Question 1

Equations are given whose graphs enclose a region. Find the area of the region. ​   ,   ,   and   ​

Accepted Answer

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

Question 2

Evaluate the improper integral if it converges, or state that it diverges. ​   ​

Accepted Answer

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

Question 3

Approximate the area under the curve defined by the function   over the interval x = 0 to x = 3 using the right-hand endpoints of three subintervals (rectangles). ​

Accepted Answer

A)   
Square units 
B)   
Square units 
C)   
Square units 
D)   
Square units 
E)   
Square units 
A)   
Square units 
B)   
Square units 
C)   
Square units 
D)   
Square units 
E)   
Square units

Question 4

Find the shaded area between the given function and the x-axis. ​   ​   ​

Accepted Answer

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

Question 5

Evaluate the definite integral   . ​

Accepted Answer

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

Question 6

Use the function   from   to   and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find   by using the 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 7

Evaluate the improper integral if it converges, or state that it diverges. ​

Accepted Answer

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

Question 8

The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half-hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are as follows.   Use the trapezoidal rule to approximate the average number of milligrams in the bloodstream during the first   hours and round your answer to two decimal places. ​

Accepted Answer

A) 433.60 mg 
B) 197.09 mg 
C) 123.18 mg 
D) 256.22 mg 
E) 335.05 mg 
A) 433.60 mg 
B) 197.09 mg 
C) 123.18 mg 
D) 256.22 mg 
E) 335.05 mg

Question 9

For what value of a is   a probability density function? ​

Accepted Answer

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

Question 10

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 11

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. ​
I. Integration by parts
   ​   ​

Accepted Answer

A) I. Integration by parts;   
B)    
C)    
D) ​   
E) not integrable by any of the techniques we have studied 
A) I. Integration by parts;   
B)    
C)    
D) ​   
E) not integrable by any of the techniques we have studied

Question 12

If the supply function for a commodity is   where p is in millions of dollars and x is the number of thousands of units. What is the producer's surplus when 8 thousand units are sold? Round your answer to the nearest million dollars. ​

Accepted Answer

A) 1,293 million dollars 
B) 3,296 million dollars 
C) 3,316 million dollars 
D) 170 million dollars 
E) 3,356 million dollars 
A) 1,293 million dollars 
B) 3,296 million dollars 
C) 3,316 million dollars 
D) 170 million dollars 
E) 3,356 million dollars

Question 13

Evaluate   . ​

Accepted Answer

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

Question 14

Use the function   from   to   and n equal subintervals with the function evaluated at the right-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 15

The probability density function for the life span of an electronics part is   where t is the number of months in service. Find the probability that any given part of this type lasts longer than 18 months. Round your answer to three decimal places. ​

Accepted Answer

A) 0.487 
B) 0.513 
C) 0.532 
D) 0.497 
E) 0.468 
A) 0.487 
B) 0.513 
C) 0.532 
D) 0.497 
E) 0.468

Question 16

With the data for selected years from 1920 to 2007, the life span of individuals in a country can be modeled by   where x = 0 represents 1900. Find the predicted average life span from 2040 to 2055. Round your answer to the nearest year. ​

Accepted Answer

A) 120 years 
B) 137 years 
C) 84 years 
D) 270 years 
E) 47 years 
A) 120 years 
B) 137 years 
C) 84 years 
D) 270 years 
E) 47 years

Question 17

Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to   in pounds per year. Suppose also that the waste decays exponentially at a rate of 9% per year. Then the amount of radioactive waste that will accumulate in b years is given by   . Evaluate this integral. ​

Accepted Answer

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

Question 18

Evaluate the integral   by integration. Round your answer to two decimal places. ​

Accepted Answer

A) 4.00 
B) 9.79 
C) 0.71 
D) 7.34 
E) 3.77 
A) 4.00 
B) 9.79 
C) 0.71 
D) 7.34 
E) 3.77

Question 19

The demand function for a product is   and the supply function is   , where p is the number of dollars and x is the number of units. Find the consumer's surplus there. Round to the nearest cent. ​

Accepted Answer

A) $373.85 
B) $486.00 
C) $583.20​ 
D) $1020.60 
E) $583.20 
A) $373.85 
B) $486.00 
C) $583.20​ 
D) $1020.60 
E) $583.20

Question 20

A drug manufacturer has developed a time-release capsule with the number of milligrams of the drug in the bloodstream given by   where x is in hours and   . Find the average number of milligrams of the drug in the bloodstream for the first 3 hours after a capsule is taken. Round your answer to the nearest milligram. ​

Accepted Answer

A) 350 mg 
B) 266 mg 
C) 233 mg 
D) 175 mg 
E) 200 mg 
A) 350 mg 
B) 266 mg 
C) 233 mg 
D) 175 mg 
E) 200 mg

Equations are given whose graphs enclose a region. Find the area of the region. , , and

Evaluate the improper integral if it converges, or state that it diverges.

Approximate the area under the curve defined by the function over the interval x = 0 to x = 3 using the right-hand endpoints of three subintervals (rectangles).

Find the shaded area between the given function and the x-axis.

Evaluate the definite integral .

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

Evaluate the improper integral if it converges, or state that it diverges.

For what value of a is a probability density function?

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

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. I. Integration by parts

If the supply function for a commodity is where p is in millions of dollars and x is the number of thousands of units. What is the producer's surplus when 8 thousand units are sold? Round your answer to the nearest million dollars.

Evaluate .

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

The probability density function for the life span of an electronics part is where t is the number of months in service. Find the probability that any given part of this type lasts longer than 18 months. Round your answer to three decimal places.

With the data for selected years from 1920 to 2007, the life span of individuals in a country can be modeled by where x = 0 represents 1900. Find the predicted average life span from 2040 to 2055. Round your answer to the nearest year.

Evaluate the integral by integration. Round your answer to two decimal places.

The demand function for a product is and the supply function is , where p is the number of dollars and x is the number of units. Find the consumer's surplus there. Round to the nearest cent.

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

Equations are given whose graphs enclose a region. Find the area of the region. ​ , , and ​

Evaluate the improper integral if it converges, or state that it diverges. ​ ​

Approximate the area under the curve defined by the function over the interval x = 0 to x = 3 using the right-hand endpoints of three subintervals (rectangles). ​

Find the shaded area between the given function and the x-axis. ​ ​ ​

Evaluate the definite integral . ​

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

Evaluate the improper integral if it converges, or state that it diverges. ​

For what value of a is a probability density function? ​

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

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. ​ I. Integration by parts ​ ​

If the supply function for a commodity is where p is in millions of dollars and x is the number of thousands of units. What is the producer's surplus when 8 thousand units are sold? Round your answer to the nearest million dollars. ​

Evaluate . ​

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

The probability density function for the life span of an electronics part is where t is the number of months in service. Find the probability that any given part of this type lasts longer than 18 months. Round your answer to three decimal places. ​

With the data for selected years from 1920 to 2007, the life span of individuals in a country can be modeled by where x = 0 represents 1900. Find the predicted average life span from 2040 to 2055. Round your answer to the nearest year. ​

Evaluate the integral by integration. Round your answer to two decimal places. ​

The demand function for a product is and the supply function is , where p is the number of dollars and x is the number of units. Find the consumer's surplus there. Round to the nearest cent. ​

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

Equations are given whose graphs enclose a region. Find the area of the region. , , and

Evaluate the improper integral if it converges, or state that it diverges.

Approximate the area under the curve defined by the function over the interval x = 0 to x = 3 using the right-hand endpoints of three subintervals (rectangles).

Find the shaded area between the given function and the x-axis.

Evaluate the definite integral .

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

Evaluate the improper integral if it converges, or state that it diverges.

For what value of a is a probability density function?

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

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. I. Integration by parts

If the supply function for a commodity is where p is in millions of dollars and x is the number of thousands of units. What is the producer's surplus when 8 thousand units are sold? Round your answer to the nearest million dollars.

Evaluate .

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

The probability density function for the life span of an electronics part is where t is the number of months in service. Find the probability that any given part of this type lasts longer than 18 months. Round your answer to three decimal places.

With the data for selected years from 1920 to 2007, the life span of individuals in a country can be modeled by where x = 0 represents 1900. Find the predicted average life span from 2040 to 2055. Round your answer to the nearest year.

Evaluate the integral by integration. Round your answer to two decimal places.

The demand function for a product is and the supply function is , where p is the number of dollars and x is the number of units. Find the consumer's surplus there. Round to the nearest cent.