphs of the given functions f and g.

A)

B)

C)

D)

E)

Question 42

Find the area of the region enclosed by these graphs and the vertical lines and .

A)
B)
C) 12
D)
E)

Question 43

Sketch the graph and find the area of the region bounded by the graph of the function f and lines y = 0, x = a and x = b.

A)

B)

C)

D)

Question 44

Sketch the graph and find the area of the region bounded below by the graph of the given function and above by the x-axis from x = a to x = b.

A)

B)

C)

D)

Question 45

Sketch the region completely enclosed by the graphs of the given functions f and g.

A)

B)

C)

D)

Question 46

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions f and g.

A)

B)

C)

D)

Question 47

Sketch the graphs of the functions f and g and find the area of the region enclosed by these graphs and the vertical lines x = a and x = b.

A)

B)

C)

D)

Question 48

Sketch the graphs of the functions f and g and find the area of the region enclosed by these graphs and the vertical lines x = a and x = b.

A)

B)

C)

D)

Question 49

Find the area of the shaded region.

A) A = 10 sq units.
B) A = 13 sq units.
C) A = 9 sq units.
D) A = 12 sq units.

Question 50

Sketch the graph of the region bounded by the graph of the function f and lines y = 0, x = a and x = b.

A)
B)
C)
D)
E)

Question 51

Sketch the region completely enclosed by the graphs of the given functions f and g.

A)

B)

C)

D)

Question 52

Find the area of the shaded region.

A)
B)
C)
D)
E)

Question 53

Find the area of the region bounded below by the graph of the function and above by the -axis from to .

A)
B)
C)
D)
E)

Question 54

Find the area of the shaded region.

A) A = 4 ln 2 sq units.
B) A = 5 ln 2 sq units.
C) A = ln 4 sq units.
D) A = 2 ln 2 sq units.

Question 55

Sketch the graph and find the area of the region bounded by the graph of the function f and lines y = 0, x = a and x = b.

A)

B)

C)

D)

Question 56

Sketch the graphs of the functions f and g and find the area of the region enclosed by these graphs and the vertical lines x = a and x = b.

A)

B)

C)

D)

Question 57

Find the area of the region bounded by the graph of the function and the lines , , and .

A)
B)
C)
D)
E)

Question 58

Sketch the graphs of the functions f and g and find the area of the region enclosed by these graphs and the vertical lines x = a and x = b.

A)

B)

C)

D)

Question 59

Sketch the graph and find the area of the region bounded by the graph of the function f and lines y = 0, x = a and x = b.

A)

B)

C)

D)

Question 60

Sketch the graph and find the area of the region bounded below by the graph of the given function and above by the x-axis from x = a to x = b.

A)

B)

C)

D)

Question 61

Evaluate the given definite integral.

A)
B)
C)
D)

Question 62

Evaluate the given definite integral.

A)
B)
C)
D)

Question 63

Because of the increasingly important role played by coal as a viable alternative energy source, the production of coal has been growing at the rate of billion metric tons/year t years from 1970 (which corresponds to t = 0). Had it not been for the energy crisis, the rate of production of coal since 1970 might have been only billion metric tons/year t years from 1970. Determine how much additional coal was produced between 1970 and the end of the century as an alternate energy source. Round your answer to one decimal place.
__________ billion metric tons

Question 64

Evaluate the given definite integral.

A)
B)
C)
D)

Question 65

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of the region bounded by the graphs of and and the vertical lines x = 0 and x = 6 is given by

A) True. Since f and g are continuous functions and on the interval [0, 6], then the area of the region bounded above by y = f(x) and below by y = g(x) on [0, 6] is given by

B) False. Since f and g are continuous functions and on the interval [0, 6], then the area of the region bounded above by y = g(x) and below by y = f(x) on [0, 6] is given by

Question 66

In an endeavor to curb population growth in a certain country, the government has decided to launch an extensive propaganda campaign. Without curbs, the government expects the rate of population growth to have been thousand people per year years from now, over the next years. However, successful implementation of the proposed campaign is expected to result in a population growth rate of thousand people per year years from now, over the next years. Assuming that the campaign is mounted, how many fewer people will there be in that country years from now then there would have been if no curbs had been imposed?
Please round your answer to the nearest whole number.
__________ thousand people

Question 67

Evaluate the given definite integral.

A)
B)
C)
D)

Question 68

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The area of the region bounded by the graphs of and and the vertical lines x = 0 and x = 4 is given by

Question 69

Find the area of the region completely enclosed by the graphs of the given functions and .

A)
B)
C)
D)
E)

Question 70

Find the area of the shaded region.

Question 71

Find the area of the shaded region. Round your answer to one decimal place.

Question 72

Find the area of the shaded region.

Question 73

In tests conducted by Auto Test Magazine on two identical models of the Phoenix Elite - one equipped with a standard engine and the other with a turbo-charger - it was found that the acceleration of the former is given by ft/sec, t seconds after starting from rest at full throttle, whereas the acceleration of the latter is given by ft/sec. How much faster is the turbo-charged model moving than the model with the standard engine at the end of a 10-second test run at full throttle?
__________ ft/sec

Question 74

In tests conducted by Auto Test Magazine on two identical models of the Phoenix Elite - one equipped with a standard engine and the other with a turbo-charger - it was found that the acceleration of the former is given by ft/sec, t sec after starting from rest at full throttle, whereas the acceleration of the latter is given by ft/sec. How much faster is the turbo-charged model moving than the model with the standard engine at the end of a 10-second test run at full throttle?

A) 65 ft/sec
B) 58 ft/sec
C) 75 ft/sec
D) 70 ft/sec

Question 75

Find the area of the shaded region.

Question 76

Evaluate the given definite integral.

A)
B)
C)
D)

Question 77

In an endeavor to curb population growth in a certain country, the government has decided to launch an extensive propaganda campaign. Without curbs, the government expects the rate of population growth to have been thousand people per year years from now, over the next years. However, successful implementation of the proposed campaign is expected to result in a population growth rate of thousand people per year years from now, over the next years. Assuming that the campaign is mounted, how many fewer people will there be in that country years from now then there would have been if no curbs had been imposed? Please round your answer to the nearest integer number of thousands.

A) thousand people
B) thousand people
C) thousand people
D) thousand people
E) thousand people

Question 78

Because of the increasingly important role played by coal as a viable alternative energy source, the production of coal has been growing at the rate of billion metric tons/year t years from 1990 (which corresponds to t = 0). Had it not been for the energy crisis, the rate of production of coal since 1990 might have been only billion metric tons/year t years from 1990. Determine how much additional coal was produced between 1990 and the end of the century as an alternate energy source.

A) 13.6 billion metric tons
B) 0.6 billion metric tons
C) 4.7 billion metric tons
D) 7.9 billion metric tons

Question 79

Evaluate the definite integral.

A)
B)
C)
D)
E)

Question 80

Evaluate the given definite integral.

A)
B)
C)
D)