Deck 6: Applications of Integration

Question 1

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

Answer 1

Question 2

Find the area of the region bounded by the given curves (a) using integration with respect to and (b) using integration with respect to

Answer 2

A.14
B.14

Question 3

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.Sketch the region and a representative rectangle.

Answer 3

Question 4

Find the work done in pushing a car a distance of while exerting a constant force of

Question 5

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Question 6

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line.Sketch the region and a representative rectangle.

Question 7

The velocity v of blood that flows in a blood vessel with radius nd length at a distance from the central axis is where is the pressure difference between the ends of the vessel and is the viscosity of the blood.Find the average velocity over the interval

Question 8

Find the number(s) such that the average value of the function on the interval is equal to

Question 9

Find the volume of the solid obtained by revolving the region under the graph of on about the

Question 10

Use a graphing utility to (a) plot the graphs of the given functions and (b) find the approximate of the points of intersection of the graphs.Then find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the RoundAnswers to two decimal places.

Question 11

Find the volume of the solid obtained by rotating the region bounded by and about the

Question 12

Find the volume common to two spheres, each with radius if the center of each sphere lies on the surface of the other sphere.

Question 13

Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the RoundAnswers to two decimal places.

Question 14

Racing cars driven by Chris and Kelly are side by side at the start of a race.The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race.Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

Question 15

The tank shown is full of water.Given that water weighs find the work (in lb-ft) required to pump the water out of the tank.

Question 16

Find the average value of the function on the given interval.

Question 17

Find the volume of the solid obtained by rotating about the x-axis the region under the curve from

Question 18

Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.Sketch the region and a representative rectangle.

Question 19

Sketch the region bounded by the graphs of the given equations and find the area of that region.

Question 20

Sketch the region bounded by the graphs of the given equations and find the area of that region.

Question 21

Find the number(s) such that the average value of the function on the interval is equal to

Question 22

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

Question 23

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x axis.

Question 24

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Question 25

Sketch the region bounded by the graphs of the given equations and find the area of that region.

Question 26

Find the average value of the function on the interval

Question 27

Find the work done in pushing a car a distance of while exerting a constant force of

Question 28

Find the volume common to two spheres, each with radius if the center of each sphere lies on the surface of the other sphere.

Question 29

Racing cars driven by Chris and Kelly are side by side at the start of a race.The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race.Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

Question 30

Use the Midpoint Rule with to approximate the area of the region bounded by the given curves.

Question 31

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.Sketch the region and a representative rectangle.

Question 32

A tank has the shape of an inverted right circular cone with a base radius of 4 m and a height of 9 m.If the tank is filled to a height of 1 m, find the work required (to the nearest joule) to empty the tank by pumping the water over the top of the tank.(The mass of water is 1 nd the force of gravity is )

Question 33

Use a graphing utility to (a) plot the graphs of the given functions, (b) find the approximate x-coordinates of the points of intersection of the graphs, and (c) find an approximation of the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the RoundAnswers to two decimal places.

Question 34

A chain weighing 4 lb/ft hangs vertically from a winch located 11 ft above the ground, and the free end of the chain is just touching the ground.Find the work done by the winch in pulling in the whole chain.

Question 35

Find the volume of the solid obtained by rotating the region bounded by and about the y-axis.

Question 36

The base of is the parabolic region Cross-sections perpendicular to the axis are squares.Find the volume of

Question 37

The velocity v of blood that flows in a blood vessel with radius and length at a distance from the central axis is where is the pressure difference between the ends of the vessel and is the viscosity of the blood.Find the average velocity over the interval

Question 38

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis.

Question 39

Find the average value of the function on the given interval.

Question 40

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Question 41

Select the correct Answer for each question.
Sketch the region bounded by the graphs of the given equations and find the area of their region.

A)
B)
C)
D)

Question 42

Select the correct Answer for each question.
Use integration to find the area of the triangle with the given vertices.(0, 0), (1, 7), (4, 2)

A)
B)26
C)28
D)13

Question 43

Select the correct Answer for each question.
Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters).You discover that both napkin rings have the same height t as shown in the figure.Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius through the center of a sphere of radius and express theAnswer in terms of

A)
B)
C)
D)
E)

Question 44

Select the correct Answer for each question.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the

A)
B)
C)
D)

Question 45

Select the correct Answer for each question.
Find the area of the region bounded by the given curves.

A)
B)
C)
D)
E)None of these

Question 46

Select the correct Answer for each question.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. the line y = 3

A)
B)
C)
D)

Question 47

Select the correct Answer for each question.
Sketch the region enclosed by Find the area of the region.

A)
B)
C)
D)
E)

Question 48

Select the correct Answer for each question.
Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

A)12.08127
B)91.504
C)3.141257
D)3.66016
E)18.3008

Question 49

Select the correct Answer for each question.
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)

Question 50

Select the correct Answer for each question.
Sketch the region bounded by the graphs of the given equations and find the area of that region.

A)
B)
C)
D)

Question 51

Select the correct Answer for each question.
Use the method of disks or washers, or the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.Sketch the region and a representative rectangle. the x-axis

A)
B)
C)
D)

Question 52

Select the correct Answer for each question.
Find the volume of the solid that is obtained by revolving the region about the line

A)
B)
C)
D)

Question 53

Select the correct Answer for each question.
Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the specified axis.

A)
B)
C)
D)
E)

Question 54

Select the correct Answer for each question.
Newton's Law of Gravitation states that two bodies with masses attract each other with a force where r is the distance between the bodies and G is the gravitation constant.If one of bodies is fixed, find the work needed to move the other from r=l to r=g .

A)
B)
C)
D)
E)

Question 55

Select the correct Answer for each question.
A spring has a natural length of cm.If a force of N is required to keep it stretched to a length of cm, how much work is required to stretch it from cm to cm?

A)
B)
C)
D)
E)

Question 56

Select the correct Answer for each question.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the

A)
B)
C)
D)

Question 57

Select the correct Answer for each question.
A tank is full of water.Find the work required to pump the water out of the outlet.

A)
B)
C)
D)
E)None of these

Question 58

Select the correct Answer for each question.
Use cylindrical shells to find the volume of the solid.A sphere of radius

A)
B)
C)
D)
E)

Question 59

Select the correct Answer for each question.
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. , y = 0; the line y = 5

A)
B)
C)
D)

Question 60

Select the correct Answer for each question.
A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm.How much work is done in stretching the spring from cm to cm?

A)
B)
C)
D)
E)

Question 61

Select the correct Answer for each question.
A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm.How much work is done in stretching the spring from cm to cm?

A)
B)
C)
D)
E)

Question 62

Select the correct Answer for each question.
Sketch the region bounded by the graphs of the given equations and find the area of their region.

A)
B)
C)
D)

Question 63

Select the correct Answer for each question.
If J of work are needed to stretch a spring from cm to cm and another J are needed to stretch it from cm to cm, what is the natural length of the spring? Round theAnswer to nearest integer.

A)
B)
C)
D)
E)

Question 64

Select the correct Answer for each question.
Find the volume of the solid that is obtained by revolving the region about the x-axis.

A)
B)
C)
D)

Question 65

Select the correct Answer for each question.
An aquarium m long, 1 m wide, and 1 m deep is full of water.Find the work needed to pump half of the water out of the aquarium.(Use the facts that the density of water is

A)
B)
C)
D)
E)

Question 66

Select the correct Answer for each question.
Use the Midpoint Rule with n = 4 to estimate the volume obtained by rotating about the region under the y-axis the region under the curve. The choices are rounded to the nearest hundredth.

A)
B)
C)
D)
E)

Question 67

Select the correct Answer for each question.
Find the volume of the solid obtained by rotating the region bounded by about the x-axis.

A)
B)
C)
D)
E)

Question 68

Select the correct Answer for each question.
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

A)
B)
C)
D)
E)None of these

Question 69

Select the correct Answer for each question.
Find the area of the shaded region.

A)
B)
C)
D)8

Question 70

Select the correct Answer for each question.
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)

Question 71

Select the correct Answer for each question.
A particle moves a distance of 160 ft along a straight line.As it moves, it is acted upon by a constant force of magnitude 10 lb in a direction opposite to that of the motion.What is the work done by the force?

A)
B)
C)
D)

Question 72

Select the correct Answer for each question.
Sketch the region bounded by the graphs of the given equations and find the area of that region.

A)
B)
C)
D)

Question 73

Select the correct Answer for each question.
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

A)
B)
C)
D)
E)None

Question 74

Select the correct Answer for each question.
Find the area of the shaded region.

A)
B)
C)
D)

Question 75

Select the correct Answer for each question.
Find the area of the region bounded by the given curves.

A)2
B)
C)4
D)
E)

Question 76

Select the correct Answer for each question.
The base of a solid is a circular disk with radius .Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

A)49
B)16
C)6
D)7
E)None of these

Question 77

Select the correct Answer for each question.
Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

A)18.3008
B)3.66016
C)91.504
D)3.141257
E)12.08127

Question 78

Select the correct Answer for each question.
Sketch a graph to estimate the x-coordinates of the points of intersection of the given curves.Then use this information to estimate the volume of the solid obtained by rotating about the y axis the region enclosed by these curves.Rounded to the nearest hundredth.

A)
B)
C)
D)
E)

Question 79

Select the correct Answer for each question.
Sketch the region bounded by the graphs of the given equations and find the area of their region.

A)
B)
C)
D)

Question 80

Select the correct Answer for each question.
The height of a monument is m.A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side meters.Find the volume of the monument.

A)
B)
C)
D)
E)