Question 1

Apply Green's Theorem to evaluate the integral.
-$$\oint _ { C } ( 6 y d x + 8 y d y )$$
$C$ : The boundary of $0 \leq x \leq \pi , 0 \leq y \leq \sin x$

Accepted Answer

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

Question 2

Find the flux of the curl of field F through the shell S.
-$\mathbf { F } = - 5 \mathrm { zi } - 3 \times \mathbf { j } - 2 \mathrm { yk } ; \mathrm { S } : \mathrm { r } ( \mathrm { r } , \theta ) = \mathrm { r } \cos \theta \mathbf { i } + \mathrm { r } \sin \theta \mathbf { j } + 6 \mathrm { rk } , 0 \leq \mathrm { r } \leq 6$ and $0 \leq \theta \leq 2 \pi$

Accepted Answer

A)  $- 108 \pi$ 
B)  $- 108$ 
C)  0 
D)  $108 \pi$ 
A)  $- 108 \pi$ 
B)  $- 108$ 
C)  0 
D)  $108 \pi$

Question 3

Evaluate the work done between point 1 and point 2 for the conservative field F.
-$$\mathrm { F } = 5 x \mathbf { i } + 5 y \mathbf { j } + 5 \mathrm { z } \mathbf { k } ; \mathrm { P } _ { 1 } ( 2,2,3 ) , \mathrm { P } _ { 2 } ( 6,7,9 )$$

Accepted Answer

A)  $\mathrm { W } = \frac { 915 } { 2 }$ 
B)  $W = - \frac { 745 } { 2 }$ 
C)  $W = 0$ 
D)  $\mathrm { W } = \frac { 745 } { 2 }$ 
A)  $\mathrm { W } = \frac { 915 } { 2 }$ 
B)  $W = - \frac { 745 } { 2 }$ 
C)  $W = 0$ 
D)  $\mathrm { W } = \frac { 745 } { 2 }$

Question 4

Using Green's Theorem, find the outward flux of F across the closed curve C.
-$\mathbf { F } = ( - 9 x + 7 y ) \mathbf { i } + ( 9 x - 3 y ) \mathbf { j } ; C$ is the region bounded above by $y = - 3 x ^ { 2 } + 7$ and below by $y = 4 x ^ { 2 }$ in the first quadrant

Accepted Answer

A)  - 64 
B)  - 56 
C)  48 
D)  72 
A)  - 64 
B)  - 56 
C)  48 
D)  72

Question 5

Find the surface area of the surface S.
-$S$ is the cap cut from the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ by the cone $z = 6 \sqrt { x ^ { 2 } + y ^ { 2 } }$.

Accepted Answer

A)  $16 \left( 1 - \frac { 6 \sqrt { 37 } } { 37 } \right)$ 
B)  $16 \left( 1 - \frac { 6 \sqrt { 37 } } { 37 } \right) \pi$ 
C)  $32 \left( 1 - \frac { 6 \sqrt { 37 } } { 37 } \right) \pi$ 
D)  $16 \left( \frac { 6 \sqrt { 37 } } { 37 } - 1 \right) \pi$ 
A)  $16 \left( 1 - \frac { 6 \sqrt { 37 } } { 37 } \right)$ 
B)  $16 \left( 1 - \frac { 6 \sqrt { 37 } } { 37 } \right) \pi$ 
C)  $32 \left( 1 - \frac { 6 \sqrt { 37 } } { 37 } \right) \pi$ 
D)  $16 \left( \frac { 6 \sqrt { 37 } } { 37 } - 1 \right) \pi$

Question 6

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-$\mathrm { F } = ( \mathrm { x } + 2 \mathrm { y } ) \mathbf { i } + ( 9 \mathrm { x } - 6 \mathrm { y } ) \mathbf { j } ; \mathrm { C }$ is the region bounded above by $\mathrm { y } = - 5 \mathrm { x } ^ { 2 } + 28$ and below by $\mathrm { y } = 2 \mathrm { x } ^ { 2 }$ in the first quadrant

Accepted Answer

A)  $\frac { 200 } { 3 }$ 
B)  $\frac { 784 } { 3 }$ 
C)  $- \frac { 550 } { 3 }$ 
D)  $- 378$ 
A)  $\frac { 200 } { 3 }$ 
B)  $\frac { 784 } { 3 }$ 
C)  $- \frac { 550 } { 3 }$ 
D)  $- 378$

Question 7

Find the surface area of the surface S.
-$\mathrm { S }$ is the portion of the surface $10 \sqrt { 7 } x + 10 y ^ { 2 } - 20 \ln y - 30 z = 0$ that lies above the rectangle $0 \leq x \leq 1$ and $1 \leq y \leq 7$ in the $x - y$ plane.

Accepted Answer

A)  $\frac { 2 } { 3 } [ 24 + \ln 7 ]$ 
B)  $\frac { 1 } { 15 } [ 24 + \ln 7 ]$ 
C)  $\frac { 20 } { 3 } [ 24 + \ln 7 ]$ 
D)  $\frac { 4 } { 3 } [ 24 + \ln 7 ]$ 
A)  $\frac { 2 } { 3 } [ 24 + \ln 7 ]$ 
B)  $\frac { 1 } { 15 } [ 24 + \ln 7 ]$ 
C)  $\frac { 20 } { 3 } [ 24 + \ln 7 ]$ 
D)  $\frac { 4 } { 3 } [ 24 + \ln 7 ]$

Question 8

Parametrize the surface S.
-$$S \text { is the portion of the cone } \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 16 } = \frac { z ^ { 2 } } { 64 } \text { that lies between } z = 3 \text { and } z = 6 \text {. }$$

Accepted Answer

The answer of Parametrize the surface S.
-\[S \text { is...

Question 9

Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( \pi , \pi , \pi ) } - 2 \sin x \cos x d x - \sin y \cos z d y - \cos y \sin z d z$$

Accepted Answer

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

Question 10

Solve the problem.
-Find a parametrization for the ellipsoid $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 81 } + \frac { z ^ { 2 } } { 4 } = 1$. (Recall that the parametrization of an ellipse $\frac { x ^ { 2 } } { 36 } +$ $\frac { y ^ { 2 } } { 81 } = 1$ is $\left. x = 6 \cos \theta , y = 9 \sin \theta , 0 \leq \theta < 2 \pi \right)$

Accepted Answer

Answers wi

Question 11

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

Accepted Answer

A) ${\nabla} f  = - z \cos ( x + y + z ) \mathbf { i } - z \cos ( x + y + z ) \mathbf { j } + ( \sin ( x + y + z ) - z \cos ( x + y + z ) ) \mathbf { k }$ 
B)  $\nabla \mathbf { f } = \cos x i \mathbf { i } + \cos y \mathbf { j } + ( \sin z + z \cos z ) \mathbf { k }$ 
C)  $\nabla \mathrm { f } = \mathrm { z } \cos ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) \mathbf { i } + \mathrm { z } \cos ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) \mathbf { j } + ( \sin ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) + \mathrm { zcos } ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) ) \mathbf { k }$ 
D) ${\nabla} f  = - \cos x \mathbf { i } - \cos y \mathbf { j } + ( \sin z - z \cos z ) \mathbf { k }$ 
A) ${\nabla} f  = - z \cos ( x + y + z ) \mathbf { i } - z \cos ( x + y + z ) \mathbf { j } + ( \sin ( x + y + z ) - z \cos ( x + y + z ) ) \mathbf { k }$ 
B)  $\nabla \mathbf { f } = \cos x i \mathbf { i } + \cos y \mathbf { j } + ( \sin z + z \cos z ) \mathbf { k }$ 
C)  $\nabla \mathrm { f } = \mathrm { z } \cos ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) \mathbf { i } + \mathrm { z } \cos ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) \mathbf { j } + ( \sin ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) + \mathrm { zcos } ( \mathrm { x } + \mathrm { y } + \mathrm { z } ) ) \mathbf { k }$ 
D) ${\nabla} f  = - \cos x \mathbf { i } - \cos y \mathbf { j } + ( \sin z - z \cos z ) \mathbf { k }$

Question 12

Parametrize the surface S.
-$$S \text { is the portion of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16 \text { between } z = - 2 \sqrt { 2 } \text { and } z = 2 \sqrt { 2 } \text {. }$$

Accepted Answer

The answer of Parametrize the surface S.
-\[S \text { is...

Question 13

Find the divergence of the field F.
-$$\mathbf { F } = \frac { y \mathbf { j } - \mathrm { zk } } { \left( \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \right) ^ { 1 / 2 } }$$

Accepted Answer

A)  $\frac { 3 \left( y ^ { 2 } + z ^ { 2 } \right) } { \left( y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ 
B)  $\frac { z ^ { 2 } - y ^ { 2 } } { \left( y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ 
C)  0 
D)  $\frac { y ^ { 2 } - z ^ { 2 } } { \left( y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ 
A)  $\frac { 3 \left( y ^ { 2 } + z ^ { 2 } \right) } { \left( y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ 
B)  $\frac { z ^ { 2 } - y ^ { 2 } } { \left( y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ 
C)  0 
D)  $\frac { y ^ { 2 } - z ^ { 2 } } { \left( y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$

Question 14

$$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$$
-$$r ( t ) = 8 t i + 7 t j + 3 t ^ { 2 } k , 0 \leq t \leq 1 ; \delta ( x , y , z ) = \frac { x } { \sqrt { 113 + 12 z } }$$

Accepted Answer

A)  $\left( \frac { 64 } { 3 } , \frac { 56 } { 3 } , 8 \right)$ 
B)  $( 8,7,3 )$ 
C)  $\left( \frac { 16 } { 3 } , \frac { 14 } { 3 } , \frac { 3 } { 2 } \right)$ 
D)  $( 16,14,6 )$ 
A)  $\left( \frac { 64 } { 3 } , \frac { 56 } { 3 } , 8 \right)$ 
B)  $( 8,7,3 )$ 
C)  $\left( \frac { 16 } { 3 } , \frac { 14 } { 3 } , \frac { 3 } { 2 } \right)$ 
D)  $( 16,14,6 )$

Question 15

Using Green's Theorem, find the outward flux of F across the closed curve C.
-F = xyi + xj; C is the triangle with vertices at (0, 0), (5, 0), and (0, 3)

Accepted Answer

A)  0 
B)  $\frac { 15 } { 2 }$ 
C)  20 
D)  $- 5$ 
A)  0 
B)  $\frac { 15 } { 2 }$ 
C)  20 
D)  $- 5$

Question 16

Find the limit.
-$$r ( t ) = \sin t j - \cos t k ; 0 \leq t \leq \frac { \pi } { 2 }$$

Accepted Answer

A)  Figure 7 
B)  Figure 4 
C)  Figure 1 
D)  Figure 2 
A)  Figure 7 
B)  Figure 4 
C)  Figure 1 
D)  Figure 2

Question 17

Find the surface area of the surface S.
-$\mathrm { S }$ is the paraboloid $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } - \mathrm { z } = 0$ below the plane $\mathrm { z } = 2$.

Accepted Answer

A)  $\frac { 13 } { 6 } \pi$ 
B)  $\frac { 9 } { 2 } \pi$ 
C)  $\frac { 13 } { 3 } \pi$ 
D)  $\frac { 13 } { 2 } \pi$ 
A)  $\frac { 13 } { 6 } \pi$ 
B)  $\frac { 9 } { 2 } \pi$ 
C)  $\frac { 13 } { 3 } \pi$ 
D)  $\frac { 13 } { 2 } \pi$

Question 18

Solve the problem.
-Assume the curl of a vector field $F$ is zero. Can one automatically conclude that the circulation $\int _ { C } \mathbf { F } \cdot \mathrm { d } \mathbf { r } = 0$ for all closed paths C? Explain or justify your answer.

Accepted Answer

No, the region containing C ma

Question 19

$$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$$
-$$r ( t ) = ( 3 \cos t ) \mathbf { i } + ( 3 \sin t ) \mathbf { j } + 3 t k , 0 \leq t \leq 2 \pi ; \delta = 5$$

Accepted Answer

A)  $30 \pi \sqrt { 2 }$ units 
B)  $6 \pi \sqrt { 2 }$ units 
C)  $90 \pi \sqrt { 2 }$ units 
D)  $10 \pi$ units 
A)  $30 \pi \sqrt { 2 }$ units 
B)  $6 \pi \sqrt { 2 }$ units 
C)  $90 \pi \sqrt { 2 }$ units 
D)  $10 \pi$ units

Question 20

Find the work done by F over the curve in the direction of increasing t.
-$$\mathbf { F } = \frac { 1 } { x + 5 } \mathbf { i } + \mathbf { j } + 6 \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \mathrm { t } ^ { 7 } \mathbf { i } + \mathrm { t } ^ { 6 } \mathbf { j } + \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq 1$$

Accepted Answer

A)  $W = \ln \left( \frac { 6 } { 5 } \right) + 12$ 
B)  $W = \ln \left( \frac { 6 } { 5 } \right) + \frac { 7 } { 2 }$ 
C)  $W = \ln \left( \frac { 6 } { 35 } \right) + \frac { 7 } { 2 }$ 
D)  $W = \ln \left( \frac { 6 } { 5 } \right)$ 
A)  $W = \ln \left( \frac { 6 } { 5 } \right) + 12$ 
B)  $W = \ln \left( \frac { 6 } { 5 } \right) + \frac { 7 } { 2 }$ 
C)  $W = \ln \left( \frac { 6 } { 35 } \right) + \frac { 7 } { 2 }$ 
D)  $W = \ln \left( \frac { 6 } { 5 } \right)$

Apply Green's Theorem to evaluate the integral. - $\oint _ { C } ( 6 y d x + 8 y d y )$ $C$ : The boundary of $0 \leq x \leq \pi , 0 \leq y \leq \sin x$

Evaluate the work done between point 1 and point 2 for the conservative field F. - $\mathrm { F } = 5 x \mathbf { i } + 5 y \mathbf { j } + 5 \mathrm { z } \mathbf { k } ; \mathrm { P } _ { 1 } ( 2,2,3 ) , \mathrm { P } _ { 2 } ( 6,7,9 )$

Using Green's Theorem, find the outward flux of F across the closed curve C. - $\mathbf { F } = ( - 9 x + 7 y ) \mathbf { i } + ( 9 x - 3 y ) \mathbf { j } ; C$ is the region bounded above by $y = - 3 x ^ { 2 } + 7$ and below by $y = 4 x ^ { 2 }$ in the first quadrant

Find the surface area of the surface S. - $S$ is the cap cut from the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ by the cone $z = 6 \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the surface area of the surface S. - $\mathrm { S }$ is the portion of the surface $10 \sqrt { 7 } x + 10 y ^ { 2 } - 20 \ln y - 30 z = 0$ that lies above the rectangle $0 \leq x \leq 1$ and $1 \leq y \leq 7$ in the $x - y$ plane.

Parametrize the surface S. - $S \text { is the portion of the cone } \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 16 } = \frac { z ^ { 2 } } { 64 } \text { that lies between } z = 3 \text { and } z = 6 \text {. }$

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( \pi , \pi , \pi ) } - 2 \sin x \cos x d x - \sin y \cos z d y - \cos y \sin z d z$

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

Parametrize the surface S. - $S \text { is the portion of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16 \text { between } z = - 2 \sqrt { 2 } \text { and } z = 2 \sqrt { 2 } \text {. }$

Find the divergence of the field F. - $\mathbf { F } = \frac { y \mathbf { j } - \mathrm { zk } } { \left( \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \right) ^ { 1 / 2 } }$

$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = 8 t i + 7 t j + 3 t ^ { 2 } k , 0 \leq t \leq 1 ; \delta ( x , y , z ) = \frac { x } { \sqrt { 113 + 12 z } }$

Using Green's Theorem, find the outward flux of F across the closed curve C. -F = xyi + xj; C is the triangle with vertices at (0, 0), (5, 0), and (0, 3)

Find the limit. - $r ( t ) = \sin t j - \cos t k ; 0 \leq t \leq \frac { \pi } { 2 }$

Find the surface area of the surface S. - $\mathrm { S }$ is the paraboloid $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } - \mathrm { z } = 0$ below the plane $\mathrm { z } = 2$ .

Solve the problem. -Assume the curl of a vector field $F$ is zero. Can one automatically conclude that the circulation $\int _ { C } \mathbf { F } \cdot \mathrm { d } \mathbf { r } = 0$ for all closed paths C? Explain or justify your answer.

$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = ( 3 \cos t ) \mathbf { i } + ( 3 \sin t ) \mathbf { j } + 3 t k , 0 \leq t \leq 2 \pi ; \delta = 5$

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Exam 17: Integrals and Vector Fields

Apply Green's Theorem to evaluate the integral. - ∮C(6ydx+8ydy)\oint _ { C } ( 6 y d x + 8 y d y )∮C​(6ydx+8ydy) CCC : The boundary of 0≤x≤π,0≤y≤sin⁡x0 \leq x \leq \pi , 0 \leq y \leq \sin x0≤x≤π,0≤y≤sinx

Find the surface area of the surface S. - SSS is the cap cut from the sphere x2+y2+z2=16x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16x2+y2+z2=16 by the cone z=6x2+y2z = 6 \sqrt { x ^ { 2 } + y ^ { 2 } }z=6x2+y2​ .

Evaluate. The differential is exact. - ∫(0,0,0)(π,π,π)−2sin⁡xcos⁡xdx−sin⁡ycos⁡zdy−cos⁡ysin⁡zdz\int _ { ( 0,0,0 ) } ^ { ( \pi , \pi , \pi ) } - 2 \sin x \cos x d x - \sin y \cos z d y - \cos y \sin z d z∫(0,0,0)(π,π,π)​−2sinxcosxdx−sinycoszdy−cosysinzdz