Deck 6: Applications of Integration

Question 1

Solve the problem.

-Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = cos t, s(0) = 1

A) s = 2 $\pi$ sin t
B) s = sin t
C) s = sin t
D) s = sin t + $\pi$

Answer 1

s = sin t

Question 2

Solve the problem.

-Given the velocity and initial position of a body moving along a coordinate line at time t, find the body's position at time t. v = sin , s( ) = 2

A) s = -2 cos + 8
B) s = -2 cos + 3
C) s = -2 cos + 4
D) s = 2 cos + 4

Answer 2

s = -2 cos + 4

Question 3

Solve the problem.

-Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 16, v(0) = 7, s(0) = 11

A) s = 8 + 7t
B) s = -8 - 7t + 11
C) s = 8 + 7t + 11
D) s = 16 + 7t + 11

Answer 3

s = 8 + 7t + 11

Question 4

Solve the problem.

-Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t. a = 20 cos 5t, v(0) = 8, s(0) = 12

A) s = - sin 5t + 8t +
B) s = cos 5t - 8t +
C) s = - cos 5t + 8t +
D) s = sin 5t + 8t +

Question 5

Solve the problem.

-A certain company has found that its expenditure rate per day (in hundreds of dollars) on a certain type of job is given by where x is the number of days since the start of the job. Find the expenditure if the job takes 5 days.

A) $ 170
B) $ 59
C) $ 17,000
D) $ 5900

Question 6

Solve the problem.

-After a new firm starts in business, it finds that its rate of profits (in hundreds of dollars per year) after t years of operation is given by Find the profit in year 8 of the operation.

A) $ 20,300
B) $ 64,000
C) $ 57,050
D) $ 46,900

Question 7

Solve the problem.

-In a certain memory experiment, subject A is able to memorize words at a rate given by In the same memory experiment, subject B is able to memorize at the rate given by How many more words does subject B memorize from t = 0 to t = 14 (during the first 14 minutes)?

A) 31
B) -8
C) 36
D) -3

Question 8

Provide an appropriate response.

-Which of the following integrals, if any, calculates the area of the shaded region?

A)
dx
B)
dx
C)
dx
D)
dx

Question 9

Find the area of the shaded region.

-

A) 937/12
B) 343/12
C) 81/12
D) 768/12

Question 10

Find the area of the shaded region.

-

A) 1153/12
B) - 343/12
C) 937/12
D) 343/12

Question 11

Find the area of the shaded region.

-

A) 9/2
B) 41/6
C) 3
D) 25/6

Question 12

Find the area of the shaded region.

-

A) 7/15
B) 76/15
C) 22/15
D) 2

Question 13

Find the area of the shaded region.

-

A) 9/2
B) 8/3
C) 19/3
D) 11/6

Question 14

Find the area of the shaded region.

-

A) 128/3
B) 32
C) 64/3
D) 32/3

Question 15

Find the area of the shaded region.

-

A) 256/5
B) 516/5
C) 2816/5
D) 512/5

Question 16

Find the area of the shaded region.

-

A) 4 +
B) 4
C) 8
D)

Question 17

Find the area of the shaded region.

-

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

Question 18

Find the area of the shaded region.

-

A)
B) 2 -
C) 1 +
D) 1 -

Question 19

Find the area enclosed by the given curves.

-y = 2x - x², y = 2x - 4

A) 37/3
B) 32/3
C) 34/3
D) 31/3

Question 20

Find the area enclosed by the given curves.

-y = x³, y = 4x

A) 4
B) 8
C) 16
D) 2

Question 21

Find the area enclosed by the given curves.

-y = x, y = x²

A) 1/12
B) 1/3
C) 1/6
D) 1/2

Question 22

Find the area enclosed by the given curves.

-y = x², y = -x² + 6

A) 8
B) 16
C) 32
D) 4

Question 23

Find the area enclosed by the given curves.

-y = - 4sin x, y = sin 2x, 0 $\le$ x $\le$ $\pi$

A) 1/2
B) 8
C) 16
D) 4

Question 24

Find the area enclosed by the given curves.

-y = sin x, y = x, $\le$ x $\le$

A) 1 -
B) -
C) -
D) +

Question 25

Find the area enclosed by the given curves.

-y = x, y = x, x = , and x =

A) $\pi$ /2
B) $\pi$
C) 3 $\pi$ /4
D) $\pi$ /4

Question 26

Find the area enclosed by the given curves.

-Find the area of the region in the first quadrant bounded by the line y = 8x, the line x = 1, the curve and the x-axis.

A) 3/4
B) 3/2
C) 6
D) 5/4

Question 27

Find the area enclosed by the given curves.

-Find the area of the region in the first quadrant bounded on the left by the line y = and on the right by the curves y = x and y = x. (Round to four decimal places.)

A) 0.3094
B) 0.5858
C) 4.3094
D) 0.4126

Question 28

Find the area enclosed by the given curves.

-Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the line above left by y = x + 4, and above right by y = - + 10.

A) 39/2
B) 73/6
C) 39/4
D) 15

Question 29

Find the area enclosed by the given curves.

-Find the area between the curves y = ln x and y = ln 2x from x = 1 to x = 5.

A) ln 32
B) ln 16 - 8
C) ln 16
D) ln 2

Question 30

Find the area enclosed by the given curves.

-Find the area of the "triangular" region in the first quadrant that is bounded above by the curve , below by the curve y = , and on the right by the line x = ln 2.

A) 1/2
B) 0
C) 11/2
D) 2 ln 2

Question 31

Find the area enclosed by the given curves.

-Find the area of the region between the curve y = 2x/(1 + ) and the interval of the x-axis.

A) 2 ln 17
B) 0
C) 2
D) ln 17

Question 32

Find the area enclosed by the given curves.

-Find the area of the region between the curve y = and the interval 0 $\le$ x $\le$ 2 on the x-axis.

A) 24/ ln 5
B) 25
C) 25 / In 5
D) 24 In 5

Question 33

Find the volume of the described solid.

-The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the parabola to the parabola .

A) 245
B) 2450
C) 1225
D) 2400

Question 34

Find the volume of the described solid.

-The solid lies between planes perpendicular to the x-axis at and . The cross sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle to the semicircle .

A) 72
B) 18
C) 144
D) 36

Question 35

Find the volume of the described solid.

-The base of the solid is the disk + $\le$ 9. The cross sections by planes perpendicular to the between and are isosceles right triangles with one leg in the disk.

A) 144
B) 90
C) 18
D) 72

Question 36

Find the volume of the described solid.

-The solid lies between planes perpendicular to the x-axis at x = - 3 and x = 3. The cross sections perpendicular to the x-axis are semicircles whose diameters run from to

A) 72 $\pi$
B) 9 $\pi$
C) 36 $\pi$
D) 18 $\pi$

Question 37

Find the volume of the described solid.

-The solid lies between planes perpendicular to the x-axis at x = - 2 and x = 2. The cross sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = to the parabola y = 8 - .

A) 32/3 $\pi$
B) 512/15 $\pi$
C) 256/15 $\pi$
D) 256/5 $\pi$

Question 38

Find the volume of the described solid.

-The base of a solid is the region between the curve y = 4cos x and the x-axis from x = 0 to . The cross sections perpendicular to the x-axis are squares with bases running from the x-axis to the curve.

A) 8 $\pi$
B) 2 $\pi$
C) 4 $\pi$
D) 15/4 $\pi$

Question 39

Find the volume of the described solid.

-The solid lies between planes perpendicular to the x-axis at x = $\pi$ /6 to x = $\pi$ /2. The cross sections perpendicular to the x-axis are circular disks with diameters running from the curve to the curve

A) -
B) -
C) ( - 1) $\pi$ +
D) (2 - 2) $\pi$ -

Question 40

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the x-axis

A) 18 $\pi$
B) 48 $\pi$
C) 168 $\pi$
D) 24 $\pi$

Question 41

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the x-axis

A) 648/5 $\pi$
B) 3159/5 $\pi$
C) 1053/5 $\pi$
D)18 $\pi$

Question 42

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the x-axis

A) 18 $\pi$
B) 21 $\pi$
C) 6 $\pi$
D) 36 $\pi$

Question 43

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the y-axis

A) 50/3 $\pi$
B) 20/3 $\pi$
C) 5 $\pi$
D) 40/3 $\pi$

Question 44

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the y-axis

A) $\pi$
B) 625 $\pi$
C) 50 $\pi$
D) 25 $\pi$

Question 45

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the y-axis

A) 5 $\pi$ + 5
B) 10 $\pi$ -
C) 20 $\pi$ - 5
D) 20 - 5 $\pi$

Question 46

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the y-axis

A) 12 $\pi$
B) 3 - 6 $\pi$
C) 6 + 3 $\pi$
D) 6 - 12 $\pi$

Question 47

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the y-axis

A) 16/3 $\pi$
B) 20/3 $\pi$
C) 32/5 $\pi$
D) 8/5 $\pi$

Question 48

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the x-axis

A) 567/5 $\pi$
B) 162/5 $\pi$
C) 648/ 5 $\pi$
D) 9 $\pi$

Question 49

Find the volume of the solid generated by revolving the shaded region about the given axis.

-About the x-axis

A) 8
B) 8 - 4 $\pi$
C) 8 + 16 $\pi$
D) 8 - 16 $\pi$

Question 50

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = x, y = 0, x = 3, x = 5

A) 98/3 $\pi$
B) 17 $\pi$
C) 8 $\pi$
D) 2/3 $\pi$

Question 51

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, x = 0, x = 9

A) 9 $\pi$
B) 81/2 $\pi$
C) 27 $\pi$
D)9/2 $\pi$

Question 52

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, x = 0, x = 4

A) 1024/5 $\pi$
B) 256 $\pi$
C) 64 $\pi$
D) 64/3 $\pi$

Question 53

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, x = 0, x = 1

A) 4 $\pi$
B) 3 $\pi$ /2
C) $\pi$
D) 2 $\pi$

Question 54

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, x = 1, x = 5

A) 2/5 $\pi$
B) $\pi$ ln 5
C) 3/5 $\pi$
D) 4/5 $\pi$

Question 55

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = x + 3, y = 0, x = -3, x = 7

A) 100 $\pi$
B) 91/2 $\pi$
C) 5 $\pi$
D) 1000/3 $\pi$

Question 56

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, x = 0, x = 8

A) 256 $\pi$
B) 16 $\pi$
C) 2048 /3 $\pi$
D) 1024 /3 $\pi$

Question 57

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, 0 $\le$ x $\le$

A) 8 $\pi$
B) 2 $\pi$
C) 4 $\pi$
D) 1/2 $\pi$

Question 58

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = 3csc x, y = 0, x = , x =

A) 9 $\pi$
B) 6 $\pi$
C) 27 $\pi$
D) 18 $\pi$

Question 59

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = 5cos $\pi$ x, y = 0, x = -0.5, x = 0.5

A) 25 $\pi$
B) 25/2 $\pi$
C) 50 $\pi$
D) 25/3 $\pi$

Question 60

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 0, x = 1, x = 7

A)
B) (ln 7)
C) $\pi$
D) $\pi$ (ln 7)

Question 61

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = 8x, y = 8, x = 0

A) 128 /3 $\pi$
B) 64/3 $\pi$
C) 4 $\pi$
D) 24 $\pi$

Question 62

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = - 3x + 6, y = 3x, x = 0

A) 54 $\pi$
B) 6 $\pi$
C) 18 $\pi$
D) 9 $\pi$

Question 63

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 7, x = 0

A) 343/4 $\pi$
B) 343/3 $\pi$
C) 343/2 $\pi$
D) 98 $\pi$

Question 64

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 36, x = 0

A) 144 $\pi$
B) 46656/5 $\pi$
C) 7776/5 $\pi$
D) 31104/5 $\pi$

Question 65

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = + 4, y = 2x + 4

A) 16/5 $\pi$
B) 8 $\pi$
C) 416/15 $\pi$
D) 224/15 $\pi$

Question 66

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = - x + 5

A) 12 $\pi$
B) 9 $\pi$
C) 27/4 $\pi$
D) 125/3 $\pi$

Question 67

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = , y = 1, x = 0 to x =

A) $\pi$ /8 -1/4
B) - $\pi$
C) + $\pi$
D) - $\pi$ /4

Question 68

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = 7csc x, y = 7 , $\le$ x $\le$

A) 49 - 98 $\pi$
B) + 14 $\pi$
C) 7 - 49 $\pi$
D) 49 + 98 $\pi$

Question 69

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = 3cos ( $\pi$ x), y = 3, x = -0.5, x = 0.5

A) 18 $\pi$
B) 9 $\pi$
C) 3 $\pi$
D) 9/2 $\pi$

Question 70

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the

-y = sec x, y = tan x, x = 0, x =

A)
B)
C)
D)

Question 71

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by x = , x = 0, y = - 2, y = 2

A) 64/5 $\pi$
B) 16/5 $\pi$
C) 8/5 $\pi$
D) 8/3 $\pi$

Question 72

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by x = , x = 0, y = 27

A) 729/5 $\pi$
B) 243/4 $\pi$
C) 243 $\pi$
D) 81 $\pi$

Question 73

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by x = , x = 0, y = 1, y = 5

A) 96/5 $\pi$
B) 64 /25 $\pi$
C) 64/5 $\pi$
D) 16/5 $\pi$

Question 74

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by x = 2tan , x = 0, y = -

A) 7 $\pi$ + 7
B) 28 $\pi$ - 7
C) 14 $\pi$ -
D) 28 - 7 $\pi$

Question 75

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by x = , 0 $\le$ y $\le$ , x = 0

A) 5 $\pi$
B) 10 $\pi$
C) $\pi$ /5
D) $\pi$ /10

Question 76

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by the triangle with vertices ( 1, 0), ( 1, 2), ( 3, 2)

A) 26/3 $\pi$
B) 20/3 $\pi$
C) 10/3 $\pi$
D) 4/3 $\pi$

Question 77

Find the volume of the solid generated by revolving the region about the y-axis.

-The region enclosed by the triangle with vertices (0, 0), ( 5, 0), ( 5, 2)

A) 100 $\pi$
B) 100/3 $\pi$
C) 25/3 $\pi$
D) 50/3 $\pi$

Question 78

Find the volume of the solid generated by revolving the region about the y-axis.

-The region in the first quadrant bounded on the left by the circle + = 4, on the right by the line , and above by the line y = 2

A) 16/3 $\pi$
B) 2 $\pi$
C) 8 $\pi$
D) 8/3 $\pi$

Question 79

Find the volume of the solid generated by revolving the region about the y-axis.

-The region in the first quadrant bounded on the left by y = , on the right by the line , and below by the x-axis

A) 4 $\pi$
B) 32/5 $\pi$
C) 64/5 $\pi$
D) 96/5 $\pi$

Question 80

Find the volume of the solid generated by revolving the region about the y-axis.

-The region in the first quadrant bounded on the left by y = , on the right by the line , and above by the line y = 2

A) 25 $\pi$
B) 25/3 $\pi$
C) 25 /2 $\pi$
D) 75 /2 $\pi$