Exam 17: Integrals and Vector Fields

Calculate the area of the surface S. - $S$ is the cap cut from the paraboloid $z = \frac { 2 } { 5 } - 10 x ^ { 2 } - 10 y ^ { 2 }$ by the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

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

Solve the problem. -Let $f ( x , y ) = \ln \left( x ^ { 2 } + y ^ { 2 } \right)$ . Which one of the following curves is a simple closed path in the domain of the function $\mathrm { f }$ ?

Apply Green's Theorem to evaluate the integral. - $\oint _ { C } \left( 7 x + y ^ { 3 } \right) d x + \left( 3 x y ^ { 2 } + 7 y \right) d y$ C: Any simple closed curve in the plane for which Green's Theorem holds

Evaluate the surface integral of G over the surface S. -S is the plane $x + y + z = 1$ above the rectangle $0 \leq x \leq 3$ and $0 \leq y \leq 5 ; G ( x , y , z ) = 6 z$

Parametrize the surface S. - $S \text { is the portion of the paraboloid } z = 5 x ^ { 2 } + 5 y ^ { 2 } \text { that lies between } z = 2 \text { and } z = 7 \text {. }$

Evaluate. The differential is exact. - $\int _ { ( 1,1,1 ) } ^ { ( 2,5,5 ) } \frac { 3 x ^ { 2 } } { y ^ { 3 } } d x - \frac { 3 x ^ { 3 } } { y ^ { 4 } } d y + \frac { 1 } { z } d z$

Test the vector field F to determine if it is conservative. - $\mathbf { F } = \left( \frac { 9 x ^ { 8 } \left( y - x ^ { 2 } \right) ^ { 9 } } { z } \right) i + \left( \frac { 5 x ^ { 9 } \left( y - x ^ { 2 } \right) ^ { 8 } } { z } \right) j - \left( \frac { x ^ { 9 } \left( y - x ^ { 2 } \right) ^ { 5 } } { z ^ { 2 } } \right) \mathbf { k }$

Solve the problem. - $\text { Show that } \mathrm { df } = - \frac { 5 \left( \mathrm { y } ^ { 7 } + \mathrm { z } \right) } { \mathrm { x } ^ { 6 } \mathrm { z } ^ { 4 } } \mathrm { dx } + \frac { 7 \mathrm { y } ^ { 6 } } { \mathrm { x } ^ { 5 } \mathrm { z } ^ { 4 } } \mathrm { dy } - \left( \frac { 4 \mathrm { y } ^ { 7 } } { \mathrm { x } ^ { 5 } \mathrm { z } ^ { 5 } } + \frac { 3 } { \mathrm { x } ^ { 5 } \mathrm { z } ^ { 4 } } \right) \mathrm { dz } \text { is exact. }$

Find the gradient field of the function. - $f ( x , y , z ) = \left( x ^ { 8 } y ^ { 2 } - y ^ { 9 } z ^ { 3 } \right) e ^ { - z ^ { 2 } }$

Solve the problem. -Find values for $a$ , b, and c so that $\nabla \times \mathbf { F } = \mathbf { 0 }$ for $\mathbf { F } = a x i + b y j + c z k$ .

Evaluate the line integral along the curve C. - $\int _ { C } ( x y + y z ) d s , C$ is the path from $( 1,1,0 )$ to $\left( e ^ { 2 } , e ^ { 2 } , 1 \right)$ given by: $C _ { 1 } : \mathbf { r } ( \mathrm { t } ) = \mathrm { e } ^ { 2 } \mathrm { t } _ { \mathbf { i } } + \mathrm { e } ^ { 2 } \mathrm { t } \mathbf { j } , 0 \leq \mathrm { t } \leq 1$ $C _ { 2 } : \mathbf { r } ( \mathrm { t } ) = \mathrm { e } ^ { 2 } \mathbf { i } + \mathrm { e } ^ { 2 } \mathbf { j } + 9 \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq 1$

$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = \left( 5 t ^ { 2 } - 1 \right) \mathbf { i } + 4 t k , - 1 \leq t \leq 1 ; \delta ( x , y , z ) = 3 \sqrt { 20 x + 36 }$

$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$ -Moment of inertia $\mathrm { I } _ { \mathrm { y } }$ about the $\mathrm { y }$ -axis, where $\mathbf { r } ( \mathrm { t } ) = ( 4 - 4 \mathrm { t } ) \mathrm { i } + 3 \mathrm { t } \mathrm { j } , 0 \leq \mathrm { t } \leq 1 ; \delta ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = 4$

Evaluate the line integral of f(x,y) along the curve C. - $f ( x , y ) = y + x , C : x ^ { 2 } + y ^ { 2 } = 100 \text { in the first quadrant from } ( 10,0 ) \text { to } ( 0,10 )$

Solve the problem. -Suppose that the parametrized plane curve $C : ( f ( u ) , g ( u ) )$ is revolved about the $x$ -axis, where $g ( u ) > 0$ and a $\leq u \leq$ b. Show that the surface area of the surface of revolution is $2 \pi \int _ { a } ^ { b } g ( u ) \sqrt { \left[ g ^ { \prime } ( u ) \right] ^ { 2 } + \left[ f ^ { \prime } ( u ) \right] ^ { 2 } } \mathrm { du }$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = 12 \mathrm { x } + 12 \mathrm { y } \mathbf { j } + 12 \mathrm { z } \mathbf { k } , \mathrm { S }$ is the surface of the sphere $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 1$ in the first octant, direction away from the origin

Evaluate the line integral along the curve C. - $\int _ { C } \left( 18 x ^ { 2 } + 5 e ^ { 5 y } + \frac { 1 } { z + 1 } \right) d s , C$ is the path from $( 0,0,0 )$ to $( 1,1,1 )$ given by: $C _ { 1 } : r ( t ) = t i , 0 \leq t \leq 1$ $C _ { 2 } : r ( t ) = i + t j , 0 \leq t \leq 1$ $C _ { 3 } : r ( t ) = i + j + t k , 0 \leq t \leq 1$

Evaluate the line integral along the curve C. - $\int _ { C } \left( x z + y ^ { 2 } \right) d s , C$ is the curve $r ( t ) = ( 8 - 2 t ) i + t j - 2 t k , 0 \leq t \leq 1$

Using Green's Theorem, calculate the area of the indicated region. -The area bounded above by $y = 3 x$ and below by $y = 10 x ^ { 2 }$