Exam 17: Integrals and Vector Fields

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = 6 x \mathbf { i } - 6 \mathrm { x } ^ { 5 } \mathrm { y } ^ { 4 } \mathbf { j } + \left( - 3 \mathrm { z } - 7 \mathrm { y } ^ { 2 } \right) \mathbf { k }$ ; the path is $\mathrm { C } _ { 1 } \cup \mathrm { C } _ { 2 } \cup \mathrm { C } _ { 3 }$ where $\mathrm { C } _ { 1 }$ is the straight line from $( 0,0,0 )$ to $( 1,0,0 ) , \mathrm { C } _ { 2 }$ is the straight line from $( 1,0,0 )$ to $( 1,1,0 )$ , and $C _ { 3 }$ is the straight line from $( 1,1,0 )$ to $( 1,1,1 )$

Find the gradient field of the function. - $f ( x , y , z ) = \left( \frac { x + y } { y + z } \right) ^ { 6 }$

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the z-axis. Shell: portion of the cone $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 0$ between $z = 1$ and $z = 2$ Density: $\delta = 1$

Evaluate the line integral along the curve C. - $\int _ { C } \left( \frac { 4 } { z } \right) ^ { 1 / 3 } \mathrm { ds } , \mathrm { C }$ is the curve $\mathbf { r } ( \mathrm { t } ) = \left( 4 \mathrm { t } ^ { 3 } \cos \mathrm { t } \right) \mathbf { i } + \left( 4 \mathrm { t } ^ { 3 } \sin \mathrm { t } \right) \mathbf { j } + 4 \mathrm { t } ^ { 3 } \mathbf { k } , 0 \leq \mathrm { t } \leq 3 \sqrt { 2 }$

Evaluate the surface integral of G over the surface S. - $S$ is the dome $z = 7 - 10 x ^ { 2 } - 10 y ^ { 2 } , z \geq 0 ; G ( x , y , z ) = \frac { 1 } { \sqrt { 400 \left( x ^ { 2 } + y ^ { 2 } \right) + 1 } }$

Evaluate the line integral of f(x,y) along the curve C. - $f ( x , y ) = \cos x + \sin y , C : y = x , 0 \leq x \leq \frac { \pi } { 2 }$

Test the vector field F to determine if it is conservative. - $\mathrm { F } = 7 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 5 } \mathrm { i } + \left( 5 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 4 } + \frac { \mathrm { z } ^ { 4 } } { \mathrm { y } ^ { 2 } } \right) \mathrm { j } - \frac { 4 \mathrm { z } ^ { 3 } } { \mathrm { y } } \mathbf { k }$

Solve the problem. -For some inexact differential forms df, a function $g ( x , y , z )$ can be found such that dh $= g ( x , y , z ) d f$ is exact. When it exists, the function $g ( x , y , z )$ is called an "integrating factor". Show that $g ( x , y , z ) = \frac { y z } { x }$ is an integrating factor for the inexact differential $\mathrm { df } = - \frac { 1 } { \mathrm { x } } \mathrm { dx } + \frac { 1 } { \mathrm { y } } \mathrm { dy } + \frac { 1 } { \mathrm { z } } \mathrm { dz }$ .

Solve the problem. -Show that the value of the integral does not depend on the path taken from A to B. $\int _ { A } ^ { B } z ^ { 2 } d x + 3 y d y + 2 x z d z$

Find the limit. - $r ( t ) = 3 i - 2 j - t k ; - 1 \leq t \leq 1$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = 7 \mathrm { x } \mathbf { i } + 7 \mathrm { yj } + 2 \mathbf { k } , \mathrm { S }$ is the surface cut from the bottom of the paraboloid $\mathrm { z } = \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 }$ by the plane $z = 2$ , direction is outward

Solve the problem. -The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell: portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that lies in the first octant Density: constant

Evaluate the line integral along the curve C. - $\int _ { C } \frac { x + y + z } { 5 } d s , C$ is the curve $r ( t ) = 4 t i + \left( 6 \cos \frac { 1 } { 2 } t \right) j + \left( 6 \sin \frac { 1 } { 2 } t \right) k , 0 \leq t \leq 2 \pi$

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle $x ^ { 2 } + y ^ { 2 } = 4$ . - $F = \frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \mathrm { i } + \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } j$

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. - $\mathbf { F } = - 9 \mathrm { y } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 3 } \mathbf { j } + 8 \mathrm { z } ^ { 3 } \mathbf { k }$ ; C: the portion of the paraboloid $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = \mathrm { z }$ cut by the cylinder $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

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

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } 8 x e ^ { 4 x ^ { 2 } + 9 y ^ { 2 } + 6 z ^ { 2 } } d x + 18 y e ^ { 4 x ^ { 2 } + 9 y ^ { 2 } + 6 z ^ { 2 } } d y + 12 z e ^ { 4 x ^ { 2 } + 9 y ^ { 2 } + 6 z ^ { 2 } } d z$