Question 1

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.

Accepted Answer

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

Question 2

Find the area of the shaded region.          
-

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

Solve the problem.
-A frustum of a right circular cone has a height of 10 m, a base of radius 6m, and a top of radius 5m. Find its volume.

Accepted Answer

A)  91$\pi$ 
B)  910 /3 $\pi$ 
C)  91/3 $\pi$ 
D)  910$\pi$ 
A)  91$\pi$ 
B)  910 /3 $\pi$ 
C)  91/3 $\pi$ 
D)  910$\pi$

Question 5

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

Accepted Answer

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

Question 6

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 $\le$

Accepted Answer

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

Question 7

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 
-About the y-

Accepted Answer

A)  12 $\pi$ 
B)  5 $\pi$ 
C)  8 $\pi$ 
D)  10 $\pi$ 
A)  12 $\pi$ 
B)  5 $\pi$ 
C)  8 $\pi$ 
D)  10 $\pi$

Question 8

Solve the problem.   
-A diving pool that is 5 m deep and full of water has a viewing window on one of its vertical walls. Find the force on a square window, 2 m on a side, with the lower edge of the window on the bottom of the pool.

Accepted Answer

A)  49,000 N 
B)  39,200 N 
C)  156,800 N 
D)  25,600 N 
A)  49,000 N 
B)  39,200 N 
C)  156,800 N 
D)  25,600 N

Question 9

Solve the problem.
-An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve   - 4 $\le$ x $\le$4, about the x-axis (dimensions are in feet). If a cubic foot holds 7.481 gallons and the helicopter gets 3 miles to the gallon, how many additional miles will the helicopter be able to fly once the tank is installed (to the nearest mile)?

Accepted Answer

A)  301 
B)  75 
C)  100 
D)  150 
A)  301 
B)  75 
C)  100 
D)  150

Question 10

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

Accepted Answer

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

Question 11

Solve the problem.
-Find a curve through the point   whose length integral, 1 $\le$ x $\le$ 2, is

Accepted Answer

A)  y =     
B)  y = 5   
C)  y =     
D)  y = 5   
A)  y =     
B)  y = 5   
C)  y =     
D)  y = 5

Question 12

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 = 6   , y = 0, x = 0, y = 1

Accepted Answer

A)  6   $\pi$ 
B)  12   $\pi$ 
C)  6(e - 1)$\pi$ 
D)  12   $\pi$ 
A)  6   $\pi$ 
B)  12   $\pi$ 
C)  6(e - 1)$\pi$ 
D)  12   $\pi$

Question 13

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

Accepted Answer

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

Question 14

Let R be the region bounded by the given curves. Find the volume of the solid generated when R is revolved about the given line.
-x = 0, y =   , and y = 2; about y = 2

Accepted Answer

A)  8$\pi$/3 
B)  9$\pi$/4 
C)  11$\pi$/4 
D)  10$\pi$/3 
A)  8$\pi$/3 
B)  9$\pi$/4 
C)  11$\pi$/4 
D)  10$\pi$/3

Question 15

Find the length of the curve.
-x =   +   from y = 1 to y = 2

Accepted Answer

A)  33/16 
B)  2 
C)  33/8 
D)  17/8 
A)  33/16 
B)  2 
C)  33/8 
D)  17/8

Question 16

Find the area enclosed by the given curves.
-y =   x2, y = -x2 + 6

Accepted Answer

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

Question 17

Find the area of the shaded region.          
-

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

A)  4$\pi$(e - 1) 
B)  $\pi$(e - 1) 
C)  2$\pi$e 
D)    (e - 1) 
A)  4$\pi$(e - 1) 
B)  $\pi$(e - 1) 
C)  2$\pi$e 
D)    (e - 1)

Question 20

Solve the problem.   
-Hooke's law is applicable to idealized (linear) springs that are not stretched or compressed too far. Consider a nonlinear spring whose restoring force is given by F(x) = 18x - 0.3   , for |x| $\le$ 9. How much work is done in stretching the spring from its equilibrium position (x = 0) to x = 1? Round to two decimal places when appropriate.

Accepted Answer

A)  473.85 J 
B)  8.93 J 
C)  236.93 J 
D)  17.7 J 
A)  473.85 J 
B)  8.93 J 
C)  236.93 J 
D)  17.7 J

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.

Find the area of the shaded region. -

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

Solve the problem. -A frustum of a right circular cone has a height of 10 m, a base of radius 6m, and a top of radius 5m. Find its volume.

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

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis

Solve the problem. -A diving pool that is 5 m deep and full of water has a viewing window on one of its vertical walls. Find the force on a square window, 2 m on a side, with the lower edge of the window on the bottom of the pool.

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

Solve the problem. -Find a curve through the point $Solve the problem. -Find a curve through the point whose length integral, 1 \le x \le 2, is$ whose length integral, 1 $\le$ x $\le$ 2, is $Solve the problem. -Find a curve through the point whose length integral, 1 \le x \le 2, is$

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 = 6 , y = 0, x = 0, y = 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 = 0, x = 9

Let R be the region bounded by the given curves. Find the volume of the solid generated when R is revolved about the given line. -x = 0, y = , and y = 2; about y = 2

Find the length of the curve. -x = + from y = 1 to y = 2

Find the area enclosed by the given curves. -y = x², y = -x² + 6

Find the area of the shaded region. -

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

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

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters

Exam 6: Applications of Integration

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.

Find the area of the shaded region. -

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

Solve the problem. -A frustum of a right circular cone has a height of 10 m, a base of radius 6m, and a top of radius 5m. Find its volume.

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

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≤

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the y-axis

Solve the problem. -A diving pool that is 5 m deep and full of water has a viewing window on one of its vertical walls. Find the force on a square window, 2 m on a side, with the lower edge of the window on the bottom of the pool.

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

Solve the problem. -Find a curve through the point whose length integral, 1 ≤\le≤ x ≤\le≤ 2, is

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 = 6 , y = 0, x = 0, y = 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 = 0, x = 9

Let R be the region bounded by the given curves. Find the volume of the solid generated when R is revolved about the given line. -x = 0, y = , and y = 2; about y = 2

Find the length of the curve. -x = + from y = 1 to y = 2

Find the area enclosed by the given curves. -y = x2, y = -x2 + 6

Find the area of the shaded region. -

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

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

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Vector Calculus

Filters

Solve the problem. -Find a curve through the point $Solve the problem. -Find a curve through the point whose length integral, 1 \le x \le 2, is$ whose length integral, 1 $\le$ x $\le$ 2, is $Solve the problem. -Find a curve through the point whose length integral, 1 \le x \le 2, is$

Find the area enclosed by the given curves. -y = x², y = -x² + 6