Question 1

Calculate the flux of the field F across the closed plane curve C.
-F = xi + yj; the curve C is the closed counterclockwise path around the rectangle with vertices at

Accepted Answer

A)  101 
B)  20 
C)  0 
D)  99 
A)  101 
B)  20 
C)  0 
D)  99

Question 2

Apply Green's Theorem to evaluate the integral.
-

Accepted Answer

A)  256 
B)  128/3 
C)  -192 
D)  0 
A)  256 
B)  128/3 
C)  -192 
D)  0

Question 3

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-F = (   +   )i + (x - y)j; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 9), and (0, 9)

Accepted Answer

A)  0 
B)  216 
C)  -216 
D)  270 
A)  0 
B)  216 
C)  -216 
D)  270

Question 4

Apply Green's Theorem to evaluate the integral.
-  ( 4y dx + 6y dy) C: The boundary of 0$\le$ x$\le$  $\pi$, 0 $\le$ y $\le$sin x

Accepted Answer

A)  -4 
B)  2 
C)  4 
D)  0 
A)  -4 
B)  2 
C)  4 
D)  0

Question 5

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

Accepted Answer

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

Question 6

Evaluate the line integral along the curve C.
-

Accepted Answer

A)  3/2 
B)  3 
C)  $\pi$ 
D)  0 
A)  3/2 
B)  3 
C)  $\pi$ 
D)  0

Question 7

Find the surface area of the surface S.
-S is the paraboloid   +   - z = 0 below the plane z = 20.

Accepted Answer

A)  182/3   $\pi$ 
B)  243/2  $\pi$ 
C)  364/3  $\pi$ 
D)  182 $\pi$ 
A)  182/3   $\pi$ 
B)  243/2  $\pi$ 
C)  364/3  $\pi$ 
D)  182 $\pi$

Question 8

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) =  -4i +  3j +  4k , S is the rectangular surface z = 0, 0 ≤ x ≤  10, and 0 ≤ y ≤  3, direction k

Accepted Answer

A)  0 
B)  420 
C)  210 
D)  140 
A)  0 
B)  420 
C)  210 
D)  140

Question 9

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

Accepted Answer

A)  W = 0 
B)  W = 25/3 
C)  W = 209/24 
D)  W = 193/24 
A)  W = 0 
B)  W = 25/3 
C)  W = 209/24 
D)  W = 193/24

Question 10

Find the mass of the wire that lies along the curve r and has density δ.
-r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 $\le$ t $\le$2$\pi$;   = 8

Accepted Answer

A)  112$\pi$   units 
B)  16$\pi$ units 
C)  784$\pi$   units 
D)  14$\pi$   units 
A)  112$\pi$   units 
B)  16$\pi$ units 
C)  784$\pi$   units 
D)  14$\pi$   units

Question 11

Using Green's Theorem, find the outward flux of F across the closed curve C.
-F = ( 2x + 6y)i + ( 4x - 4y)j; C is the region bounded above by y = -3   + 72 and below by   in the first quadrant

Accepted Answer

A)  -288 
B)  - 2070 
C)  414 
D)  426 
A)  -288 
B)  - 2070 
C)  414 
D)  426

Question 12

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

Accepted Answer

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

Question 13

Solve the problem.
-The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: upper hemisphere of   +   +   = 16 cut by the plane z = 0
Density:   = 1

Accepted Answer

A)    = 16 $\pi$ 
B)    = 1024 /3  $\pi$ 
C)    = 32  $\pi$ 
D)    = 128  $\pi$ 
A)    = 16 $\pi$ 
B)    = 1024 /3  $\pi$ 
C)    = 32  $\pi$ 
D)    = 128  $\pi$

Question 14

Evaluate the surface integral of G over the surface S.
-S is the dome z = 3 - 10   - 10   , z $\neq$ 0; G(x, y, z) =

Accepted Answer

A)  18 $\pi$ 
B)  3 $\pi$ 
C)  3/10  $\pi$ 
D)  18 
A)  18 $\pi$ 
B)  3 $\pi$ 
C)  3/10  $\pi$ 
D)  18

Question 15

Solve the problem.
-The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: "nose" of the paraboloid   +   = 2z cut by the plane z = 2
Density:   =

Accepted Answer

A)    = 8 $\pi$ 
B)    = 16/3 $\pi$ 
C)    = 64/3 $\pi$ 
D)    = 128$\pi$ 
A)    = 8 $\pi$ 
B)    = 16/3 $\pi$ 
C)    = 64/3 $\pi$ 
D)    = 128$\pi$

Question 16

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) =   ; S is the surface of the parabolic cylinder  36 yS1U1P12S1S1P0  +  4z =  32 bounded by the planes x = 0 ,   x = 1,  y = 0, and z = 0

Accepted Answer

A)  1/9 
B)  4/9 
C)  32/81 
D)  16/9 
A)  1/9 
B)  4/9 
C)  32/81 
D)  16/9

Question 17

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

Accepted Answer

A)  0 
B)  68 
C)  60 
D)  120 
A)  0 
B)  68 
C)  60 
D)  120

Question 18

Apply Green's Theorem to evaluate the integral.
-

Accepted Answer

A)  -18 
B)  -27$\pi$ 
C)  27$\pi$ 
D)  -72 
A)  -18 
B)  -27$\pi$ 
C)  27$\pi$ 
D)  -72

Question 19

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = xyi +   j - 2yzk ; D: the solid wedge cut from the first quadrant by the plane   and the parabolic cylinder x = 16 - 9

Accepted Answer

A)  22208/405 
B)  19136/405 
C)  18112/405 
D)  36224/405 
A)  22208/405 
B)  19136/405 
C)  18112/405 
D)  36224/405

Question 20

Find the surface area of the surface S.
-S is the portion of the surface 4x + 3z = 2 that lies above the rectangle 5 $\le$ x $\le$ 7 and   in the

Accepted Answer

A)  400/3 
B)  80 
C)  80/3 
D)  720/7 
A)  400/3 
B)  80 
C)  80/3 
D)  720/7

Calculate the flux of the field F across the closed plane curve C. -F = xi + yj; the curve C is the closed counterclockwise path around the rectangle with vertices at

Apply Green's Theorem to evaluate the integral. -

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = ( + )i + (x - y)j; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 9), and (0, 9)

Apply Green's Theorem to evaluate the integral. - $Apply Green's Theorem to evaluate the integral. - ( 4y dx + 6y dy) C: The boundary of 0 \le x \le \pi , 0 \le y \le sin x$ ( 4y dx + 6y dy) C: The boundary of 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ sin x

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

Evaluate the line integral along the curve C. -

Find the surface area of the surface S. -S is the paraboloid + - z = 0 below the plane z = 20.

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = -4i + 3j + 4k , S is the rectangular surface z = 0, 0 ≤ x ≤ 10, and 0 ≤ y ≤ 3, direction k

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

Find the mass of the wire that lies along the curve r and has density δ. -r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 $\le$ t $\le$ 2 $\pi$ ; $Find the mass of the wire that lies along the curve r and has density δ. -r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 \le t \le 2 \pi ; = 8$ = 8

Using Green's Theorem, find the outward flux of F across the closed curve C. -F = ( 2x + 6y)i + ( 4x - 4y)j; C is the region bounded above by y = -3 + 72 and below by in the first quadrant

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

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: upper hemisphere of + + = 16 cut by the plane z = 0 Density: = 1

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: "nose" of the paraboloid + = 2z cut by the plane z = 2 Density: =

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = ; S is the surface of the parabolic cylinder 36 y² + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0

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

Apply Green's Theorem to evaluate the integral. -

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = xyi + j - 2yzk ; D: the solid wedge cut from the first quadrant by the plane and the parabolic cylinder x = 16 - 9

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Exam 17: Vector Calculus

Calculate the flux of the field F across the closed plane curve C. -F = xi + yj; the curve C is the closed counterclockwise path around the rectangle with vertices at

Apply Green's Theorem to evaluate the integral. -

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = ( + )i + (x - y)j; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 9), and (0, 9)

Apply Green's Theorem to evaluate the integral. - ( 4y dx + 6y dy) C: The boundary of 0 ≤\le≤ x ≤\le≤ π\piπ , 0 ≤\le≤ y ≤\le≤ sin x

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

Evaluate the line integral along the curve C. -

Find the surface area of the surface S. -S is the paraboloid + - z = 0 below the plane z = 20.

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = -4i + 3j + 4k , S is the rectangular surface z = 0, 0 ≤ x ≤ 10, and 0 ≤ y ≤ 3, direction k

Find the work done by F over the curve in the direction of increasing t. -F = xyi + 8j + 3xk; C: r(t) = cos 8ti + sin 8tj + tk, 0 ≤\le≤ t ≤\le≤

Find the mass of the wire that lies along the curve r and has density δ. -r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 ≤\le≤ t ≤\le≤ 2 π\piπ ; = 8

Using Green's Theorem, find the outward flux of F across the closed curve C. -F = ( 2x + 6y)i + ( 4x - 4y)j; C is the region bounded above by y = -3 + 72 and below by in the first quadrant

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

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: upper hemisphere of + + = 16 cut by the plane z = 0 Density: = 1

Evaluate the surface integral of G over the surface S. -S is the dome z = 3 - 10 - 10 , z ≠\neq= 0; G(x, y, z) =

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: "nose" of the paraboloid + = 2z cut by the plane z = 2 Density: =

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = ; S is the surface of the parabolic cylinder 36 y2 + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0

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

Apply Green's Theorem to evaluate the integral. -

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = xyi + j - 2yzk ; D: the solid wedge cut from the first quadrant by the plane and the parabolic cylinder x = 16 - 9

Find the surface area of the surface S. -S is the portion of the surface 4x + 3z = 2 that lies above the rectangle 5 ≤\le≤ x ≤\le≤ 7 and in the

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

Apply Green's Theorem to evaluate the integral. - $Apply Green's Theorem to evaluate the integral. - ( 4y dx + 6y dy) C: The boundary of 0 \le x \le \pi , 0 \le y \le sin x$ ( 4y dx + 6y dy) C: The boundary of 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ sin x

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

Find the mass of the wire that lies along the curve r and has density δ. -r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 $\le$ t $\le$ 2 $\pi$ ; $Find the mass of the wire that lies along the curve r and has density δ. -r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 \le t \le 2 \pi ; = 8$ = 8

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = ; S is the surface of the parabolic cylinder 36 y² + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0