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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 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: = A) = 8 \pi B) = 16/3 \pi C) = 64/3 \pi D) = 128 \pi + 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: = A) = 8 \pi B) = 16/3 \pi C) = 64/3 \pi D) = 128 \pi = 2z cut by the plane z = 2 Density: 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: = A) = 8 \pi B) = 16/3 \pi C) = 64/3 \pi D) = 128 \pi = 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: = A) = 8 \pi B) = 16/3 \pi C) = 64/3 \pi D) = 128 \pi


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

E) A) and D)
F) A) and C)

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


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

E) B) and D)
F) B) and C)

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Evaluate the surface integral of G over the surface S. -S is the dome z = 3 - 10 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) = A) 18 \pi B) 3 \pi C) 3/10 \pi D) 18 - 10 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) = A) 18 \pi B) 3 \pi C) 3/10 \pi D) 18 , z ≠\neq 0; G(x, y, z) = 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) = A) 18 \pi B) 3 \pi C) 3/10 \pi D) 18


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

E) All of the above
F) C) and D)

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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 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 A) -288 B) - 2070 C) 414 D) 426 + 72 and below by 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 A) -288 B) - 2070 C) 414 D) 426 in the first quadrant


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

E) A) and B)
F) A) and C)

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


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

E) A) and C)
F) B) and C)

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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 A) -4 B) 2 C) 4 D) 0 ( 4y dx + 6y dy) C: The boundary of 0 ≤\le x ≤\le π\pi , 0 ≤\le y ≤\le sin x


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

E) A) and D)
F) All of the above

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Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = 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 y<sup>2</sup> + 4z = 32 bounded by the planes x = 0 , x = 1, y = 0, and z = 0 A) 1/9 B) 4/9 C) 32/81 D) 16/9 ; 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


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

E) A) and D)
F) A) and C)

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Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = ( 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) A) 0 B) 216 C) -216 D) 270 + 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) A) 0 B) 216 C) -216 D) 270 ) i + (x - y) j; C is the rectangle with vertices at (0, 0) , ( 3, 0) , ( 3, 9) , and (0, 9)


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

E) A) and C)
F) All of the above

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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\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 A) 112 \pi units B) 16 \pi units C) 784 \pi units D) 14 \pi units = 8


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

E) All of the above
F) A) and D)

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Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = xyi + 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 A) 22208/405 B) 19136/405 C) 18112/405 D) 36224/405 j - 2yzk ; D: the solid wedge cut from the first quadrant by the plane 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 A) 22208/405 B) 19136/405 C) 18112/405 D) 36224/405 and the parabolic cylinder x = 16 - 9 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 A) 22208/405 B) 19136/405 C) 18112/405 D) 36224/405


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

E) A) and B)
F) None of the above

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Apply Green's Theorem to evaluate the integral. -Apply Green's Theorem to evaluate the integral. - A) 256 B) 128/3 C) -192 D) 0


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

E) A) and D)
F) B) and C)

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


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

E) A) and B)
F) B) and C)

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Evaluate the line integral along the curve C. - Evaluate the line integral along the curve C. - A) 3/2 B) 3 C) \pi D) 0


A) 3/2
B) 3
C) π\pi
D) 0

E) A) and D)
F) B) and C)

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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 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 A) = 16 \pi B) = 1024 /3 \pi C) = 32 \pi D) = 128 \pi + 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 A) = 16 \pi B) = 1024 /3 \pi C) = 32 \pi D) = 128 \pi + 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 A) = 16 \pi B) = 1024 /3 \pi C) = 32 \pi D) = 128 \pi = 16 cut by the plane z = 0 Density: 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 A) = 16 \pi B) = 1024 /3 \pi C) = 32 \pi D) = 128 \pi = 1


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

E) A) and D)
F) None of the above

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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 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

E) B) and C)
F) A) and B)

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


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

E) A) and B)
F) None of the above

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Using Green's Theorem, find the outward flux of F across the closed curve C. -F = - 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 A) 0 B) 68 C) 60 D) 120 i + 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 A) 0 B) 68 C) 60 D) 120 j ; C is the region defined by the polar coordinate inequalities 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 A) 0 B) 68 C) 60 D) 120 and 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 A) 0 B) 68 C) 60 D) 120


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

E) B) and D)
F) A) and D)

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Apply Green's Theorem to evaluate the integral. - Apply Green's Theorem to evaluate the integral. - A) -18 B) -27 \pi C) 27 \pi D) -72


A) -18
B) -27 π\pi
C) 27 π\pi
D) -72

E) None of the above
F) A) and B)

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Find the surface area of the surface S. -S is the paraboloid Find the surface area of the surface S. -S is the paraboloid + - z = 0 below the plane z = 20. A) 182/3 \pi B) 243/2 \pi C) 364/3 \pi D) 182 \pi + Find the surface area of the surface S. -S is the paraboloid + - z = 0 below the plane z = 20. A) 182/3 \pi B) 243/2 \pi C) 364/3 \pi D) 182 \pi - z = 0 below the plane z = 20.


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

E) A) and B)
F) A) and C)

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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


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

E) B) and C)
F) A) and B)

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