Deck 7: Applications of Integration

Question 1

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

A)
B)
C)
D)
E)

Answer 1

B

Question 2

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. ​

A)
B)
C)
D)
E)

Answer 2

A

Question 3

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

A)
B)
C)
D)
E)

Answer 3

A

Question 4

Find the fluid force of a square vertical plate submerged in water, where meters and the weight-density of water is 9800 newtons per cubic meter.

A)
B)
C)
D)
E)

Question 5

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

A)
B)
C)
D)
E)

Question 6

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis. Verify your results using the integration capabilities of a graphing utility.

A)
B)
C)
D)
E)

Question 7

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

A)
B)
C)
D)
E)

Question 8

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .

A)
B)
C)
D)
E)

Question 9

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

A)
B)
C)
D)
E)

Question 10

Find the center of mass of the point masses lying on the x-axis.

A)
B)
C)
D)
E)

Question 11

The area of the top side of a piece of sheet metal is 10 square feet. The sheet metal is submerged horizontally in 6 feet of water. Find the fluid force on the top side. Round your answer to one decimal place.

A) 3,744.0 lb
B) 3,684.0 lb
C) 3,624.0 lb
D) 3,852.0 lb
E) 4,032.0 lb

Question 12

Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .

A)
B)
C)
D)
E)

Question 13

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by about the x-axis.

A)
B)
C)
D)
E)

Question 14

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

A)
B)
C)
D)
E)

Question 15

Find the arc length from clockwise to along the circle . Round your answer to four decimal places.

A) 2.5232
B) 0.2804
C) 0.2432
D) 1.764
E) 2.1892

Question 16

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

A)
B)
C)
D)
E)

Question 17

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume ρ =1.) .

A)
B)
C)
D)
E)

Question 18

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

A)
B)
C)
D)
E)

Question 19

A barn is 100 feet long and 40 feet wide. A cross section of the roof is the inverted catenary . Find the number of square feet of roofing on the barn. Round your answer to the nearest integer.

A) 4681 square feet
B) 4701 square feet
C) 2351 square feet
D) 2000 square feet
E) 4000 square feet

Question 20

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations , and about the line .

A)
B)
C)
D)
E)

Question 21

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

A)
B)
C)
D)
E)

Question 22

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.

A)
B)
C)
D)
E)

Question 23

A rectangular plate of height h feet and base b feet is submerged vertically in a tank of fluid that weighs w pounds per cubic foot. The center of the plate is k feet below the surface of the fluid. The fluid force on the surface of the plate is given by . Find the fluid force on the rectangular plate as shown in the figure given feet and feet and fluid is water. Round your answer to one decimal place.

A)
B)
C)
D)
E)

Question 24

A solid is generated by revolving the region bounded by and about the y-axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-third of the volume is removed. Find the diameter of the hole. Round your answer to three decimal places.

A) 9.238
B) 4.619
C) 3.893
D) 15.07
E) 7.787

Question 25

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

A)
B)
C)
D)
E)

Question 26

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of about the y-axis. Round your answer to three decimal places.

A) 58.659
B) 117.319
C) 235.685
D) 5.323
E) 2.661

Question 27

Consider a beam of length L = 5 feet with a fulcrum x feet from one end as shown in the figure. In order to move a 550-pound object, a person weighing 184 pounds wants to balance it on the beam. Find x (the distance between the person and the fulcrum) such that the system is equilibrium. Round your answer to two decimal places.

A) 5.25 feet
B) 3.50 feet
C) 3.60 feet
D) 4.75 feet
E) 3.75 feet

Question 28

Find the arc length of the graph of the function over the interval [1,2].

A)
B)
C)
D)
E)

Question 29

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form.

A)
B)
C)
D)
E)

Question 30

A circular plate of radius r feet is submerged vertically in a tank of fluid that weighs w pounds per cubic foot. The center of the circle is feet below the surface of the fluid. The fluid force on the surface of the plate is given by Find the fluid force on the circular plate as shown in the figure given feet and feet and fluid is water. Round your answer to one decimal place.

A)
B)
C)
D)
E)

Question 31

Use the disk or shell method to find the volume of the solid generated by revolving the region in the first quadrant bounded by the graph of the equation about the given line. (i) the x-axis; (ii) the y-axis

A)
B)
C)
D)
E)

Question 32

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

A)
B)
C)
D)
E)

Question 33

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.

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

Question 34

A force of 25 pounds stretches a spring 11 inches in an exercise machine. Find the work done in stretching the spring 3 feet from its natural position. ​

A) ft-lb
B) ft-lb
C) ft-lb
D) ft-lb
E) ft-lb

Question 35

Find the center of mass of the given system of point masses. m_i
10
5
8
4

A)
B)
C)
D)
E)

Question 36

A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of and the x-axis about the x-axis, where x and y are measured in meters. Find the volume of the tank. Round your answer to two decimal places.

A) 0.09 m³
B) 0.03 m³
C) 0.01 m³
D) 0.17 m³
E) 0.02 m³

Question 37

A bridge has a main span of about 1350 meters. Each of its towers has a height of about 146 meters. Approximate the length of a parabolic cable along the main span if the length C of the cable is given by . Round your answer to one decimal place.

A) 695.5 meters
B) 736.4 meters
C) 2046.5 meters
D) 1391 meters
E) 2782 meters

Question 38

A quantity of a gas with an initial volume of 2 cubic feet and a pressure of 4500 pounds per square foot expands to a volume of 7 cubic feet. Find the work done by the gas. Round your answer to two decimal places. Assume the temperature of the gas in this process remain constant.

A) 5,637.43 ft-lb
B) 11,274.87 ft-lb
C) 3,214.29 ft-lb
D) 2,066.33 ft-lb
E) 14,484.94 ft-lb

Question 39

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

A)
B)
C)
D)
E)

Question 40

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

A)
B)
C)
D)
E)

Question 41

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

A)
B)
C)
D)
E)

Question 42

Find the center of mass of the point masses lying on the x-axis.

A)
B)
C)
D)
E)

Question 43

Find the fluid force of a triangular vertical plate submerged in water, where the dimensions are given in meters and the weight-density of water is 9800 newtons per cubic meter.

A)
B)
C)
D)
E)

Question 44

An overhead garage door has two springs, one on each side of the door. A force of 16 pounds is required to stretch each spring 1 foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 16 feet and springs are at their natural length when the door is open. Find the work done by pair of strings.

A) 512 ft-lb
B) 1,024 ft-lb
C) 8,192 ft-lb
D) 4,096 ft-lb
E) 2,048 ft-lb

Question 45

Find the arc length of the graph of the function over the interval [0,5].

A)
B)
C)
D)
E)

Question 46

A 22-foot chain that weighs 2 pounds per foot hangs from a winch 22 feet above ground level. Find the work done by the winch in winding up the entire chain. ​

A) 290.40 ft-lb
B) 871.20 ft-lb
C) 726.00 ft-lb
D) 387.20 ft-lb
E) 484.00 ft-lb

Question 47

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.

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

Question 48

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

Question 49

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. ​

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

Question 50

A quantity of gas with an initial volume of 5 cubic feet and a pressure of 1700 pounds per square foot expands to a volume of 9 cubic feet. Find the work done by the gas for the given volume and pressure. Round your answer to two decimal places. Assume the temperature of the gas in this process remain constant.

A) 2,997.71 ft-lb
B) 7,494.28 ft-lb
C) 8,993.14 ft-lb
D) 4,996.19 ft-lb
E) 3,996.95 ft-lb

Question 51

If the accumulation function F(x) is given by , evaluate F(4)

A)
B)
C)
D)
E)

Question 52

A tank on a water tower is a sphere of radius 25 feet. Determine the depth of the water when the tank is filled to one-fourth of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.) Round your answer to two decimal places.

A) 16.99 feet
B) 9.44 feet
C) 23.91 feet
D) 16.32 feet
E) 23.89 feet

Question 53

A cylindrical gasoline tank 4 feet in diameter and 5 feet long is carried on the bank of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done in pumping the entire gasoline contents that weighs 42 pounds per cubic feet of the fuel tank into a tractor.

Question 54

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

A)
B)
C)
D)
E)

Question 55

Find the center of mass of the given system of point masses. m_i
10
8
10

A)
B)
C)
D)
E)

Question 56

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line .

A)
B)
C)
D)
E)

Question 57

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the -axis.

A)
B)
C)
D)
E)

Question 58

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?

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

Question 59

Find the arc length of the graph of the function over the interval [14, 16].

A)
B)
C)
D)
E)

Question 60

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

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

Question 61

A tank with a base of 4 feet by 5 feet and a height of 4 feet is full of water. The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge in order to empty all of the tank. Round your answer to one decimal place.

A) 1,996.8 ft-lb
B) 1,248.0 ft-lb
C) 2,496.0 ft-lb
D) 499.2 ft-lb
E) 9,984.0 ft-lb

Question 62

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.

A)
B)
C)
D)
E)

Question 63

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

A)
B)
C)
D)
E)

Question 64

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations , and about the -axis. Round your answer to four decimal places.

A) 5.3098
B) 12.2824
C) 20.6444
D) 15.9387
E) 37.9232

Question 65

Find the arc length of the graph of the function over the interval [1,216].

A)
B)
C)
D)
E)

Question 66

Find the area of the region bounded by the graphs of the equations. .

A)
B)
C)
D)
E)

Question 67

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 10-ton satellite to a height of 200 miles above Earth. Assume that the Earth has a radius of 4000 miles.

A) 2,857.14 mi-ton
B) 1,904.76 mi-ton
C) 1,142.86 mi-ton
D) 1,523.81 mi-ton
E) 4,190.48 mi-ton

Question 68

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form

A)
B)
C)
D)
E)

Question 69

Use the integration capabilities of a graphing utility to approximate the arc length of the graph of on the interval . Round your answer to three decimal places.

A) 7.440
B) 8.840
C) 5.270
D) 7.640
E) 3.820

Question 70

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). Find the area of the face of the section superimposed on the rectangular coordinate system. Round your answer to three decimal places. ​ ​

A) 6.198 m²
B) 8.883 m²
C) 6.766 m²
D) 8.134 m²
E) 5.267 m²

Question 71

Suppose that and model the revenue (in billions of dollars) for a large corporation. The model R₁ gives projected annual revenues from 2008 through 2015, with t = 8 corresponding to 2008, and R₂ gives projected revenues if there is a decrease in the rate of growth of corporate sales over the period. Approximate the total reduction in revenue if corporate sales are actually closer to the model R₂. Round your answer to three decimal places.

A) $1.540 billion
B) $10.780 billion
C) $157.780 billion
D) $24.010 billion
E) $17.710 billion

Question 72

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.

A)
B)
C)
D)
E)

Question 73

A lunar module weighs 16 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 70 miles? Consider the radius of the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth.

A) ft-lb
B) ft-lb
C) ft-lb
D) ft-lb
E) ft-lb

Question 74

An open tank has the shape of a right circular cone. The tank is 12 feet across the top and 7 feet high. How much work is done in emptying the tank by pumping the water over the top edge? Note: The density of water is 62.4 lbs per cubic foot.

A) 9,172.80 ft-lb
B) 5,503.68 ft-lb
C) 7,338.24 ft-lb
D) 13,759.20 ft-lb
E) 16,511.04 ft-lb

Question 75

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.

A)
B)
C)
D)
E)

Question 76

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by about the y-axis. Round your answer to three decimal places.

A) 6.529
B) 0.533
C) 2.42
D) 0.923
E) 2.769

Question 77

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.

A)
B)
C)
D)
E)

Question 78

The surface of a machine part is the region between the graphs of and as shown in the figure. Find the area of the surface of the machine part. Round your answer to five decimal places.

A) 4.16667
B) 2.08333
C) 0.00213
D) 104.16667
E) 52.08333

Question 79

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

A)
B)
C)
D)
E)

Question 80

A porthole on a vertical side of a submarine (submerged in seawater) is square . Find the fluid force on the porthole, assuming that the center of the square is feet below the surface.

A)
B)
C)
D)
E)