Question 1

Evaluate $ \iint_{s}(x-8 y+z) d S $, where $S : z = 16 - x , 0 \leq x \leq 16,0 \leq y \leq 16$

Accepted Answer

A)  $- 7,373 \sqrt { 3 }$ 
B)  $- 12,288 \sqrt { 2 }$ 
C)  0 
D)  $- 19,661 \sqrt { 2 }$ 
E)  $- 15,974 \sqrt { 3 }$ 
A)  $- 7,373 \sqrt { 3 }$ 
B)  $- 12,288 \sqrt { 2 }$ 
C)  0 
D)  $- 19,661 \sqrt { 2 }$ 
E)  $- 15,974 \sqrt { 3 }$

Question 2

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. $$\overrightarrow { \mathbf { F } } ( x , y , z ) = 7 x ^ { 6 } y ^ { 8 } z ^ { 9 } \hat { \mathbf { i } } + 8 x ^ { 7 } y ^ { 7 } z ^ { 9 } \hat { \mathbf { j } } + 9 x ^ { 7 } y ^ { 8 } z ^ { 8 } \hat { \mathbf { k } }$$

Accepted Answer

A)  conservative with potential function $f ( x , y , z ) = x ^ { 8 } y ^ { 8 } z ^ { 8 }$ 
B)  conservative with potential function $f ( x , y , z ) = x ^ { 7 } y ^ { 8 } z ^ { 9 }$ 
C)  conservative with potential function $f ( x , y , z ) = x ^ { 9 } y ^ { 8 } z ^ { 7 }$ 
D)  conservative with potential function $f ( x , y , z ) = x ^ { 9 } y ^ { 9 } z ^ { 9 }$ 
E)  not conservative 
A)  conservative with potential function $f ( x , y , z ) = x ^ { 8 } y ^ { 8 } z ^ { 8 }$ 
B)  conservative with potential function $f ( x , y , z ) = x ^ { 7 } y ^ { 8 } z ^ { 9 }$ 
C)  conservative with potential function $f ( x , y , z ) = x ^ { 9 } y ^ { 8 } z ^ { 7 }$ 
D)  conservative with potential function $f ( x , y , z ) = x ^ { 9 } y ^ { 9 } z ^ { 9 }$ 
E)  not conservative

Question 3

$\text { Let } \mathbf { F } ( x , y , z ) = \arctan \frac { x } { y } \mathbf { i } + \ln \sqrt { x ^ { 2 } + y ^ { 2 } } \mathbf { j } + \mathbf { k } \text { and let } C \text { be the triangle }$ with vertices of $( 0,0,0 ) , ( 6,6,6 )$, and $( 0,0,12 )$, oriented counterclockwise. Use Stokes's Theorem to evaluate $\int _ { c } \mathbf { F } \cdot d \mathbf { r }$.

Accepted Answer

A)  0 
B)  6 
C)  9 
D)  18 
E)  12 
A)  0 
B)  6 
C)  9 
D)  18 
E)  12

Question 4

$\text { Let } \mathbf { F } ( x , y , z ) = z ^ { 2 } \mathbf { i } + 2 x \mathbf { j } + y ^ { 2 } \mathbf { k } \text { and let } S \text { be the graph of }$ $z = 1 - x ^ { 2 } - y ^ { 2 } , z \geq 0$,oriented counterclockwise. Use Stokes's Theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$.

Accepted Answer

A)  0 
B)  $2 \pi$ 
C)  $\pi$ 
D)  1 
E)  2 
A)  0 
B)  $2 \pi$ 
C)  $\pi$ 
D)  1 
E)  2

Question 5

Find the curl of the vector field $\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }$

Accepted Answer

A)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$ 
B)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$ 
C)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$ 
D)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$ 
E)  $\operatorname { curl } ( \overrightarrow { \mathrm { F } } ) = x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$ 
A)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$ 
B)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$ 
C)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = - x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( 8 - y z ) \hat { \mathbf { k } }$ 
D)  $\operatorname { curl } ( \overrightarrow { \mathbf { F } } ) = x y \hat { \mathbf { i } } + \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$ 
E)  $\operatorname { curl } ( \overrightarrow { \mathrm { F } } ) = x y \hat { \mathbf { i } } - \hat { \mathbf { j } } + ( y z - 8 ) \hat { \mathbf { k } }$

Question 6

A tractor engine has a steel component with a circular base modeled by the vector-valued function $\mathbf { r } ( t ) = 6 \cos t \mathbf { i } + 6 \sin t \mathbf { j }$. Its height is given by $z = 1 + y ^ { 2 }$. (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places.

Accepted Answer

A)  $753.98 \mathrm {~cm} ^ { 2 }$ 
B)  $735.13 \mathrm {~cm} ^ { 2 }$ 
C)  $716.28 \mathrm {~cm} ^ { 2 }$ 
D)  $245.04 \mathrm {~cm} ^ { 2 }$ 
E)  $697.43 \mathrm {~cm} ^ { 2 }$ 
A)  $753.98 \mathrm {~cm} ^ { 2 }$ 
B)  $735.13 \mathrm {~cm} ^ { 2 }$ 
C)  $716.28 \mathrm {~cm} ^ { 2 }$ 
D)  $245.04 \mathrm {~cm} ^ { 2 }$ 
E)  $697.43 \mathrm {~cm} ^ { 2 }$

Question 7

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. $\overrightarrow { \mathrm { F } } ( x , y ) = \frac { 3 y } { x } \hat { \mathbf { i } } - \frac { x ^ { 3 } } { y ^ { 3 } } \hat { \mathbf { j } }$

Accepted Answer

A)  conservative with potential function $f ( x , y ) = \frac { 3 x ^ { 3 } } { y }$ 
B)  conservative with potential function $f ( x , y ) = \frac { x ^ { 3 } } { 3 y }$ 
C)  conservative with potential function $f ( x , y ) = \frac { x ^ { 3 } } { 4 y }$ 
D)  conservative with potential function $f ( x , y ) = \frac { 4 x ^ { 3 } } { y }$ 
E)  not conservative 
A)  conservative with potential function $f ( x , y ) = \frac { 3 x ^ { 3 } } { y }$ 
B)  conservative with potential function $f ( x , y ) = \frac { x ^ { 3 } } { 3 y }$ 
C)  conservative with potential function $f ( x , y ) = \frac { x ^ { 3 } } { 4 y }$ 
D)  conservative with potential function $f ( x , y ) = \frac { 4 x ^ { 3 } } { y }$ 
E)  not conservative

Question 8

Use a computer algebra system to evaluate $ \iint_{s} x y d S $ where $ S $ is $z = \frac { 1 } { 2 } x y , 0 \leq x \leq 4,0 \leq y \leq 4$. Round your answer to two decimal places.

Accepted Answer

A)  $124.82$ 
B)  $1.07$ 
C)  $62.41$ 
D)  $141.01$ 
E)  $64.00$ 
A)  $124.82$ 
B)  $1.07$ 
C)  $62.41$ 
D)  $141.01$ 
E)  $64.00$

Question 9

Evaluate the line integral along the given path. $$\int ( 4 x - 5 y ) d s \quad C : \overrightarrow { \mathbf { r } } ( t ) = 3 t \hat { \mathbf { i } } + 2 t \hat { \mathbf { j } } , \quad 0 \leq y \leq 6$$

Accepted Answer

A)  $30 \sqrt { 13 }$ 
B)  $40 \sqrt { 13 }$ 
C)  $42 \sqrt { 13 }$ 
D)  $26 \sqrt { 13 }$ 
E)  $36 \sqrt { 13 }$ 
A)  $30 \sqrt { 13 }$ 
B)  $40 \sqrt { 13 }$ 
C)  $42 \sqrt { 13 }$ 
D)  $26 \sqrt { 13 }$ 
E)  $36 \sqrt { 13 }$

Question 10

Find the moments of inertia for a wire that lies along $\overrightarrow { \mathbf { r } } ( t ) = 7 \cos t \tilde { \mathbf { i } } + 7 \sin t \tilde { \mathbf { j } } , 0 \leq t \leq 2 \pi$, with density $\rho ( x , y ) = 1$

Accepted Answer

A)  $I _ { x } = 343 \pi , I _ { y } = 343 \pi$ 
B)  $I _ { x } = 7 \pi , I _ { y } = 49 \pi$ 
C)  $I _ { x } = 49 \pi , I _ { y } = 49 \pi$ 
D)  $L _ { x } = 343 \pi , I _ { y } = 7 \pi$ 
E)  $I _ { x } = 7 \pi , I _ { y } = 7 \pi$ 
A)  $I _ { x } = 343 \pi , I _ { y } = 343 \pi$ 
B)  $I _ { x } = 7 \pi , I _ { y } = 49 \pi$ 
C)  $I _ { x } = 49 \pi , I _ { y } = 49 \pi$ 
D)  $L _ { x } = 343 \pi , I _ { y } = 7 \pi$ 
E)  $I _ { x } = 7 \pi , I _ { y } = 7 \pi$

Question 11

For the vector field $\mathbf { F } ( x , y ) = A x ^ { 5 } y ^ { 5 } \mathbf { i } + 5 x ^ { 6 } y ^ { 4 } \mathbf { j } \text {, find the value of } A \text { for which the }$ field is conservative.

Accepted Answer

A)  5 
B)  6 
C)  4 
D)  25 
E)  $\mathbf { F } ( x , y )$ is not conservative for any value of $A$. 
A)  5 
B)  6 
C)  4 
D)  25 
E)  $\mathbf { F } ( x , y )$ is not conservative for any value of $A$.

Question 12

Find a vector-valued function for the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121$

Accepted Answer

A)  $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + \sqrt { u ^ { 2 } + 121 } \mathbf { k }$ 
B)  $\mathbf { r } ( u , v ) = u \sin v \mathbf { i } + 11 u \cos v \mathbf { j } + \sqrt { u ^ { 2 } + 1 } \mathbf { k }$ 
C) $\mathbf { r } ( u , v ) = 11 u \sin v \mathbf { i } + u \cos v \mathbf { j } + \sqrt { u ^ { 2 } - 1 \mathbf { k } }$ 
D) $\mathbf { r } ( u , v ) = 11 u \cos v \mathbf { i } + 11 u \sin v \mathbf { j } + 11 \sqrt { u ^ { 2 } - 1 } \mathbf { k }$ 
E)  $\mathbf { r } ( u , v ) = 11 u \cos \boldsymbol { i } + 11 u \sin v \mathbf { j } + 11 \sqrt { u ^ { 2 } - 121 } \mathbf { k }$ 
A)  $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + \sqrt { u ^ { 2 } + 121 } \mathbf { k }$ 
B)  $\mathbf { r } ( u , v ) = u \sin v \mathbf { i } + 11 u \cos v \mathbf { j } + \sqrt { u ^ { 2 } + 1 } \mathbf { k }$ 
C) $\mathbf { r } ( u , v ) = 11 u \sin v \mathbf { i } + u \cos v \mathbf { j } + \sqrt { u ^ { 2 } - 1 \mathbf { k } }$ 
D) $\mathbf { r } ( u , v ) = 11 u \cos v \mathbf { i } + 11 u \sin v \mathbf { j } + 11 \sqrt { u ^ { 2 } - 1 } \mathbf { k }$ 
E)  $\mathbf { r } ( u , v ) = 11 u \cos \boldsymbol { i } + 11 u \sin v \mathbf { j } + 11 \sqrt { u ^ { 2 } - 121 } \mathbf { k }$

Question 13

Use Divergence Theorem to evaluate $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$  and find the outward flux of $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j } + z ^ { 2 } \mathbf { k }$ through the surface $S$ of the solid bounded by the planes $x = 0 , x = 12 , y = 0 , y = 12 , z = 0$ and $z = 12$.

Accepted Answer

A)  6,912 
B)  576 
C)  62,208 
D)  20,736 
E)  5,184 
A)  6,912 
B)  576 
C)  62,208 
D)  20,736 
E)  5,184

Question 14

Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path C given in polar coordinates is $A = \frac { 1 } { 2 } \int _ { C } r ^ { 2 } d \theta " \text { to find the area of the region }$ bounded by the graphs of the polar equation $r = 15 ( 1 - \cos \theta )$.

Accepted Answer

A)  $\frac { 75 \pi } { 2 }$ 
B)  $\frac { 675 \pi } { 2 }$ 
C)  $\frac { 1125 \pi } { 2 }$ 
D)  $\frac { 45 \pi } { 2 }$ 
E)  $\frac { 15 \pi } { 2 }$ 
A)  $\frac { 75 \pi } { 2 }$ 
B)  $\frac { 675 \pi } { 2 }$ 
C)  $\frac { 1125 \pi } { 2 }$ 
D)  $\frac { 45 \pi } { 2 }$ 
E)  $\frac { 15 \pi } { 2 }$

Question 15

Find a piecewise smooth parametrization of the path C given in the following graph.

Accepted Answer

A)  $\mathbf { r } ( t ) = 5 \cos t \mathbf { i } + 5 \sin t \mathbf { j }$ 
B)  $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j }$ 
C)  $\mathbf { r } ( t ) = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j }$ 
D)  $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j }$ 
E)  $\mathbf { r } ( t ) = 7 \cos t \mathbf { i } + 7 \sin t \mathbf { j }$ 
A)  $\mathbf { r } ( t ) = 5 \cos t \mathbf { i } + 5 \sin t \mathbf { j }$ 
B)  $\mathbf { r } ( t ) = 4 \cos t \mathbf { i } + 4 \sin t \mathbf { j }$ 
C)  $\mathbf { r } ( t ) = 2 \cos t \mathbf { i } + 2 \sin t \mathbf { j }$ 
D)  $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j }$ 
E)  $\mathbf { r } ( t ) = 7 \cos t \mathbf { i } + 7 \sin t \mathbf { j }$

Question 16

Find the rectangular equation for the surface by eliminating parameters from the vector-valued function. Identify the surface. $\overrightarrow { \mathbf { r } } ( u , v ) = u \hat { \mathbf { i } } + v \hat { \mathbf { j } } + \frac { v } { 10 } \hat { \mathbf { k } }$

Accepted Answer

A)  the plane $z = \frac { x } { 10 }$ 
B)  the cylinder $x ^ { 2 } + y ^ { 2 } = 100$ 
C)  the plane $x = \frac { y } { 10 }$ 
D)  the plane $z = \frac { y } { 10 }$ 
E)  the plane $y = \frac { x } { 10 }$ 
A)  the plane $z = \frac { x } { 10 }$ 
B)  the cylinder $x ^ { 2 } + y ^ { 2 } = 100$ 
C)  the plane $x = \frac { y } { 10 }$ 
D)  the plane $z = \frac { y } { 10 }$ 
E)  the plane $y = \frac { x } { 10 }$

Question 17

Find the divergence of the vector field at the given point. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 8 x y z \hat { \mathbf { i } } + 8 y \hat { \mathbf { j } } + 8 x \hat { \mathbf { k } } , ( 8,9,8 )$

Accepted Answer

A)  582 
B)  584 
C)  602 
D)  590 
E)  586 
A)  582 
B)  584 
C)  602 
D)  590 
E)  586

Question 18

Set up and evaluate a line integral to find the area of the region R bounded by the graph of $x ^ { 2 } + y ^ { 2 } = 25$

Accepted Answer

A)  $\frac { 1 } { 2 } \int _ { C } x d y - y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
B)  $\frac { 1 } { 2 } \int _ { C } y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
C)  $\int _ { C } x d y = \iint _ { R } 1 d A = 10 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
D)  $\int _ { C } x d y - y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
E)  $\frac { 1 } { 2 } \int _ { C } x d y = \iint _ { R } 1 d A = 10 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
A)  $\frac { 1 } { 2 } \int _ { C } x d y - y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
B)  $\frac { 1 } { 2 } \int _ { C } y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
C)  $\int _ { C } x d y = \iint _ { R } 1 d A = 10 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
D)  $\int _ { C } x d y - y d x = \iint _ { R } 1 d A = 25 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$ 
E)  $\frac { 1 } { 2 } \int _ { C } x d y = \iint _ { R } 1 d A = 10 \pi$ where $C : x ^ { 2 } + y ^ { 2 } = 25$

Question 19

Use Green's Theorem to evaluate the integral $\int _ { C } ( y - x ) d x + ( 2 x - y ) d y \text { where } C \text { is }$ the boundary of the region lying inside the rectangle bounded by $x = - 8 , x = 8 , y = - 4 , y = 4$, and outside the square bounded by $x = - 1 , x = 1 , y = - 1$, and $y = 1$.

Accepted Answer

A)  159 
B)  187 
C)  175 
D)  165 
E)  124 
A)  159 
B)  187 
C)  175 
D)  165 
E)  124

Question 20

Find a piecewise smooth parametrization of the path C given in the following graph.

Accepted Answer

A)  $\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } + t \mathbf { j } & 0 \leq t \leq 2 \ \left( 4 + t ^ { 2 } \right) \mathbf { i } - 4 \mathbf { j } & 2 \leq t \leq 4 \ \left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 10 \end{array} \right.$ 
B)  $\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } - t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \ ( 4 + t ) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \ ( 8 + t ) \mathbf { j } & 4 \leq t \leq 8 \end{array} \right.$ 
C) 
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } 
t \mathbf { i } + t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \
( 4 + t ) \mathbf { i } - 4 \mathbf { j } & 2 \leq t \leq 4 \
\left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 8
\end{array} \right.$ 
D) 
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } 
t ^ { 2 } + t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \
( 4 - t ) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \
( 8 - t ) \mathbf { j } & 4 \leq t \leq 8
\end{array} \right.$ 
E) 
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } 
t \mathbf { i } + t \mathbf { j } & 0 \leq t \leq 2 \
\left( 4 - t ^ { 2 } \right) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \
\left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 10
\end{array} \right.$ 
A)  $\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } + t \mathbf { j } & 0 \leq t \leq 2 \ \left( 4 + t ^ { 2 } \right) \mathbf { i } - 4 \mathbf { j } & 2 \leq t \leq 4 \ \left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 10 \end{array} \right.$ 
B)  $\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } - t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \ ( 4 + t ) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \ ( 8 + t ) \mathbf { j } & 4 \leq t \leq 8 \end{array} \right.$ 
C) 
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } 
t \mathbf { i } + t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \
( 4 + t ) \mathbf { i } - 4 \mathbf { j } & 2 \leq t \leq 4 \
\left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 8
\end{array} \right.$ 
D) 
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } 
t ^ { 2 } + t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \
( 4 - t ) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \
( 8 - t ) \mathbf { j } & 4 \leq t \leq 8
\end{array} \right.$ 
E) 
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } 
t \mathbf { i } + t \mathbf { j } & 0 \leq t \leq 2 \
\left( 4 - t ^ { 2 } \right) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \
\left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 10
\end{array} \right.$

Evaluate $\iint_{s}(x-8 y+z) d S$ , where $S : z = 16 - x , 0 \leq x \leq 16,0 \leq y \leq 16$

Find the curl of the vector field $\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }$

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. $\overrightarrow { \mathrm { F } } ( x , y ) = \frac { 3 y } { x } \hat { \mathbf { i } } - \frac { x ^ { 3 } } { y ^ { 3 } } \hat { \mathbf { j } }$

Use a computer algebra system to evaluate $\iint_{s} x y d S$ where $S$ is $z = \frac { 1 } { 2 } x y , 0 \leq x \leq 4,0 \leq y \leq 4$ . Round your answer to two decimal places.

Evaluate the line integral along the given path. $\int ( 4 x - 5 y ) d s \quad C : \overrightarrow { \mathbf { r } } ( t ) = 3 t \hat { \mathbf { i } } + 2 t \hat { \mathbf { j } } , \quad 0 \leq y \leq 6$

Find the moments of inertia for a wire that lies along $\overrightarrow { \mathbf { r } } ( t ) = 7 \cos t \tilde { \mathbf { i } } + 7 \sin t \tilde { \mathbf { j } } , 0 \leq t \leq 2 \pi$ , with density $\rho ( x , y ) = 1$

For the vector field $\mathbf { F } ( x , y ) = A x ^ { 5 } y ^ { 5 } \mathbf { i } + 5 x ^ { 6 } y ^ { 4 } \mathbf { j } \text {, find the value of } A \text { for which the }$ field is conservative.

Find a vector-valued function for the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121$

Find a piecewise smooth parametrization of the path C given in the following graph.

Find the rectangular equation for the surface by eliminating parameters from the vector-valued function. Identify the surface. $\overrightarrow { \mathbf { r } } ( u , v ) = u \hat { \mathbf { i } } + v \hat { \mathbf { j } } + \frac { v } { 10 } \hat { \mathbf { k } }$

Find the divergence of the vector field at the given point. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 8 x y z \hat { \mathbf { i } } + 8 y \hat { \mathbf { j } } + 8 x \hat { \mathbf { k } } , ( 8,9,8 )$

Set up and evaluate a line integral to find the area of the region R bounded by the graph of $x ^ { 2 } + y ^ { 2 } = 25$

Find a piecewise smooth parametrization of the path C given in the following graph.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Exact First-Order Equations

Filters

Exam 15: Vector Fields

Evaluate ∬s(x−8y+z)dS \iint_{s}(x-8 y+z) d S ∬s​(x−8y+z)dS , where S:z=16−x,0≤x≤16,0≤y≤16S : z = 16 - x , 0 \leq x \leq 16,0 \leq y \leq 16S:z=16−x,0≤x≤16,0≤y≤16

Find the curl of the vector field F→(x,y,z)=(8y−z)i^+xyzj^+4ezk^\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }F(x,y,z)=(8y−z)i^+xyzj^​+4ezk^

Use a computer algebra system to evaluate ∬sxydS \iint_{s} x y d S ∬s​xydS where S S S is z=12xy,0≤x≤4,0≤y≤4z = \frac { 1 } { 2 } x y , 0 \leq x \leq 4,0 \leq y \leq 4z=21​xy,0≤x≤4,0≤y≤4 . Round your answer to two decimal places.

Evaluate the line integral along the given path. ∫(4x−5y)dsC:r→(t)=3ti^+2tj^,0≤y≤6\int ( 4 x - 5 y ) d s \quad C : \overrightarrow { \mathbf { r } } ( t ) = 3 t \hat { \mathbf { i } } + 2 t \hat { \mathbf { j } } , \quad 0 \leq y \leq 6∫(4x−5y)dsC:r(t)=3ti^+2tj^​,0≤y≤6

Find a vector-valued function for the hyperboloid x2+y2−z2=121x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121x2+y2−z2=121

Find a piecewise smooth parametrization of the path C given in the following graph.

Find the divergence of the vector field at the given point. F→(x,y,z)=8xyzi^+8yj^+8xk^,(8,9,8)\overrightarrow { \mathbf { F } } ( x , y , z ) = 8 x y z \hat { \mathbf { i } } + 8 y \hat { \mathbf { j } } + 8 x \hat { \mathbf { k } } , ( 8,9,8 )F(x,y,z)=8xyzi^+8yj^​+8xk^,(8,9,8)

Set up and evaluate a line integral to find the area of the region R bounded by the graph of x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

Find a piecewise smooth parametrization of the path C given in the following graph.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Exact First-Order Equations

Filters

Evaluate $\iint_{s}(x-8 y+z) d S$ , where $S : z = 16 - x , 0 \leq x \leq 16,0 \leq y \leq 16$

Find the curl of the vector field $\overrightarrow { \mathbf { F } } ( x , y , z ) = ( 8 y - z ) \hat { \mathbf { i } } + x y z \hat { j } + 4 e ^ { z } \hat { \mathbf { k } }$

Use a computer algebra system to evaluate $\iint_{s} x y d S$ where $S$ is $z = \frac { 1 } { 2 } x y , 0 \leq x \leq 4,0 \leq y \leq 4$ . Round your answer to two decimal places.

Evaluate the line integral along the given path. $\int ( 4 x - 5 y ) d s \quad C : \overrightarrow { \mathbf { r } } ( t ) = 3 t \hat { \mathbf { i } } + 2 t \hat { \mathbf { j } } , \quad 0 \leq y \leq 6$

Find a vector-valued function for the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121$

Find the divergence of the vector field at the given point. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 8 x y z \hat { \mathbf { i } } + 8 y \hat { \mathbf { j } } + 8 x \hat { \mathbf { k } } , ( 8,9,8 )$

Set up and evaluate a line integral to find the area of the region R bounded by the graph of $x ^ { 2 } + y ^ { 2 } = 25$