Deck 5: Applications of Integration

Question 1

Find the integral.

A)
B)
C)
D)

Answer 1

Question 2

Find the integral.

A)
B)
C)
D)

Answer 2

Question 3

Find the integral.

A)
B)
C)
D)

Answer 3

Question 4

Find the integral using the indicated substitution. ,

A)
B)
C)
D)

Question 5

Find the average value of the function on the interval . Round your answer to 3 decimal places.

A)0.45
B)9.342
C)
D)0.432
E)18

Question 6

Evaluate the integral.

A)
B)
C)
D)

Question 7

Find the integral using the indicated substitution. ,

A)
B)
C)
D)

Question 8

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 9

Find the integral.

A)
B)
C)
D)

Question 10

Find the integral.

A)
B)
C)
D)

Question 11

The temperature of a metal rod, m long, is x (in degree Celsius) at a distance x meters from one end of the rod. What is the average temperature of the rod?

A)
B)
C)
D)
E)

Question 12

Find the integral.

A)
B)
C)
D)

Question 13

Find the indefinite integral

A)
B)
C)
D)

Question 14

Find the indefinite integral.

A)
B)
C)
D)

Question 15

A steel girder weighing 200 lb is hoisted from ground level to the roof of a 40-ft building using a chain that weighs 5 lb/running foot. Find the work done.

A)4,000 ft-lb
B)8,100 ft-lb
C)12,000 ft-lb
D)8,000 ft-lb

Question 16

Find the integral.

A)
B)
C)
D)

Question 17

Find the indefinite integral

A)
B)
C)
D)

Question 18

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 19

Find the indefinite integral.

A)
B)
C)
D)

Question 20

Find the indefinite integral.

A)
B)
C)
D)

Question 21

Find the number(s) a such that the average value of the function on the interval is equal to 10.

A)
B)
C)
D)
E)

Question 22

Find the average value of the function on the interval .

Question 23

Find the integral.

A)
B)
C)
D)

Question 24

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 25

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 the answer to nearest integer.

A)
B)
C)
D)
E)

Question 26

Find the average value of the function on the interval .

A)
B)
C)
D)
E)

Question 27

Evaluate the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 28

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

Question 29

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

Question 30

A heavy rope, ft long, weighs lb/ft and hangs over the edge of a building 110 ft high. How much work is done in pulling the rope to the top of the building?

A)
B)
C)
D)
E)

Question 31

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 32

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 33

Find the average value of the function on the interval .

Question 34

The linear density of a m long rod is where x is measured in meters from one end of the rod. Find the average density of the rod.

Question 35

Find the integral.

A)
B)
C)
D)

Question 36

Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 37

Find the indefinite integral.

A)
B)
C)
D)

Question 38

Find the indefinite integral.

A)
B) + 8x + C
C)
D) + 8x + C

Question 39

Find the indefinite integral.

A)2x + + C
B)2x + + C
C)2x + + C
D)2x + + C

Question 40

In a certain city the temperature hours after 7 A.M. was modeled by the function Find the average temperature to three decimal places during the period from 7 A.M. to 7 P.M.

Question 41

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 h as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of .

A)
B)
C)
D)
E)

Question 42

Find the derivative of the function

A)
B)
C)
D)

Question 43

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 3 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 1000 kg/m³ and the force of gravity is 9.8 m/sec².)

Question 44

In a steam engine the pressure and volume of steam satisfy the equation , where k is a constant. (This is true for adiabatic expansion, that is, expansion in which there is no heat transfer between the cylinder and its surroundings.) Calculate the work done by the engine (in ft-lb) during a cycle when the steam starts at a pressure of and a volume of and expands to a volume of Use the fact that the work done by the gas when the volume expands from to volume is .

Question 45

An aquarium m long, m wide, and 1 m deep is full of water. Find the work (in J) needed to pump half of the water out of the aquarium. (Use the facts that the density of water is and )

Question 46

Use cylindrical shells to find the volume of the solid. A sphere of radius .

A)
B)
C)
D)
E)

Question 47

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region about the indicated axis or line. y-axis

A) $\pi$
B) $\pi$
C) $\pi$
D) $\pi$

Question 48

Find the derivative of the function.

A)
B)
C)
D)

Question 49

Evaluate the limit by interpreting it as the limit of a Riemann sum of a function on the interval [a, b].

A)
B)-
C)-
D)

Question 50

If J of work are needed to stretch a spring from 8 cm to 14 cm and another J are needed to stretch it from 14 cm to 19 cm, what is the natural length of the spring?

Question 51

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line. y = 1 - x², y = 0; the line y = 4

A) $\pi$
B) $\pi$
C) $\pi$
D) $\pi$

Question 52

The tank shown is full of water. Given that water weighs 62.5 lb/ft and R = 5, find the work (in lb-ft) required to pump the water out of the tank.

Question 53

Evaluate the integral

A)
B)
C)
D)

Question 54

Evaluate the integral

A)
B)
C)
D)

Question 55

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.

A)
B)
C)
D)
E)

Question 56

A chain weighing 3 lb/ft hangs vertically from a winch located 13 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 57

Evaluate the integral

A)
B)
C)
D)

Question 58

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x + 1, x = 0, y = 0, x = $\pi$

A) $\pi$ ² +2 $\pi$
B) $\pi$ ²+2 $\pi$
C) $\pi$ ²+4 $\pi$
D) $\pi$ ²+4 $\pi$

Question 59

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations and inequalities about the y-axis. x² - y² = 36, x $\ge$ 0, y = -6, y = 6

A) 432 $\pi$
B) 288 $\pi$
C) 864 $\pi$
D) 576 $\pi$

Question 60

Evaluate the integral.

A)14
B)10
C)-11
D)-14

Question 61

Find the volume of a pyramid with height and base an equilateral triangle with side a = .

A)
B)
C)
D)
E)

Question 62

The Mean Value Theorem for Integrals says that if is continuous on [ , ], then there exists a number m in [ , ] such that .

Question 63

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.
y = 2x², y = 4x - 2, y = 8; the y-axis

Question 64

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 65

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 66

Use to evaluate the integral.

A)22
B)
C)
D)54

Question 67

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)

Question 68

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.
y = , y = 0, x = 2, x = 5; the y-axis

Question 69

The given expression is the limit of a Riemann sum of a function f on [a, b]. Write this expression as a definite integral on [a, b].

A)
B)
C)
D)

Question 70

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method. about the y-axis

A)
B)
C)
D)
E)

Question 71

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. y = 25 - x², y = 0; the line

A) $\pi$

B) $\pi$

C) $\pi$

D) $\pi$

Question 72

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.
y = 3x², y = 0, x = 1; the y-axis

Question 73

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 74

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

A)
B)
C)
D)
E)

Question 75

The volume of the frustum of a pyramid with square base of side b = , square top of side a = , and height h = is .

Question 76

Sketch a plane region and indicate the axis about which it is revolved so that the resulting solid of revolution (found using the shell method) is given by the integral. 2 $\pi$

A) y-axis
B) y-axis
C) x-axis
D) x-axis

Question 77

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 y-axis. Round answers to two decimal places.
y = x, y = x⁵ - x², x $\ge$ 0

Question 78

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

Question 79

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

A)
B)
C)
D)
E)

Question 80

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. y = , y = -x + 5; the y-axis

A)250 $\pi$
B) $\pi$

C) $\pi$

D)250 $\pi$