Question 1

Find the area of the surface given by $\mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }$ $0 \leq u \leq 2$ and $0 \leq v \leq 4$

Accepted Answer

A)  $81 \sqrt { 2 }$ 
B)  $38 \sqrt { 2 }$ 
C)  $72 \sqrt { 2 }$ 
D)  $45 \sqrt { 2 }$ 
E)  $22 \sqrt { 2 }$ 
A)  $81 \sqrt { 2 }$ 
B)  $38 \sqrt { 2 }$ 
C)  $72 \sqrt { 2 }$ 
D)  $45 \sqrt { 2 }$ 
E)  $22 \sqrt { 2 }$

Question 2

$\text { Let } F ( x , y , z ) = 2 x \hat { \mathbf { i } } - 2 y \hat { \mathbf { j } } + z ^ { 2 } \hat { \mathbf { k } } \text { and let } S \text { be the cylinder } x ^ { 2 } + y ^ { 2 } = 4,0 \leq z \leq 3 \text {. }$  Verify the Divergence Theorem by evaluating $\iint _ { S } \mathbf { F } \cdot \mathbf { N } d s$ as a surface integral and as a triple integral.

Accepted Answer

A)  $18 \pi$ 
B)  $36 \pi$ 
C)  $12 \pi$ 
D)  $108 \pi$ 
E)  $54 \pi$ 
A)  $18 \pi$ 
B)  $36 \pi$ 
C)  $12 \pi$ 
D)  $108 \pi$ 
E)  $54 \pi$

Question 3

$\text { Let } \mathbf { F } ( x , y , z ) = x y \mathbf { i } + z \mathbf { j } + ( x + y ) \mathbf { k } \text { and let } S \text { be the surface bounded by the planes }$ $y = 4$ and $z = 4 - x$ and the coordinate planes. Verify the Divergence Theorem by evaluating $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$  as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

Accepted Answer

A)  $76.25$ 
B)  $85.33$ 
C)  $156.25$ 
D)  $64.00$ 
E)  $20.25$ 
A)  $76.25$ 
B)  $85.33$ 
C)  $156.25$ 
D)  $64.00$ 
E)  $20.25$

Question 4

Use Stokes's Theorem to evaluate $\mathbf { F } ( x , y , z ) = x y z \mathbf { i } + y \mathbf { j } + z \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 49$ and $S$ is the first-octant portion of $z = x ^ { 2 }$ over $x ^ { 2 } + y ^ { 2 } = 49$
Use a computer algebra system to verify your result.

Accepted Answer

A)  $\frac { 16,807 } { 15 }$ 
B)  $- \frac { 16,807 } { 15 }$ 
C)  $- \frac { 33,614 } { 15 }$ 
D)  0 
E)  $\frac { 33,614 } { 15 }$ 
A)  $\frac { 16,807 } { 15 }$ 
B)  $- \frac { 16,807 } { 15 }$ 
C)  $- \frac { 33,614 } { 15 }$ 
D)  0 
E)  $\frac { 33,614 } { 15 }$

Question 5

Find the divergence of the vector field F given by $$\mathbf { F } ( x , y , z ) = \sin ( 7 x ) \mathbf { i } + \cos ( 6 y ) \mathbf { j } + z ^ { 3 } \hat { \mathbf { k } }$$

Accepted Answer

A)  $7 \cos ( x ) - 6 \sin ( y ) + 3 z ^ { 2 }$ 
B)  $\cos ( x ) + \sin ( y ) + z ^ { 4 }$ 
C)  $7 \cos ( 7 x ) + 6 \sin ( 6 y ) + 3 z ^ { 4 }$ 
D)  $7 \cos ( 7 x ) - 6 \sin ( 6 y ) + 3 z ^ { 2 }$ 
E)  $\cos ( 7 x ) - \sin ( 6 y ) + z ^ { 2 }$ 
A)  $7 \cos ( x ) - 6 \sin ( y ) + 3 z ^ { 2 }$ 
B)  $\cos ( x ) + \sin ( y ) + z ^ { 4 }$ 
C)  $7 \cos ( 7 x ) + 6 \sin ( 6 y ) + 3 z ^ { 4 }$ 
D)  $7 \cos ( 7 x ) - 6 \sin ( 6 y ) + 3 z ^ { 2 }$ 
E)  $\cos ( 7 x ) - \sin ( 6 y ) + z ^ { 2 }$

Question 6

Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. $$\overrightarrow { \mathbf { r } } ( u , v ) = ( 2 u + v ) \hat { \mathbf { i } } + ( u - v ) \hat { \mathbf { j } } + v \hat { \mathbf { k } } , ( 4 , - 4,4 )$$

Accepted Answer

A)  $( x - 4 ) - 2 ( y + 4 ) - 3 ( z - 4 ) = 0$ 
B)  $2 ( x - 4 ) + 2 ( y + 4 ) - 3 ( z - 4 ) = 0$ 
C)  $z = 0$ 
D)  $( x - 4 ) - 2 ( y + 4 ) - 3 ( z - 4 ) = 0$. 
E)  $2 ( x - 4 ) - 2 ( y + 4 ) + 1 ( z - 4 ) = 0$. 
A)  $( x - 4 ) - 2 ( y + 4 ) - 3 ( z - 4 ) = 0$ 
B)  $2 ( x - 4 ) + 2 ( y + 4 ) - 3 ( z - 4 ) = 0$ 
C)  $z = 0$ 
D)  $( x - 4 ) - 2 ( y + 4 ) - 3 ( z - 4 ) = 0$. 
E)  $2 ( x - 4 ) - 2 ( y + 4 ) + 1 ( z - 4 ) = 0$.

Question 7

Find the value of the line integral $ \int_{c} \mathbf{F} \cdot d \mathbf{r} $, where $ \mathbf{F}(x, y)=12 x y \mathbf{i}+6 x^{2} \mathbf{j} $ and $\mathbf { r } ( t ) = 8 t \mathbf { i } + 8 t ^ { 2 } \mathbf { j } , 0 \leq t \leq 1$
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)

Accepted Answer

A)  3,072 
B)  192 
C)  6,144 
D)  6,144. 
E)  6,912. 
A)  3,072 
B)  192 
C)  6,144 
D)  6,144. 
E)  6,912.

Question 8

Evaluate $\iint _ { S } 4 x y d S$, where $S$ is $z = 5 - x - y$, first octant.

Accepted Answer

A)  $\frac { 125 \sqrt { 2 } } { 2 }$ 
B)  $\frac { 4,625 \sqrt { 2 } } { 2 }$ 
C)  $\frac { 4,625 \sqrt { 3 } } { 2 }$ 
D)  $\frac { 625 \sqrt { 3 } } { 6 }$ 
A)  $\frac { 125 \sqrt { 2 } } { 2 }$ 
B)  $\frac { 4,625 \sqrt { 2 } } { 2 }$ 
C)  $\frac { 4,625 \sqrt { 3 } } { 2 }$ 
D)  $\frac { 625 \sqrt { 3 } } { 6 }$

Question 9

Use the Divergence Theorem to evaluate $ \iint_{S} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathrm{N}} d s $ Verify your answer by evaluating the integral as a triple integral. 
$F ( x , y , z ) = 2 x \hat { \mathbf { i } } - 2 y \hat { \mathbf { j } } + z ^ { 2 } \hat { \mathbf { k } }$
$S$ : cube bounded by the planes $x = 0 , x = 2 , y = 0 , y = 2 , z = 0 , z = 2$

Accepted Answer

A)  16 
B)  4 
C)  32 
D)  8 
E)  64 
A)  16 
B)  4 
C)  32 
D)  8 
E)  64

Question 10

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) $f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }$

Accepted Answer

A)  $\nabla f ( x , y ) = ( 6 y + 8 x ) \hat { \mathbf { i } } - ( 20 x + 8 y ) \hat { \mathbf { j } }$ 
B)  $\nabla f ( x , y ) = 6 y \hat { \mathbf { i } } - 20 x \hat { \mathbf { j } }$ 
C)  $\nabla f ( x , y ) = 6 y \hat { \mathbf { i } } + 20 x \hat { \mathbf { j } }$ 
D)  $\nabla f ( x , y ) = ( 20 x + 8 y ) \hat { \mathbf { i } } + ( 6 y + 8 x ) \hat { \mathbf { j } }$ 
E)  $\nabla f ( x , y ) = 20 x \hat { \mathbf { i } } + 6 y \hat { \mathbf { j } }$ 
A)  $\nabla f ( x , y ) = ( 6 y + 8 x ) \hat { \mathbf { i } } - ( 20 x + 8 y ) \hat { \mathbf { j } }$ 
B)  $\nabla f ( x , y ) = 6 y \hat { \mathbf { i } } - 20 x \hat { \mathbf { j } }$ 
C)  $\nabla f ( x , y ) = 6 y \hat { \mathbf { i } } + 20 x \hat { \mathbf { j } }$ 
D)  $\nabla f ( x , y ) = ( 20 x + 8 y ) \hat { \mathbf { i } } + ( 6 y + 8 x ) \hat { \mathbf { j } }$ 
E)  $\nabla f ( x , y ) = 20 x \hat { \mathbf { i } } + 6 y \hat { \mathbf { j } }$

Question 11

Sketch the vector field $\mathbf { F } ( x , y ) = x \mathbf { i } + 3 y \mathbf { j }$

Accepted Answer

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

Question 12

Find the flux $\overrightarrow { \mathrm { F } } \text { of through } S , \iint _ { S } \overrightarrow { \mathrm { F } } \cdot \overrightarrow { \mathrm { N } } d S \text {, where } \overrightarrow { \mathrm { N }}$ is the upward unit normal  vector to S. 
$\overrightarrow { \mathrm { F } } ( x , y , z ) = x \hat { \mathbf { i } } + y \hat { \mathbf { j } } + z \hat { \mathbf { k } }$
$S : z = 64 - x ^ { 2 } - y ^ { 2 } , z \geq 0$

Accepted Answer

A)  $4,301 \pi$ 
B)  $6,144 \pi$ 
C)  $7,987 \pi$ 
D)  $3072 \pi$ 
E)  $1843.2 \pi$ 
A)  $4,301 \pi$ 
B)  $6,144 \pi$ 
C)  $7,987 \pi$ 
D)  $3072 \pi$ 
E)  $1843.2 \pi$

Question 13

Evaluate $ \iint_{S} f(x, y, z) d S $, Where $f(x,y,z)= {x^2}+{y^2}+{z^2}$ and S is given by $ z=x+y, x^{2}+y^{2} \leq 100 $

Accepted Answer

A)  10,000 
B)  $10,000 \pi$ 
C)  $10,000 \sqrt { 3 } \pi$ 
D)  $10,000 \sqrt { 2 } \pi$ 
E)  $10,000 \sqrt { 3 }$ 
A)  10,000 
B)  $10,000 \pi$ 
C)  $10,000 \sqrt { 3 } \pi$ 
D)  $10,000 \sqrt { 2 } \pi$ 
E)  $10,000 \sqrt { 3 }$

Question 14

Find the work done by the force field $\overrightarrow { \mathbf { F } } \text { in moving an object from } P \text { to } O \text {. }$ $\overrightarrow { \mathbf { F } } ( x , y ) = 18 x ^ { 5 } y ^ { 5 } \hat { \mathbf { i } } + \left( 15 x ^ { 6 } y ^ { 4 } - 1 \right) \hat { \mathbf { j } } , P ( 0,0 ) , Q ( 3,2 )$

Accepted Answer

A)  27,992 
B)  55,985 
C)  83,978 
D)  41,989 
E)  69,982 
A)  27,992 
B)  55,985 
C)  83,978 
D)  41,989 
E)  69,982

Question 15

Green's Theorem to evaluate the integral $\int _ { C } 11 x y d x + ( x + y ) d y$
for the path $C$ : boundary of the region lying between the graphs of $y = 0$ and $y = 121 - x ^ { 2 }$.

Accepted Answer

A)  0 
B)  $\frac { 5324 } { 3 }$ 
C)  121 
D)  $\frac { 484 } { 3 }$ 
E)  11 
A)  0 
B)  $\frac { 5324 } { 3 }$ 
C)  121 
D)  $\frac { 484 } { 3 }$ 
E)  11

Question 16

Find the value of the line integral $ \int_{c} \mathbf{F} \cdot d \mathbf{r} $ on the closed path consisting of line segments from $( 0,2 )$ to $( 0,0 )$, from $( 0,0 )$ to $( 2,0 )$, and then from $( 2,0 )$ to $( 0,2 )$, where $\mathbf { F } ( x , y ) = 3 y e ^ { 3 x y } \mathbf { i } + 3 x e ^ { 3 x y } \mathbf { j }$.
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)

Accepted Answer

A)  99 
B)  12 
C)  36 
D)  30 
E)  0 
A)  99 
B)  12 
C)  36 
D)  30 
E)  0

Question 17

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function $y = x ^ { \frac { 3 } { 7 } } , 0 \leq x \leq 4$ about the $x$-axis.

Accepted Answer

A)  $x = u ^ { \frac { 3 } { 7 } } , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ { \frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$ 
B) $x = u , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ {\frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$ 
C) $x = u , y = u ^ { \frac { 3 } { 7 } } \sin v , z = u ^ { \frac { 3 } { 7 } } \cos v , 0 \leq u \leq 4 , v \geq 0$ 
D) $x = u ^ { \frac { 3 } { 7 } } , y = u ^ { \frac { 3 } { 7 } } \sin v , z = u ^ { \frac { 3 } { 7 } } \cos v , u \geq 4,0 \leq v \leq 2 \pi$ 
E) $x = v , y = v ^ { \frac { 3 } { 7 } } \cos u , z = v ^ { \frac { 3 } { 7 } } \sin u , u \geq 4 , v \geq 0$ 
A)  $x = u ^ { \frac { 3 } { 7 } } , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ { \frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$ 
B) $x = u , y = u ^ { \frac { 3 } { 7 } } \cos v , z = u ^ {\frac { 3 } { 7 } } \sin v , 0 \leq u \leq 4,0 \leq v \leq 2 \pi$ 
C) $x = u , y = u ^ { \frac { 3 } { 7 } } \sin v , z = u ^ { \frac { 3 } { 7 } } \cos v , 0 \leq u \leq 4 , v \geq 0$ 
D) $x = u ^ { \frac { 3 } { 7 } } , y = u ^ { \frac { 3 } { 7 } } \sin v , z = u ^ { \frac { 3 } { 7 } } \cos v , u \geq 4,0 \leq v \leq 2 \pi$ 
E) $x = v , y = v ^ { \frac { 3 } { 7 } } \cos u , z = v ^ { \frac { 3 } { 7 } } \sin u , u \geq 4 , v \geq 0$

Question 18

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 $A = \frac { 1 } { 2 } \int _ { C } r ^ { 2 } d \theta ^ { \prime \prime }$ to find the area of the region bounded by the graphs of the polar equation $r = \frac { 10 } { 2 - \cos \theta }$. Round your answer to two decimal places.

Accepted Answer

A)  $132.92$ 
B)  $134.92$ 
C)  $133.92$ 
D)  $120.92$ 
E)  $167.92$ 
A)  $132.92$ 
B)  $134.92$ 
C)  $133.92$ 
D)  $120.92$ 
E)  $167.92$

Question 19

Use Green's Theorem to evaluate the line integral $\int _ { C } \left( e ^ { - x ^ { 2 } / 2 } - y \right) d x + \left( e ^ { - y ^ { 2 } / 2 } + x \right) d y$ where $C$ is the boundary of the region lying between the graphs of the circle $x = 8 \cos \theta , y = 8 \sin \theta$ and the ellipse $x = 6 \cos \theta , y = 4 \sin \theta$.

Accepted Answer

A)  $120 \pi$ 
B)  $162 \pi$ 
C)  $172 \pi$ 
D)  $80 \pi$ 
E)  $91 \pi$ 
A)  $120 \pi$ 
B)  $162 \pi$ 
C)  $172 \pi$ 
D)  $80 \pi$ 
E)  $91 \pi$

Question 20

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

Accepted Answer

A)  $- 475.02$ 
B)  $- 496.00$ 
C)  $- 203.00$ 
D)  $- 971.38$ 
E)  $- 43.75$ 
A)  $- 475.02$ 
B)  $- 496.00$ 
C)  $- 203.00$ 
D)  $- 971.38$ 
E)  $- 43.75$

Find the area of the surface given by $\mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }$ $0 \leq u \leq 2$ and $0 \leq v \leq 4$

Use Stokes's Theorem to evaluate $\mathbf { F } ( x , y , z ) = x y z \mathbf { i } + y \mathbf { j } + z \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 49$ and $S$ is the first-octant portion of $z = x ^ { 2 }$ over $x ^ { 2 } + y ^ { 2 } = 49$ Use a computer algebra system to verify your result.

Find the divergence of the vector field F given by $\mathbf { F } ( x , y , z ) = \sin ( 7 x ) \mathbf { i } + \cos ( 6 y ) \mathbf { j } + z ^ { 3 } \hat { \mathbf { k } }$

Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. $\overrightarrow { \mathbf { r } } ( u , v ) = ( 2 u + v ) \hat { \mathbf { i } } + ( u - v ) \hat { \mathbf { j } } + v \hat { \mathbf { k } } , ( 4 , - 4,4 )$

Evaluate $\iint _ { S } 4 x y d S$ , where $S$ is $z = 5 - x - y$ , first octant.

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) $f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }$

Sketch the vector field $\mathbf { F } ( x , y ) = x \mathbf { i } + 3 y \mathbf { j }$

Evaluate $\iint_{S} f(x, y, z) d S$ , Where $f(x,y,z)= {x^2}+{y^2}+{z^2}$ and S is given by $z=x+y, x^{2}+y^{2} \leq 100$

Green's Theorem to evaluate the integral $\int _ { C } 11 x y d x + ( x + y ) d y$ for the path $C$ : boundary of the region lying between the graphs of $y = 0$ and $y = 121 - x ^ { 2 }$ .

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function $y = x ^ { \frac { 3 } { 7 } } , 0 \leq x \leq 4$ about the $x$ -axis.

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

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

Find the area of the surface given by r(u,v)=9ui−vj+vk, where \mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }r(u,v)=9ui−vj+vk, where 0≤u≤20 \leq u \leq 20≤u≤2 and 0≤v≤40 \leq v \leq 40≤v≤4

Find the divergence of the vector field F given by F(x,y,z)=sin⁡(7x)i+cos⁡(6y)j+z3k^\mathbf { F } ( x , y , z ) = \sin ( 7 x ) \mathbf { i } + \cos ( 6 y ) \mathbf { j } + z ^ { 3 } \hat { \mathbf { k } }F(x,y,z)=sin(7x)i+cos(6y)j+z3k^

Evaluate ∬S4xydS\iint _ { S } 4 x y d S∬S​4xydS , where SSS is z=5−x−yz = 5 - x - yz=5−x−y , first octant.

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) f(x,y)=10x2+8xy+3y2f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }f(x,y)=10x2+8xy+3y2

Sketch the vector field F(x,y)=xi+3yj\mathbf { F } ( x , y ) = x \mathbf { i } + 3 y \mathbf { j }F(x,y)=xi+3yj

Evaluate ∬Sf(x,y,z)dS \iint_{S} f(x, y, z) d S ∬S​f(x,y,z)dS , Where f(x,y,z)=x2+y2+z2f(x,y,z)= {x^2}+{y^2}+{z^2}f(x,y,z)=x2+y2+z2 and S is given by z=x+y,x2+y2≤100 z=x+y, x^{2}+y^{2} \leq 100 z=x+y,x2+y2≤100

Green's Theorem to evaluate the integral ∫C11xydx+(x+y)dy\int _ { C } 11 x y d x + ( x + y ) d y∫C​11xydx+(x+y)dy for the path CCC : boundary of the region lying between the graphs of y=0y = 0y=0 and y=121−x2y = 121 - x ^ { 2 }y=121−x2 .

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function y=x37,0≤x≤4y = x ^ { \frac { 3 } { 7 } } , 0 \leq x \leq 4y=x73​,0≤x≤4 about the xxx -axis.

Use a computer algebra system to evaluate ∬sx2−4xydS \iint_{s} x^{2}-4 x y d S ∬s​x2−4xydS where S S S is z=4−x2−y2 z=4-x^{2}-y^{2} z=4−x2−y2 , 0≤x≤2,0≤y≤20 \leq x \leq 2,0 \leq y \leq 20≤x≤2,0≤y≤2 . Round your answer to two decimal places.

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

Find the area of the surface given by $\mathbf { r } ( u , v ) = 9 u \mathbf { i } - v \mathbf { j } + v \mathbf { k } \text {, where }$ $0 \leq u \leq 2$ and $0 \leq v \leq 4$

Find the divergence of the vector field F given by $\mathbf { F } ( x , y , z ) = \sin ( 7 x ) \mathbf { i } + \cos ( 6 y ) \mathbf { j } + z ^ { 3 } \hat { \mathbf { k } }$

Evaluate $\iint _ { S } 4 x y d S$ , where $S$ is $z = 5 - x - y$ , first octant.

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) $f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }$

Sketch the vector field $\mathbf { F } ( x , y ) = x \mathbf { i } + 3 y \mathbf { j }$

Evaluate $\iint_{S} f(x, y, z) d S$ , Where $f(x,y,z)= {x^2}+{y^2}+{z^2}$ and S is given by $z=x+y, x^{2}+y^{2} \leq 100$

Green's Theorem to evaluate the integral $\int _ { C } 11 x y d x + ( x + y ) d y$ for the path $C$ : boundary of the region lying between the graphs of $y = 0$ and $y = 121 - x ^ { 2 }$ .

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function $y = x ^ { \frac { 3 } { 7 } } , 0 \leq x \leq 4$ about the $x$ -axis.

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