Question 1

Evaluate the definite integral   . ​

Accepted Answer

A)    
B) ​   
C)    
D)    
E)    
A)    
B) ​   
C)    
D)    
E)    E

Question 2

Suppose that the Carter Car Service franchise finds that the income generated by its stores can be modeled by assuming that the income is a continuous stream with a monthly rate of flow at time t given by   (dollars per month). Find the total income from a Carter Car Service store for years 3 through 5 of operation and round to the nearest penny. ​

Accepted Answer

A) $973,587.71 
B) $2,307,500.23 
C) $1,059,312.41 
D) $1,684,306.74 
E) $1,186,131.50 
A) $973,587.71 
B) $2,307,500.23 
C) $1,059,312.41 
D) $1,684,306.74 
E) $1,186,131.50 D

Question 3

Use the Trapezoidal Rule to approximate   with n = 6. Round your answer to two decimal places. ​

Accepted Answer

A) 12.51 
B) 3.65 
C) 5.21 
D) 2.34 
E) 6.25 
A) 12.51 
B) 3.65 
C) 5.21 
D) 2.34 
E) 6.25 E

Question 4

Evaluate the integral   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

Use the function   from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S). ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

Find the value of the sum   , if   . ​

Accepted Answer

A) 64 
B) 60 
C) 20 
D) 8 
E) 32 
A) 64 
B) 60 
C) 20 
D) 8 
E) 32

Question 7

Find the producer's surplus for a product if its demand function is   and its supply function is   where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars. ​

Accepted Answer

A) 53 million dollars 
B) 44 million dollars 
C) 40 million dollars 
D) 36 million dollars 
E) 47 million dollars 
A) 53 million dollars 
B) 44 million dollars 
C) 40 million dollars 
D) 36 million dollars 
E) 47 million dollars

Question 8

Evaluate the integral   .

Accepted Answer

A)    
B)    
C)    
D)    
E) integral diverges 
A)    
B)    
C)    
D)    
E) integral diverges

Question 9

Use Simpson's Rule to approximate   with n = 4. Round your answer to 3 decimal places. ​

Accepted Answer

A) 0.562 
B) 0.066 
C) 1.617 
D) 0.733 
E) 2.123 
A) 0.562 
B) 0.066 
C) 1.617 
D) 0.733 
E) 2.123

Question 10

Assume that a store finds that its sales revenue changes at a rate given by   dollars per day, where t is the number of days after an advertising campaign ends and   . Find the total sales for the second week after the campaign ends   to   . ​

Accepted Answer

A) $5,733 
B) $6,909 
C) $4,179 
D) $12,117 
E) $1,575 
A) $5,733 
B) $6,909 
C) $4,179 
D) $12,117 
E) $1,575

Question 11

Find the area of the region enclosed by the graphs of   and   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

The following table shows the rate of oil consumption (in thousands of barrels per year) by a certain city. Estimate the total consumption of oil by the city from 1999 -2004 by using 5 equal subdivisions and left-hand endpoints to estimate the area under the graph that corresponds to the table from 1999 to 2004. ​   ​

Accepted Answer

A) 17.708 thousand barrels 
B) 16.656 thousand barrels 
C) 16.812 thousand barrels 
D) 17.606 thousand barrels 
E) 15.733 thousand barrels 
A) 17.708 thousand barrels 
B) 16.656 thousand barrels 
C) 16.812 thousand barrels 
D) 17.606 thousand barrels 
E) 15.733 thousand barrels

Question 13

A franchise models the profit from its store as a continuous income stream with a monthly rate of flow at time t given by   (dollars per month). When a new store opens, its manager is judged against the model, with special emphasis on the second half of the first year. Find the total profit for the second 6-month period   to   . Round your answer to the nearest dollar. ​

Accepted Answer

A) $61,831 
B) $51,679 
C) $40,973 
D) $30,821 
E) $46,232 
A) $61,831 
B) $51,679 
C) $40,973 
D) $30,821 
E) $46,232

Question 14

Find the value of the given sum and round to four decimal places. ​   ​

Accepted Answer

A) 3.2405 
B) 2.3000 
C) 2.5571 
D) 2.0000 
E) 2.6250 
A) 3.2405 
B) 2.3000 
C) 2.5571 
D) 2.0000 
E) 2.6250

Question 15

Evaluate the integral   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 16

Suppose that a printing firm considers the production of its presses as a continuous income stream. If the annual rate of flow at time t is given by   in thousands of dollars per year, and if money is worth 7% compounded continuously, find the present value and future value of the presses over the next 10 years. Round your answer to the nearest dollar. ​

Accepted Answer

A) Present Value: $212,562; Future Value: $428,047 
B) Present Value: $181,071; Future Value: $364,632 
C) Present Value: $119,242; Future Value: $240,124 
D) Present Value: $193,878; Future Value: $390,422 
E) Present Value: $127,833; Future Value: $257,424 
A) Present Value: $212,562; Future Value: $428,047 
B) Present Value: $181,071; Future Value: $364,632 
C) Present Value: $119,242; Future Value: $240,124 
D) Present Value: $193,878; Future Value: $390,422 
E) Present Value: $127,833; Future Value: $257,424

Question 17

Use integration by parts to evaluate   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 18

Use integration by parts to evaluate   . Note that evaluation may require integration by parts more than once. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

Suppose the number of daily sales of a product was found to be given by   , x days after the start of an advertising campaign for this product. Find the average daily sales during the first 25 days of the campaign-that is, from   to   . Round your answer to the nearest dollar. ​

Accepted Answer

A) $133 
B) $130 
C) $127 
D) $265 
E) $255 
A) $133 
B) $130 
C) $127 
D) $265 
E) $255

Question 20

The rate of depreciation of a building is given by   dollars per year,   . Use the definite integral to find the total depreciation over the first 20 years. ​

Accepted Answer

A) $1,680,000 
B) $84,000 
C) $840,000 
D) $480,000 
E) $3,360,000 
A) $1,680,000 
B) $84,000 
C) $840,000 
D) $480,000 
E) $3,360,000

Evaluate the definite integral .

Use the Trapezoidal Rule to approximate with n = 6. Round your answer to two decimal places.

Evaluate the integral .

Use the function from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S).

Find the value of the sum , if .

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars.

Evaluate the integral .

Use Simpson's Rule to approximate with n = 4. Round your answer to 3 decimal places.

Assume that a store finds that its sales revenue changes at a rate given by dollars per day, where t is the number of days after an advertising campaign ends and . Find the total sales for the second week after the campaign ends to .

Find the area of the region enclosed by the graphs of and .

Find the value of the given sum and round to four decimal places.

Evaluate the integral .

Use integration by parts to evaluate .

Use integration by parts to evaluate . Note that evaluation may require integration by parts more than once.

Suppose the number of daily sales of a product was found to be given by , x days after the start of an advertising campaign for this product. Find the average daily sales during the first 25 days of the campaign-that is, from to . Round your answer to the nearest dollar.

The rate of depreciation of a building is given by dollars per year, . Use the definite integral to find the total depreciation over the first 20 years.

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Functions of Two or More Variables

Filters

Exam 14: Definite Integrals: Techniques of Integration

Evaluate the definite integral . ​

Use the Trapezoidal Rule to approximate with n = 6. Round your answer to two decimal places. ​

Evaluate the integral . ​

Use the function from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S). ​

Find the value of the sum , if . ​

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars. ​

Evaluate the integral .

Use Simpson's Rule to approximate with n = 4. Round your answer to 3 decimal places. ​

Assume that a store finds that its sales revenue changes at a rate given by dollars per day, where t is the number of days after an advertising campaign ends and . Find the total sales for the second week after the campaign ends to . ​

Find the area of the region enclosed by the graphs of and . ​

Find the value of the given sum and round to four decimal places. ​ ​

Evaluate the integral . ​

Use integration by parts to evaluate . ​

Use integration by parts to evaluate . Note that evaluation may require integration by parts more than once. ​

Suppose the number of daily sales of a product was found to be given by , x days after the start of an advertising campaign for this product. Find the average daily sales during the first 25 days of the campaign-that is, from to . Round your answer to the nearest dollar. ​

The rate of depreciation of a building is given by dollars per year, . Use the definite integral to find the total depreciation over the first 20 years. ​

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Functions of Two or More Variables

Filters

Evaluate the definite integral .

Use the Trapezoidal Rule to approximate with n = 6. Round your answer to two decimal places.

Evaluate the integral .

Use the function from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S).

Find the value of the sum , if .

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars.

Use Simpson's Rule to approximate with n = 4. Round your answer to 3 decimal places.

Assume that a store finds that its sales revenue changes at a rate given by dollars per day, where t is the number of days after an advertising campaign ends and . Find the total sales for the second week after the campaign ends to .

Find the area of the region enclosed by the graphs of and .

Find the value of the given sum and round to four decimal places.

Evaluate the integral .

Use integration by parts to evaluate .

Use integration by parts to evaluate . Note that evaluation may require integration by parts more than once.

Suppose the number of daily sales of a product was found to be given by , x days after the start of an advertising campaign for this product. Find the average daily sales during the first 25 days of the campaign-that is, from to . Round your answer to the nearest dollar.

The rate of depreciation of a building is given by dollars per year, . Use the definite integral to find the total depreciation over the first 20 years.