Exam 17: Integrals and Vector Fields

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. - $F = - \frac { 1 } { 8 \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 4 } } \mathrm { i } ; \mathrm { C }$ is the region defined by the polar coordinate inequalities $1 \leq \mathrm { r } \leq 3$ and $0 \leq \theta \leq \pi$

Evaluate the line integral of f(x,y) along the curve C. - $f ( x , y ) = 2 y ^ { 2 } , C : y = e ^ { - x } , 0 \leq x \leq 4$

$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$ -Moment of inertia $\mathrm { I } _ { \mathrm { Z } }$ about the $\mathrm { z }$ -axis, where $r ( t ) = ( 4 \sin 3 t ) i + ( 4 \cos 3 t ) j + e ^ { 2 t } k , 0 \leq t \leq 1 ; \delta ( x , y , z ) = z ^ { 2 }$

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = \frac { \mathrm { y } } { \mathrm { z } } \mathbf { i } + \frac { \mathrm { x } } { \mathrm { z } } \mathbf { j } + \frac { \mathrm { x } } { \mathrm { y } } \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \mathrm { t } ^ { 7 } \mathbf { i } + \mathrm { t } ^ { 5 } \mathbf { j } + \mathrm { t } ^ { 2 } \mathbf { k } , 0 \leq \mathrm { t } \leq 1$

Calculate the circulation of the field F around the closed curve C. - $\mathbf { F } = \mathrm { xy } \mathbf { i } + 3 \mathbf { j }$ , curve $\mathrm { C }$ is $\mathbf { r } ( \mathrm { t } ) = 4 \cos \mathrm { ti } + 4 \sin \mathrm { tj } , 0 \leq \mathrm { t } \leq 2 \pi$

Find the surface area of the surface S. - $S$ is the portion of the paraboloid $z = 81 - x ^ { 2 } - y ^ { 2 }$ that lies above the ring $16 \leq x ^ { 2 } + y ^ { 2 } \leq 36$ in the $x y$ -plane.

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

Find the limit. - $\mathbf { r } ( \mathrm { t } ) = 2 \cos \mathrm { ti } + \sin \mathrm { tk } ; 0 \leq t \leq \pi$

Solve the problem. -Find a field $G = P ( x , y ) i + Q ( x , y ) j$ in the $x y$ -plane with the property that at any point $( a , b ) z ( 0,0 ) , G$ is a vector of magnitude $\sqrt { \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 } }$ tangent to the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = \mathrm { a } ^ { 2 } + \mathrm { b } ^ { 2 }$ and pointing in the clockwise direction.

Calculate the area of the surface S. - $S$ is the portion of the paraboloid $z = 3 x ^ { 2 } + 3 y ^ { 2 }$ that lies between $z = 4$ and $z = 5$ .

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } = 7 x \mathbf { i } + 7 y \mathbf { j } + \mathrm { z } \mathbf { k }$ ; $\mathrm { S }$ is portion of the plane $x + y + z = 4$ for which $0 \leq x \leq 3$ and $0 \leq y \leq 4$ ; direction is outward (away from origin)

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = xyi + yzj + xzk , S is the surface of the rectangular prism formed from the coordinate planes and the planes x = 1, y = 4, and z = 4, direction is outward

Evaluate the surface integral of the function g over the surface S. - $G ( x , y , z ) = \frac { y } { \sqrt { 36 y ^ { 2 } + 1 } } ; S$ is the surface of the parabolic cylinder $30 y ^ { 2 } + 10 z = 80$ bounded by the planes $x = 0 , x =$ 1 , $y = 0$ , and $z = 0$

Solve the problem. -Consider a small region inside an elastic material such as gelatin. As the material "jiggles", this small region oscillates about its equilibrium position $\left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ . The force that tends to restore the small region to its equilibrium position can be approximated as $F = - k \left( x - x _ { 0 } \right) i - k \left( y - y _ { 0 } \right) \mathbf { j } - k ( z - z 0 ) \mathbf { k }$ Find a potential function $f$ for this force field.

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = - 7 \mathrm { zi } + 6 x \mathbf { j } + 4 y \mathbf { k } ; S$ is the portion of the cone $z = 6 \sqrt { x ^ { 2 } + y ^ { 2 } }$ below the plane $z = 2$

Calculate the area of the surface S. - $S$ is the portion of the cone $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 9 } = \frac { z ^ { 2 } } { 25 }$ that lies between $z = 4$ and $z = 5$ .

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = x y \mathbf { i } + \mathrm { y } ^ { 2 } \mathbf { j } - 2 y z \mathbf { k }$ ; $\mathrm { D }$ : the solid wedge cut from the first quadrant by the plane $\mathrm { y } + \mathrm { z } = 2$ and the parabolic cylinder $x = 1 - 16 y ^ { 2 }$

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = -6i + 2j + 3k , S is the rectangular surface z = 0, 0 x 3, and 0 y 5, direction k

Solve the problem. -The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell: cylinder $x ^ { 2 } + z ^ { 2 } = 64$ bounded by $y = 0$ and $y = 6$ Density: constant

Calculate the circulation of the field F around the closed curve C. - $\mathbf { F } = x ^ { 2 } y ^ { 3 } \mathbf { i } + x ^ { 2 } y ^ { 3 } \mathbf { j } ;$ curve $C$ is the counterclockwise path around the rectangle with vertices at $( 0,0 ) , ( 4,0 ) , ( 4,2 )$ , and $( 0,2 )$