Question 1

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by   , and   about the   -axis.

Accepted Answer

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

Question 2

Use the shell method to find the volume of the solid generated by revolving the plane region about the line   .

Accepted Answer

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

Question 3

Electrical wires suspended between two towers form a catenary modeled by the equation   where x and y are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable. Round your answer to three decimal places.

Accepted Answer

A)  61.723 m 
B)  40.000 m 
C)  47.008 m 
D)  23.504 m 
E)  30.862 m 
A)  61.723 m 
B)  40.000 m 
C)  47.008 m 
D)  23.504 m 
E)  30.862 m

Question 4

A force of 255 Newtons stretches a spring 20 centimeters. How much work is done in stretching the spring from 10 centimeters to 40 centimeters?

Accepted Answer

A)  97.625000 joules 
B)  100.625000 joules 
C)  95.625000 joules 
D)  90.625000 joules 
E)  105.625000 joules 
A)  97.625000 joules 
B)  100.625000 joules 
C)  95.625000 joules 
D)  90.625000 joules 
E)  105.625000 joules

Question 5

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of   about the x-axis.

Accepted Answer

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

Question 6

The chief financial officer of a company reports that profits for the past fiscal year were $17.8 million. The officer predicts that profits for the next 8 years will grow at a continuous annual rate somewhere between   % and 8%. Estimate the cumulative difference in total profit over the 8 years based on the predicted range of growth rates. Round your answer to three decimal places.

Accepted Answer

A)  $614.225 billion 
B)  $38.411 billion 
C)  $35.132 billion 
D)  $2.134 billion 
E)  $133.803 billion 
A)  $614.225 billion 
B)  $38.411 billion 
C)  $35.132 billion 
D)  $2.134 billion 
E)  $133.803 billion

Question 7

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Accepted Answer

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

Question 8

Find the center of mass of the given system of point masses.

Accepted Answer

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

Question 9

Consider a beam of length L = 12 feet with a fulcrum x feet from one end as shown in the figure. Two objects weighing 36 pounds and 108 pounds are placed at opposite ends of the beam. Find x (the distance between the fulcrum and the object weighing 36 pounds) such that the system is equilibrium.

Accepted Answer

A)  10 feet 
B)  11 feet 
C)  8 feet 
D)  9 feet 
E)  7 feet 
A)  10 feet 
B)  11 feet 
C)  8 feet 
D)  9 feet 
E)  7 feet

Question 10

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal places.

Accepted Answer

A)  249.60 lb 
B)  374.40 lb 
C)  748.80 lb 
D)  187.20 lb 
E)  124.80 lb 
A)  249.60 lb 
B)  374.40 lb 
C)  748.80 lb 
D)  187.20 lb 
E)  124.80 lb

Question 11

Set up the definite integral that gives the area of the region bounded by the graph of   and   .

Accepted Answer

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

Question 12

Find the area of the surface generated by revolving the curve about the x-axis. ​   .
​

Accepted Answer

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

Question 13

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is full, assuming that the diameter is   feet and the gasoline weighs   pounds per cubic foot. (Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.) Round your answer to two decimal places.

Accepted Answer

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

Question 14

Find   for the lamina of uniform density   bounded by the graphs of the equations   and   .

Accepted Answer

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

Question 15

Find the area of the region bounded by the graphs of the algebraic functions.

Accepted Answer

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

Question 16

Find the volume of the solid generated by rotating the circle   about the x-axis.

Accepted Answer

A)  V = 24π. 
B)  V = 72π2 
C)  V = 72π. 
D)  V = 72π2. 
E)  V = 24π2 
A)  V = 24π. 
B)  V = 72π2 
C)  V = 72π. 
D)  V = 72π2. 
E)  V = 24π2

Question 17

Find   ,   , and   for the lamina of uniform density   bounded by the graphs of the equations   . ​

Accepted Answer

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

Question 18

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete weighs 5280 pounds. Find the weight of the section. Round your answer to the nearest pound. ​

Accepted Answer

A)  93,805 pounds 
B)  85,891 pounds 
C)  45,062 pounds 
D)  55,622 pounds 
E)  71,450 pounds 
A)  93,805 pounds 
B)  85,891 pounds 
C)  45,062 pounds 
D)  55,622 pounds 
E)  71,450 pounds

Question 19

Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y.

Accepted Answer

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

Question 20

Find the arc length of the graph of the function   over the interval   .

Accepted Answer

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

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by , and about the -axis.

Use the shell method to find the volume of the solid generated by revolving the plane region about the line .

Electrical wires suspended between two towers form a catenary modeled by the equation where x and y are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable. Round your answer to three decimal places.

A force of 255 Newtons stretches a spring 20 centimeters. How much work is done in stretching the spring from 10 centimeters to 40 centimeters?

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of about the x-axis.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the center of mass of the given system of point masses.

Consider a beam of length L = 12 feet with a fulcrum x feet from one end as shown in the figure. Two objects weighing 36 pounds and 108 pounds are placed at opposite ends of the beam. Find x (the distance between the fulcrum and the object weighing 36 pounds) such that the system is equilibrium.

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal places.

Set up the definite integral that gives the area of the region bounded by the graph of and .

Find the area of the surface generated by revolving the curve about the x-axis. .

Find for the lamina of uniform density bounded by the graphs of the equations and .

Find the area of the region bounded by the graphs of the algebraic functions.

Find the volume of the solid generated by rotating the circle about the x-axis.

Find , , and for the lamina of uniform density bounded by the graphs of the equations .

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete weighs 5280 pounds. Find the weight of the section. Round your answer to the nearest pound.

Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y.

Find the arc length of the graph of the function over the interval .

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Exam 7: Applications of Integration

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by , and about the -axis.

Use the shell method to find the volume of the solid generated by revolving the plane region about the line .

Electrical wires suspended between two towers form a catenary modeled by the equation where x and y are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable. Round your answer to three decimal places.

A force of 255 Newtons stretches a spring 20 centimeters. How much work is done in stretching the spring from 10 centimeters to 40 centimeters?

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of about the x-axis.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Find the center of mass of the given system of point masses.

Consider a beam of length L = 12 feet with a fulcrum x feet from one end as shown in the figure. Two objects weighing 36 pounds and 108 pounds are placed at opposite ends of the beam. Find x (the distance between the fulcrum and the object weighing 36 pounds) such that the system is equilibrium.

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal places.

Set up the definite integral that gives the area of the region bounded by the graph of and .

Find the area of the surface generated by revolving the curve about the x-axis. ​ . ​

Find for the lamina of uniform density bounded by the graphs of the equations and .

Find the area of the region bounded by the graphs of the algebraic functions.

Find the volume of the solid generated by rotating the circle about the x-axis.

Find , , and for the lamina of uniform density bounded by the graphs of the equations . ​

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete weighs 5280 pounds. Find the weight of the section. Round your answer to the nearest pound. ​

Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y.

Find the arc length of the graph of the function over the interval .

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Anal

Filters

Find the area of the surface generated by revolving the curve about the x-axis. .

Find , , and for the lamina of uniform density bounded by the graphs of the equations .

Concrete sections for the new building have the dimensions (in meters) and shape as shown in the figure (the picture is not necessarily drawn to scale). One cubic meter of concrete weighs 5280 pounds. Find the weight of the section. Round your answer to the nearest pound.