Question 1

Solve the problem.
-The radial flow field of an incompressible fluid is shown below. Which of the closed paths would exhibit a non-zero flux?

Accepted Answer

A)  A, B, and C 
B)  C 
C)  A and B 
D)  none of these 
A)  A, B, and C 
B)  C 
C)  A and B 
D)  none of these

Question 2

Using Green's Theorem, calculate the area of the indicated region.
-The circle $\mathbf { r } ( \mathrm { t } ) = ( 11 \cos \mathrm { t } ) \mathbf { i } + ( 11 \sin \mathrm { t } ) \mathbf { j } , 0 \leq \mathrm { t } \leq 2 \pi$

Accepted Answer

A)  $121 \pi$ 
B)  $2 \pi$ 
C)  $11 \pi$ 
D)  $22 \pi$ 
A)  $121 \pi$ 
B)  $2 \pi$ 
C)  $11 \pi$ 
D)  $22 \pi$

Question 3

Find the equation for the plane tangent to the parametrized surface S at the point P.
-$$\mathrm { S } \text { is the cone } \mathbf { r } ( \mathrm { r } , \theta ) = \mathrm { r } \cos \theta \mathbf { i } + \mathrm { r } \sin \theta \mathbf { j } + 4 \mathrm { rk } ; \mathrm { P } = ( - 5 \sqrt { 2 } , 5 \sqrt { 2 } , 40 ) \text {. }$$

Accepted Answer

The answer of Find the equation for the plane tangent...

Question 4

Evaluate the surface integral of the function g over the surface S.
-$G ( x , y , z ) = x ^ { 2 } y ^ { 2 } z ^ { 2 } ; S$ is the surface of the rectangular prism formed from the coordinate planes and the planes $x$ $= 1 , y = 1$, and $z = 3$

Accepted Answer

A)  1 
B)  7 
C)  3 
D)  21 
A)  1 
B)  7 
C)  3 
D)  21

Question 5

Test the vector field F to determine if it is conservative.
-F = xyi + yj + zk

Accepted Answer

A)  Not conservative 
B)  Conservative 
A)  Not conservative 
B)  Conservative

Question 6

Find the flux of the vector field F across the surface S in the indicated direction.
-$\mathbf { F } = \mathrm { x } ^ { 5 } \mathrm { yi } - \mathrm { zk }$; $\mathrm { S }$ is portion of the cone $\mathrm { z } = 4 \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ between $\mathrm { z } = 0$ and $\mathrm { z } = 3$; direction is outward

Accepted Answer

A)  $- \frac { 27 } { 8 } \pi$ 
B)  $- \frac { 9 } { 16 }$ 
C)  $\frac { 9 } { 8 } \pi$ 
D)  $- \frac { 9 } { 8 } \pi$ 
A)  $- \frac { 27 } { 8 } \pi$ 
B)  $- \frac { 9 } { 16 }$ 
C)  $\frac { 9 } { 8 } \pi$ 
D)  $- \frac { 9 } { 8 } \pi$

Question 7

Solve the problem.
-The velocity field $F$ of a fluid is the spin field $F = - \frac { k y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { i } + \frac { k x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { j }$. Following the smooth curve $y = f ( x )$ from $( a , f ( a ) )$ to $( b , f ( b ) )$, show that the flux across the curve is $\int _ { C } F \cdot n d s = k \left[ \left( a ^ { 2 } + ( f ( a ) ) ^ { 2 } \right) ^ { 1 / 2 } - \left( b ^ { 2 } + ( f ( b ) ) ^ { 2 } \right) ^ { 1 / 2 } \right]$

Accepted Answer

To solve the problem, we can start by pa

Question 8

Find the limit.
-$$r ( t ) = \frac { 3 } { 2 } t i + ( 2 - t ) k ; 0 \leq t \leq 2$$

Accepted Answer

A)  Figure 5 
B)  Figure 8 
C)  Figure 1 
D)  Figure 3 
A)  Figure 5 
B)  Figure 8 
C)  Figure 1 
D)  Figure 3

Question 9

Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( \pi / 4 , \pi / 20 , \pi / 16 ) } 4 \sin 4 x d x + 5 \sec ^ { 2 } 5 y d y - 8 \cos 8 z d z$$

Accepted Answer

A)  1 
B)  0 
C)  4 
D)  2 
A)  1 
B)  0 
C)  4 
D)  2

Question 10

Find the work done by F over the curve in the direction of increasing t.
-$$\mathbf { F } = \mathrm { zi } + 3 \mathrm { x } \mathbf { j } + 6 y \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \mathrm { ti } + \mathrm { tj } + \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq 1$$

Accepted Answer

A)  $W = 5$ 
B)  $W = \frac { 15 } { 2 }$ 
C)  $W = 15$ 
D)  $W = 30$ 
A)  $W = 5$ 
B)  $W = \frac { 15 } { 2 }$ 
C)  $W = 15$ 
D)  $W = 30$

Question 11

Evaluate. The differential is exact.
-$$\int _ { ( 1,1,1 ) } ^ { ( 5,3,4 ) } \frac { 1 } { x } d x + \frac { 1 } { y } d y + \frac { 1 } { z } d z$$

Accepted Answer

A)  ln 60 
B)  ln 20 
C)  ln 12 
D)  0 
A)  ln 60 
B)  ln 20 
C)  ln 12 
D)  0

Question 12

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-$F = \sin y i + x z j + 5 z k$; D: the thick sphere $4 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9$

Accepted Answer

A)  $135 \pi$ 
B)  $\frac { 76 } { 3 } \pi$ 
C)  $380 \pi$ 
D)  $\frac { 380 } { 3 } \pi$ 
A)  $135 \pi$ 
B)  $\frac { 76 } { 3 } \pi$ 
C)  $380 \pi$ 
D)  $\frac { 380 } { 3 } \pi$

Question 13

Find the flux of the curl of field F through the shell S.
-$$\mathbf { F } = ( \mathrm { z } - \mathrm { y } ) \mathbf { i } + ( \mathrm { x } - \mathrm { z } ) \mathbf { j } + ( \mathrm { z } - \mathrm { x } ) \mathbf { k } ; \mathrm { S } : \mathrm { r } ( \mathrm { r } , \theta ) = \mathrm { r } \cos \theta \mathbf { i } + \mathrm { r } \sin \theta \mathbf { j } + \left( 9 - \mathrm { r } ^ { 2 } \right) \mathbf { k } , 0 \leq \mathrm { r } \leq 3 \text { and } 0 \leq \theta \leq 2 \pi$$

Accepted Answer

A)  $- 36 \pi$ 
B)  $36 \pi$ 
C)  $- 18 \pi$ 
D)  $18 \pi$ 
A)  $- 36 \pi$ 
B)  $36 \pi$ 
C)  $- 18 \pi$ 
D)  $18 \pi$

Question 14

Find the flux of the curl of field F through the shell S.
-$\mathbf { F } = x ^ { 2 } \mathbf { i } + 6 x \mathbf { j } + 9 \mathbf { k } ; \mathrm { S }$ is the upper hemisphere of $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 16$

Accepted Answer

A)  $16 \pi$ 
B)  16 
C)  $96 \pi$ 
D)  $\pi$ 
A)  $16 \pi$ 
B)  16 
C)  $96 \pi$ 
D)  $\pi$

Question 15

Using Green's Theorem, find the outward flux of F across the closed curve C.
-F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), (9, 0), and (0, 9)

Accepted Answer

A)  81 
B)  0 
C)  729 
D)  162 
A)  81 
B)  0 
C)  729 
D)  162

Question 16

Find the potential function f for the field F.
-$$F = \frac { x } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathbf { i } + \frac { y } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } j - \frac { 2 \sqrt { x ^ { 2 } + y ^ { 2 } } } { z ^ { 3 } } \mathbf { k }$$

Accepted Answer

A)  $f ( x , y , z ) = \frac { \sqrt { x ^ { 2 } + y ^ { 2 } } } { z ^ { 2 } } + C$ 
B)  $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } } + \ln z + C$ 
C)  $f ( x , y , z ) = \frac { 1 } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } + C$ 
D)  $f ( x , y , z ) = \frac { x + y } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } + C$ 
A)  $f ( x , y , z ) = \frac { \sqrt { x ^ { 2 } + y ^ { 2 } } } { z ^ { 2 } } + C$ 
B)  $f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } } + \ln z + C$ 
C)  $f ( x , y , z ) = \frac { 1 } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } + C$ 
D)  $f ( x , y , z ) = \frac { x + y } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } + C$

Question 17

Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( 3,7,9 ) } ( 3 x + 1 ) e ^ { 3 x } d x + z d y + y d z$$

Accepted Answer

A)  $3 \mathrm { e } ^ { 9 } + 64$ 
B)  $3 e ^ { 9 } + 63$ 
C)  $3 \mathrm { e } ^ { 9 } + 16$ 
D)  0 
A)  $3 \mathrm { e } ^ { 9 } + 64$ 
B)  $3 e ^ { 9 } + 63$ 
C)  $3 \mathrm { e } ^ { 9 } + 16$ 
D)  0

Question 18

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) = x + z; S is the surface of the wedge formed from the coordinate planes and the planes x + z = 3 and y = 5

Accepted Answer

A)  $63 + 45 \sqrt { 2 }$ 
B)  $72 + 45 \sqrt { 2 }$ 
C)  $63 + 15 \sqrt { 2 }$ 
D)  $\frac { 81 } { 2 } + 45 \sqrt { 2 }$ 
A)  $63 + 45 \sqrt { 2 }$ 
B)  $72 + 45 \sqrt { 2 }$ 
C)  $63 + 15 \sqrt { 2 }$ 
D)  $\frac { 81 } { 2 } + 45 \sqrt { 2 }$

Question 19

Find the potential function f for the field F.
-$$\mathbf { F } = - \left( \frac { \mathrm { x } } { \left( \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \right) ^ { 3 / 2 } } \right) \mathbf { i } - \left( \frac { \mathrm { y } } { \left( \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \right) ^ { 3 / 2 } } \right) \mathbf { j } - \left( \frac { \mathrm { z } } { \left( \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \right) ^ { 3 / 2 } } \right) \mathbf { k }$$

Accepted Answer

A)  $f ( x , y , z ) = - \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } + C$ 
B)  $f ( x , y , z ) = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } + C$ 
C)  $f ( x , y , z ) = \frac { 3 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } + C$ 
D)  $f ( x , y , z ) = \frac { 2 } { 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 5 / 2 } } + C$ 
A)  $f ( x , y , z ) = - \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } + C$ 
B)  $f ( x , y , z ) = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } + C$ 
C)  $f ( x , y , z ) = \frac { 3 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } + C$ 
D)  $f ( x , y , z ) = \frac { 2 } { 5 \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 5 / 2 } } + C$

Question 20

Find the surface area of the surface S.
-S is the area cut from the plane $z = 3 y$ by the cylinder $x ^ { 2 } + y ^ { 2 } = 36$.

Accepted Answer

A)  $36 \sqrt { 10 }$ 
B)  $36 \sqrt { 10 } \pi$ 
C)  $6 \sqrt { 10 } \pi$ 
D)  $12 \sqrt { 10 } \pi$ 
A)  $36 \sqrt { 10 }$ 
B)  $36 \sqrt { 10 } \pi$ 
C)  $6 \sqrt { 10 } \pi$ 
D)  $12 \sqrt { 10 } \pi$

Solve the problem. -The radial flow field of an incompressible fluid is shown below. Which of the closed paths would exhibit a non-zero flux?

Using Green's Theorem, calculate the area of the indicated region. -The circle $\mathbf { r } ( \mathrm { t } ) = ( 11 \cos \mathrm { t } ) \mathbf { i } + ( 11 \sin \mathrm { t } ) \mathbf { j } , 0 \leq \mathrm { t } \leq 2 \pi$

Evaluate the surface integral of the function g over the surface S. - $G ( x , y , z ) = x ^ { 2 } y ^ { 2 } z ^ { 2 } ; S$ is the surface of the rectangular prism formed from the coordinate planes and the planes $x$ $= 1 , y = 1$ , and $z = 3$

Test the vector field F to determine if it is conservative. -F = xyi + yj + zk

Find the limit. - $r ( t ) = \frac { 3 } { 2 } t i + ( 2 - t ) k ; 0 \leq t \leq 2$

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( \pi / 4 , \pi / 20 , \pi / 16 ) } 4 \sin 4 x d x + 5 \sec ^ { 2 } 5 y d y - 8 \cos 8 z d z$

Evaluate. The differential is exact. - $\int _ { ( 1,1,1 ) } ^ { ( 5,3,4 ) } \frac { 1 } { x } d x + \frac { 1 } { y } d y + \frac { 1 } { z } d z$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $F = \sin y i + x z j + 5 z k$ ; D: the thick sphere $4 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9$

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = x ^ { 2 } \mathbf { i } + 6 x \mathbf { j } + 9 \mathbf { k } ; \mathrm { S }$ is the upper hemisphere of $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 16$

Using Green's Theorem, find the outward flux of F across the closed curve C. -F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), (9, 0), and (0, 9)

Find the potential function f for the field F. - $F = \frac { x } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathbf { i } + \frac { y } { z ^ { 2 } \sqrt { x ^ { 2 } + y ^ { 2 } } } j - \frac { 2 \sqrt { x ^ { 2 } + y ^ { 2 } } } { z ^ { 3 } } \mathbf { k }$

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( 3,7,9 ) } ( 3 x + 1 ) e ^ { 3 x } d x + z d y + y d z$

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x + z; S is the surface of the wedge formed from the coordinate planes and the planes x + z = 3 and y = 5

Find the surface area of the surface S. -S is the area cut from the plane $z = 3 y$ by the cylinder $x ^ { 2 } + y ^ { 2 } = 36$ .

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Filters

Exam 17: Integrals and Vector Fields