Exam 21: Parameters, Coordinates, Integrals

Find the parametric equation of the plane through the point (-4, 2, 4)and parallel to the lines $\vec { r } ( t ) = ( 1 - 2 t ) \vec { i } + ( 5 + 2 t ) \vec { j } + ( 3 - 4 t ) \vec { k }$ and $\vec { s } ( t ) = ( 3 - 4 t ) \vec { i } + 4 t \vec { j } + ( 4 - 2 t ) \vec { k }$ Select all that apply.

Consider the parametric surface $\vec { r } ( s , t ) = s \sin \left( \frac { \pi } { 2 } \right) \vec { i } + s \cos \left( \frac { \pi } { 2 } \right) \vec { j } + 4 t \vec { k }$ Does it contain the x-axis?

Compute $\int _ { S } ( z \vec { i } + x \vec { j } + y \vec { k } ) \cdot \overrightarrow { d A }$ , where S is oriented in the positive $\vec { j }$ direction and given, for 0 $\le$ s $\le$ 1, 0 $\le$ t $\le$ 2, by x = s, y = t², z =8 t.

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (2, 0, 7). Find the xyz-equation of the plane, P, containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder).

Consider the plane $\vec { r } ( s , t ) = ( 2 + s - 4 t ) \vec { i } + ( 5 - s + 4 t ) \vec { j } + ( 6 - 4 t - s ) \vec { k }$ Does it contain the point (1, 6, -1)?

Find parametric equations for the cylinder $49 x ^ { 2 } + 5 y ^ { 2 } = 245 , - 7 \leq z \leq 5$

Let ${ \vec { F } } = x ( 1 + z ) \vec { i } + 7 y ( 1 + z ) \vec { j }$ Show that the parametric surface S given by x = s cos t, y = s sin t, z = s, for 1 $\le$ s $\le$ 2, 0 $\le$ t $\le$ 2 $\pi$ , oriented downward can also be written as the surface $z = \sqrt { x ^ { 2 } + y ^ { 2 } } , 1 \leq z \leq 2$ . Which of the following iterated integrals calculates the flux of $\vec { F }$ across S? Select all that apply.

Find parametric equations for the cylinder $y ^ { 2 } + z ^ { 2 } = 49$

Let R be the region in the first quadrant bounded between the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the two axes.Then $\int _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A = \frac { \pi } { 8 }$ Let $\bar { R }$ be the region in the first quadrant bounded between the ellipse $25 x ^ { 2 } + 9 y ^ { 2 } = 1$ and the two axes. Use the change of variable x = s/5, y = t/3 to evaluate the integral $\int _ { x } \left( 75 x ^ { 2 } + 27 y ^ { 2 } \right) d A$

Using cylindrical coordinates, find parametric equations for the cylinder $x ^ { 2 } + y ^ { 2 } = 9$ Select all that apply.

Consider the plane $\vec { r } ( s , t ) = ( 3 + s - 5 t ) \vec { i } + ( 5 - s + 5 t ) \vec { j } + ( 5 - 10 t + s ) \vec { k }$ Find a normal vector to the plane.

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (4, 0, 7). Find two unit vectors $\vec { u }$ and $\vec { v }$ in the plane, P, containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder)which are perpendicular to each other.

The following equations represent a curve or a surface.Select the best geometric description. (Note: $\rho$ , $\varphi$ , $\theta$ are spherical coordinates; r, $\theta$ , z are cylindrical coordinates.)

Let ${ \vec { F } } = x ( 1 + z ) \vec { i } + 2 y ( 1 + z ) \vec { j }$ Find the flux of $\vec { F }$ across the parametric surface S given by x = s cos t, y = s sin t, z = s, for 1 $\le$ s $\le$ 2, 0 $\le$ t $\le$ 2 $\pi$ , oriented downward.

Let $\vec { v } _ { 1 } = 2 \vec { i } - 5 \vec { j } + \vec { k }$ and $\vec { v } _ { 2 } = 5 \vec { i } + \vec { j } + \vec { k }$ Find a parametric equation for the plane through the point (1, 2, -1)and containing the vectors $\vec { v } _ { 1 }$ and $\vec { v } _ { 2 }$ Select all that apply.

Let ${ \vec { F } } = x \vec { i } + 5 y \vec { j } + z \vec { k }$ and S be parametric surface $\vec { r } ( s , t ) = s \vec { i } + t \vec { j } + ( 4 s + t - 4 ) \vec { k } , \quad 0 \leq s \leq 1,0 \leq t \leq 2$ oriented upward. Use the formula for a flux integral over a parametric surface to find $\int _ { S } \vec { F } \cdot d \vec { A }$ .

Find parametric equations for the sphere $( x - 4 ) ^ { 2 } + ( y + 4 ) ^ { 2 } + ( z - 8 ) ^ { 2 } = 16$

Consider the change of variables x = s + 4t, y = s - 5t. Find the absolute value of the Jacobian $\left| \frac { \partial ( x , y ) } { \partial ( s , t ) } \right|$ .

Compute the flux of the vector field $\overrightarrow { \vec { H } } = \vec { y } i + 2 z \vec { j }$ over the surface S, which is oriented upward and given, for 0 $\le$ s $\le$ 1, 0 $\le$ t $\le$ 2 by $x = s + t ^ { 2 } , y = s ^ { 2 } , z = 4 t$

Consider the parametric surface $\vec { r } ( s , t ) = s \sin \left( \frac { \pi } { 2 } \right) \vec { i } + s \cos \left( \frac { \pi } { 2 } \right) \vec { j } + 4 t \vec { k }$ Does it contain the point (0, -2, 0)?