Exam 17: Integrals and Vector Fields

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

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the $z$ -axis. Shell: upper hemisphere of $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ cut by the plane $z = 0$ Density: $\delta = 4$

Find the limit. - $r ( t ) = ( 3 - 2 t ) \mathbf { i } + t j ; 0 \leq t \leq \frac { 3 } { 2 }$

Find the gradient field of the function. - $f ( x , y , z ) = \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { x ^ { 5 } }$

Evaluate. The differential is exact. - $\int _ { ( 1,1,1 ) } ^ { ( 2,2,2 ) } - \frac { 3 z ^ { 8 } } { x ^ { 4 } y ^ { 2 } } d x - \frac { 2 z ^ { 8 } } { x ^ { 3 } y ^ { 3 } } d y + \frac { 8 z ^ { 7 } } { x ^ { 3 } y ^ { 2 } } d$

Calculate the area of the surface S. - $S$ is the lower portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49$ cut by the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Solve the problem. -The radial flow field of an incompressible fluid is shown below. For which of the closed paths is the circulation not necessarily zero?

Solve the problem. -Imagine a force field in which the force is always perpendicular to dr. What is special about the work done in moving a particle in such a field?

Calculate the flux of the field F across the closed plane curve C. - $\mathbf { F } = x \mathbf { i } + y \mathbf { j } ;$ the curve $C$ is the counterclockwise path around the circle $x ^ { 2 } + y ^ { 2 } = 16$

Find the equation for the plane tangent to the parametrized surface S at the point P. - $\mathrm { S } \text { is the paraboloid } r ( \theta , r ) = r \cos \theta i + r \sin \theta j + 5 r ^ { 2 } \mathbf { k } ; \mathrm { P } \text { is the point corresponding to } ( \theta , r ) = \left( \frac { \pi } { 4 } , 1 \right) \text {. }$

Test the vector field F to determine if it is conservative. - $\mathbf { F } = - \csc ^ { 2 } x \csc y \mathbf { i } - \cot x \cot y \csc y \mathbf { j } - \cos x \mathbf { k }$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } ( x , y , z ) = x z i + y z j + k , S$ is the cap cut from the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ by the plane $z = 3$ , direction is outward

Parametrize the surface S. - $S \text { is the cap cut from the paraboloid } z = \frac { 1 } { 12 } - 9 x ^ { 2 } - 9 y ^ { 2 } \text { by the cone } z = \sqrt { x ^ { 2 } + y ^ { 2 } } \text {. }$

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = 3 \mathrm { y } \mathbf { i } + \sqrt { \mathrm { z } } \mathbf { j } + ( 4 \mathrm { x } + 3 \mathrm { z } ) \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \mathrm { t } \mathbf { i } + \mathrm { t } ^ { 2 } \mathbf { j } + \mathrm { t } , 0 \leq \mathrm { t } \leq 2$

Find the surface area of the surface S. - $S$ is the intersection of the plane $3 x + 4 y + 12 z = 7$ and the cylinder with sides $y = 4 x ^ { 2 }$ and $y = 8 - 4 x ^ { 2 }$ .

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. - $\mathbf { F } = \left( x - e ^ { \mathrm { x } } \cos y \right) \mathbf { i } + \left( x + \mathrm { e } ^ { \mathrm { x } } \sin \mathrm { y } \right) \mathbf { j } ; \mathrm { C }$ is the lobe of the lemniscate $\mathrm { r } ^ { 2 } = \sin 2 \theta$ that lies in the first quadrant.

$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$ -Moment of inertia $I _ { x }$ about the $x$ -axis, where $\mathbf { r } ( t ) = \frac { 10 } { 3 } \sqrt { 2 } t 3 / 2 \mathbf { i } + ( 5 t \cos t ) j + ( 5 t \sin t ) \mathbf { k } , 0 \leq t \leq 1$ ; $\delta ( x , y , z ) = \frac { 9 } { y ^ { 2 } + z ^ { 2 } }$

Solve the problem. -In thermodynamics, the differential form of the internal energy of a system is $\mathrm { dU } = \mathrm { T } \mathrm { dS } - \mathrm { P } \mathrm { dV }$ , where $\mathrm { U }$ is the internal energy, $\mathrm { T }$ is the temperature, $\mathrm { S }$ is the entropy, $\mathrm { P }$ is the pressure, and $\mathrm { V }$ is the volume of the system. The First Law of Thermodynamics asserts that dU is an exact differential. Using this information, justify the thermodynamic relation $\frac { \partial \mathrm { T } } { \partial \mathrm { V } } = - \frac { \partial \mathrm { P } } { \partial \mathrm { S } }$ .

Evaluate the surface integral of G over the surface S. - $S \text { is the cylinder } y ^ { 2 } + z ^ { 2 } = 36 , z \geq 0 \text { and } 4 \leq x \leq 6 ; G ( x , y , z ) = z$

$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = \left( \frac { \sqrt { 13 } } { 2 } t ^ { 2 } - 7 \right) i + 6 t j , 0 \leq t \leq 1 ; \delta = 3 t$