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 =   sin   , s(   ) = 2

Accepted Answer

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

Question 2

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method.
-The region bounded by y = 6x -   and y = x about the y-axis

Accepted Answer

A)  625/4 $\pi$ 
B)  625/12$\pi$ 
C)  625/8 $\pi$ 
D)  625/6$\pi$ 
A)  625/4 $\pi$ 
B)  625/12$\pi$ 
C)  625/8 $\pi$ 
D)  625/6$\pi$

Question 3

Find the volume of the solid generated by revolving the region about the given line.
-The region bounded above by the line   , below by the curve   , and on the right by the line   , about the line

Accepted Answer

A)  64/ 3 $\pi$ 
B)  8192 /15 $\pi$ 
C)  1024/5$\pi$ 
D)  7168/15$\pi$ 
A)  64/ 3 $\pi$ 
B)  8192 /15 $\pi$ 
C)  1024/5$\pi$ 
D)  7168/15$\pi$

Question 4

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the volume of paint needed. Assume that x and y are measured in meters.
-The spherical zone generated when the upper portion of the circle   +   = 81 on the interval   is revolved about the x-axis.

Accepted Answer

A)  27/100 $\pi$   
B)  27/200   
C)  27/200  $\pi$   
D)  270  $\pi$   
A)  27/100 $\pi$   
B)  27/200   
C)  27/200  $\pi$   
D)  270  $\pi$

Question 5

Determine the area of the surface generated when the curve is revolved about the indicated axis. 
-x =   -   , for 1 $\le$ y $\le$ 7; about the y-axis

Accepted Answer

A)  130,943 /48 
B)  130,943/24 
C)  130,943/24 $\pi$ 
D)  130,943/48 $\pi$ 
A)  130,943 /48 
B)  130,943/24 
C)  130,943/24 $\pi$ 
D)  130,943/48 $\pi$

Question 6

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis.
-y = 4   , y = 4x, for x $\ge$ 0

Accepted Answer

A)  32/21$\pi$ 
B)  64/3 $\pi$ 
C)  16/15 $\pi$ 
D)  64/21 $\pi$ 
A)  32/21$\pi$ 
B)  64/3 $\pi$ 
C)  16/15 $\pi$ 
D)  64/21 $\pi$

Question 7

Find the area of the shaded region.          
-

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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)

Accepted Answer

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

Question 10

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.

Accepted Answer

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

Question 11

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 =

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Solve the problem.   
-The following figure shows the shape and dimensions of a small dam. Assuming the water level is at the top of the dam, find the total force on the face of the dam. Round to the nearest whole number.

Accepted Answer

A)  40,833,333 N 
B)  20,416,667 N 
C)  81,666,667 N 
D)  163,333,333 N 
A)  40,833,333 N 
B)  20,416,667 N 
C)  81,666,667 N 
D)  163,333,333 N

Question 14

Find the volume of the solid generated by revolving the region about the given line.
-The region in the second quadrant bounded above by the curve y = 4 -   , below by the x-axis, and on the right by the y-axis, about the line x = 1

Accepted Answer

A)  56/3 $\pi$ 
B)  32/3 $\pi$ 
C)  256 /15 $\pi$ 
D)  8$\pi$ 
A)  56/3 $\pi$ 
B)  32/3 $\pi$ 
C)  256 /15 $\pi$ 
D)  8$\pi$

Question 15

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

Accepted Answer

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

Question 16

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis.
-x = 18 -   , x =   , y = 0

Accepted Answer

A)  81 $\pi$ 
B)  81 /4$\pi$ 
C)  162 $\pi$ 
D)  81/2 $\pi$ 
A)  81 $\pi$ 
B)  81 /4$\pi$ 
C)  162 $\pi$ 
D)  81/2 $\pi$

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Set up an integral for the length of the curve.
-y =   ,  $\le$ x $\le$

Accepted Answer

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

Question 20

Solve the problem.   
-A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 45 N. How much work is done in compressing the spring 2.5 m from its equilibrium position?

Accepted Answer

A)  112.5 J 
B)  56.25 J 
C)  562.5 J 
D)  281.25 J 
A)  112.5 J 
B)  56.25 J 
C)  562.5 J 
D)  281.25 J

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

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by y = 6x - and y = x about the y-axis

Find the volume of the solid generated by revolving the region about the given line. -The region bounded above by the line , below by the curve , and on the right by the line , about the line

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the volume of paint needed. Assume that x and y are measured in meters. -The spherical zone generated when the upper portion of the circle + = 81 on the interval is revolved about the x-axis.

Find the area of the shaded region. -

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

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)

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.

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 =

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

Solve the problem. -The following figure shows the shape and dimensions of a small dam. Assuming the water level is at the top of the dam, find the total force on the face of the dam. Round to the nearest whole number.

Find the volume of the solid generated by revolving the region about the given line. -The region in the second quadrant bounded above by the curve y = 4 - , below by the x-axis, and on the right by the y-axis, about the line x = 1

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

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. -x = 18 - , x = , y = 0

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

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

Set up an integral for the length of the curve. -y = $Set up an integral for the length of the curve. -y = , \le x \le$ , $Set up an integral for the length of the curve. -y = , \le x \le$ $\le$ x $\le$ $Set up an integral for the length of the curve. -y = , \le x \le$

Solve the problem. -A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 45 N. How much work is done in compressing the spring 2.5 m from its equilibrium position?

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Exam 6: Applications of Integration