Question 1

Find the work done by F over the curve in the direction of increasing t.
-F = 10zi + 2xj + 9yk; C: r(t) = ti + tj + tk, 0 $\le$ t $\le$ 1

Accepted Answer

A)  W = 42 
B)  W = 21/2 
C)  W = 7 
D)  W = 21 
A)  W = 42 
B)  W = 21/2 
C)  W = 7 
D)  W = 21

Question 2

Using Green's Theorem, find the outward flux of F across the closed curve C.
-F =(   +   )i + (x - y)j ; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 10), and

Accepted Answer

A)  120 
B)  60 
C)  330 
D)  -270 
A)  120 
B)  60 
C)  330 
D)  -270

Question 3

Calculate the area of the surface S.
-S is the portion of the paraboloid z = 3   + 3   that lies between z = 3 and z = 4.

Accepted Answer

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

Question 4

Using Green's Theorem, calculate the area of the indicated region.
-The area bounded above by y = 3   and below by y = 5

Accepted Answer

A)  27/250 
B)  27/500 
C)  27/1000 
D)  27/125 
A)  27/250 
B)  27/500 
C)  27/1000 
D)  27/125

Question 5

Find the flux of the curl of field F through the shell S.
-F = -5   yi + 5x   j +   k; S is the portion of the paraboloid 2 -   -   = z that lies above the

Accepted Answer

A)  80 $\pi$ 
B)  40/3  $\pi$ 
C)  80 
D)  10 $\pi$ 
A)  80 $\pi$ 
B)  40/3  $\pi$ 
C)  80 
D)  10 $\pi$

Question 6

Find the mass of the wire that lies along the curve r and has density δ.
-r(t) =   i + 7tj, 0 $\le$ t $\le$ 1;   = 3t

Accepted Answer

A)  169/15 units 
B)  57 units 
C)  169/10 units 
D)  3/2 units 
A)  169/15 units 
B)  57 units 
C)  169/10 units 
D)  3/2 units

Question 7

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-F = -   i; C is the region defined by the polar coordinate inequalities 1 $\le$ r $\le$4 and

Accepted Answer

A)  65/96 
B)  21/32 
C)  0 
D)  - 21/32 
A)  65/96 
B)  21/32 
C)  0 
D)  - 21/32

Question 8

Evaluate the surface integral of the function g over the surface S.
-G(x,y,z) =   xS1U1P12S1S1P0 + yS1U1P12S1S1P0   + zS1U1P12S1S1P0 ; S is the surface of the cube formed from the coordinate planes and the planes x =2 ,  y = 2 and  z = 2 .

Accepted Answer

A)  80/3 
B)  112/3 
C)  80 
D)  112 
A)  80/3 
B)  112/3 
C)  80 
D)  112

Question 9

Solve the problem.
-The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell: portion of the sphere   +   +   = 9 that lies in the first octant
Density: constant

Accepted Answer

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

Question 10

Find the flux of the vector field F across the surface S in the indicated direction.
-  S is the portion of the parabolic cylinder y =  2xS1U1P12S1S1P0  for which 0 ≤ z ≤  3 and -2 ≤ x ≤  2; direction is outward (away from the y-z plane)

Accepted Answer

A)  192 
B)  64 
C)  - 64 
D)  -192 
A)  192 
B)  64 
C)  - 64 
D)  -192

Question 11

Calculate the circulation of the field F around the closed curve C.
-F = (-x - y)i + (x + y)j , curve C is the counterclockwise path around the circle with radius  2 centered at   (2,1)

Accepted Answer

A)  16$\pi$ 
B)  8(1 + $\pi$) + 24 
C)  8(1 + $\pi$) 
D)  8$\pi$ 
A)  16$\pi$ 
B)  8(1 + $\pi$) + 24 
C)  8(1 + $\pi$) 
D)  8$\pi$

Question 12

Evaluate the line integral along the curve C.
-

Accepted Answer

A)  1/0 
B)  5/0   
C)  1/0   
D)  2/0 
A)  1/0 
B)  5/0   
C)  1/0   
D)  2/0

Question 13

Evaluate the surface integral of G over the surface S.
-S is the plane x + y + z = 2 above the rectangle 0 $\le$ x$\le$ 3 and 0 $\le$ y $\le$ 3; G(x,y,z) = 3z

Accepted Answer

A)  135 
B)  27   
C)  - 27 
D)  -27   
A)  135 
B)  27   
C)  - 27 
D)  -27

Question 14

Calculate the area of the surface S.
-S is the cap cut from the paraboloid z =   - 9   - 9   by the cone z =   .

Accepted Answer

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

Question 15

Find the work done by F over the curve in the direction of increasing t.
-F = -8yi + 8xj + 3   k; C: r(t) = cos ti + sin tj, 0 $\le$ t $\le$ 6

Accepted Answer

A)  W = 96 
B)  W = 0 
C)  W = 48 
D)  W = 144 
A)  W = 96 
B)  W = 0 
C)  W = 48 
D)  W = 144

Question 16

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = sin yi + xzj + 3zk ; D: the thick sphere 4 $\le$   +   +  $\le$ 9

Accepted Answer

A)  76$\pi$ 
B)  228$\pi$ 
C)  81 $\pi$ 
D)    $\pi$ 
A)  76$\pi$ 
B)  228$\pi$ 
C)  81 $\pi$ 
D)    $\pi$

Question 17

Using Green's Theorem, find the outward flux of F across the closed curve C.
-F = -   i ; C is the region defined by the polar coordinate inequalities   and

Accepted Answer

A)  0 
B)  37/2592 
C)  37/5184 
D)  - 37/2592 
A)  0 
B)  37/2592 
C)  37/5184 
D)  - 37/2592

Question 18

Find the surface area of the surface S.
-S is the portion of the paraboloid z = 49 -   -   that lies above the ring 1 $\le$   +   $\le$ 36 in the

Accepted Answer

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

Question 19

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = 9x   i + 10yj - 3   k; D: the solid wedge cut from the first quadrant by the plane   and the elliptic cylinder   + 49   = 196

Accepted Answer

A)  160/3 
B)  2240/3 
C)  1120/3 
D)  70/3 
A)  160/3 
B)  2240/3 
C)  1120/3 
D)  70/3

Question 20

Find the gradient field F of the function f.        
-f(x, y, z) =

Accepted Answer

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

Find the work done by F over the curve in the direction of increasing t. -F = 10zi + 2xj + 9yk; C: r(t) = ti + tj + tk, 0 $\le$ t $\le$ 1

Using Green's Theorem, find the outward flux of F across the closed curve C. -F =( + )i + (x - y)j ; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 10), and

Calculate the area of the surface S. -S is the portion of the paraboloid z = 3 + 3 that lies between z = 3 and z = 4.

Using Green's Theorem, calculate the area of the indicated region. -The area bounded above by y = 3 and below by y = 5

Find the flux of the curl of field F through the shell S. -F = -5 yi + 5x j + k; S is the portion of the paraboloid 2 - - = z that lies above the

Evaluate the surface integral of the function g over the surface S. -G(x,y,z) = x² + y² + z² ; S is the surface of the cube formed from the coordinate planes and the planes x =2 , y = 2 and z = 2 .

Solve the problem. -The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell: portion of the sphere + + = 9 that lies in the first octant Density: constant

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder y = 2x² for which 0 ≤ z ≤ 3 and -2 ≤ x ≤ 2; direction is outward (away from the y-z plane)

Calculate the circulation of the field F around the closed curve C. -F = (-x - y)i + (x + y)j , curve C is the counterclockwise path around the circle with radius 2 centered at (2,1)

Evaluate the line integral along the curve C. -

Evaluate the surface integral of G over the surface S. -S is the plane x + y + z = 2 above the rectangle 0 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 3; G(x,y,z) = 3z

Calculate the area of the surface S. -S is the cap cut from the paraboloid z = - 9 - 9 by the cone z = .

Find the work done by F over the curve in the direction of increasing t. -F = -8yi + 8xj + 3 $Find the work done by F over the curve in the direction of increasing t. -F = -8yi + 8xj + 3 k; C: r(t) = cos ti + sin tj, 0 \le t \le 6$ k; C: r(t) = cos ti + sin tj, 0 $\le$ t $\le$ 6

Using Green's Theorem, find the outward flux of F across the closed curve C. -F = - i ; C is the region defined by the polar coordinate inequalities and

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = 9x i + 10yj - 3 k; D: the solid wedge cut from the first quadrant by the plane and the elliptic cylinder + 49 = 196

Find the gradient field F of the function f. -f(x, y, z) =

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of 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

Filters

Exam 17: Vector Calculus

Find the work done by F over the curve in the direction of increasing t. -F = 10zi + 2xj + 9yk; C: r(t) = ti + tj + tk, 0 ≤\le≤ t ≤\le≤ 1