Deck 21: Parameters, Coordinates, Integrals

Question 1

Find parametric equations for the sphere

Answer 1

Question 2

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.

A) $x = - 4 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 4 - 4 u - 2 v$
B) $x = - 4 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 4 + 4 u - 2 v$ .
C) $x = - 4 - 2 u + 4 v , y = 2 + 2 u - 4 v , z = 4 - 4 u + 2 v$
D) $x = - 2 u - 4 v , y = 2 u + 4 v , z = - 4 u - 2 v$
E) $x = - 4 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 4 - 4 u - 2 v$ ..

Answer 2

$x = - 4 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 4 - 4 u - 2 v$
$x = - 4 - 2 u + 4 v , y = 2 + 2 u - 4 v , z = 4 - 4 u + 2 v$
$x = - 4 - 2 u - 4 v , y = 2 + 2 u + 4 v , z = 4 - 4 u - 2 v$ ..

Question 3

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 (5, 0, 4).
Let P be the plane containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder).
In each case, give a parameterization and specify the range of values your parameters must take on.
(i)the circle in which the cylinder, S, cuts the plane, P.
(ii)the surface of the cylinder S.

Answer 3

(i)
(ii)

Question 4

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)?

Question 5

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 and 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.

Question 6

Let R be the region in the first quadrant bounded between the circle and the two axes.Then Let be the region in the first quadrant bounded between the ellipse and the two axes.
Use the change of variable x = s/5, y = t/3 to evaluate the integral

Question 7

Consider the change of variables x = s + 3t, y = s - 2t.
Let R be the region bounded by the lines 2x + 3y = 1, 2x + 3y = 4, x - y = -3, and x - y = 2.Find the region T in the st-plane that corresponds to region R.
Use the change of variables to evaluate .

Question 8

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.)

A)Part of a line through the origin
B)Disk
C)Part of a cylinder.
D)Part of a cone.

Question 9

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

A) $\vec { r } ( s , \theta ) = 7 \cos \overrightarrow { \theta i } + 7 \sin \theta \vec { j }$
B) $\vec { r } ( s , \theta ) = 7 \cos \theta \vec { i } + 7 \sin \theta \vec { j } + s \vec { k }$
C) $\vec { r } ( s , \theta ) = s \vec { i } + 49 \cos \theta \vec { j } + 49 \sin \theta \vec { k }$
D) $\vec { r } ( s , \theta ) = \operatorname { si } + 7 \cos \theta \vec { j } + 7 \sin \theta \vec { k }$
E) $\vec { r } ( s , \theta ) = \operatorname { si } + 7 \cos \theta \vec { j } + 7 \cos \theta \vec { k }$

Question 10

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.

A) $x = 1 + 2 t + 5 s , y = 2 - 5 t + s , z = - 1 + t + s$
B) $x = 1 + 2 t - 5 s , y = 2 - 5 t - s , z = - 1 + t - s$
C) $x = - 1 + 2 t + 5 s , y = - 2 - 5 t + s , z = 1 + t + s$
D) $x = 1 - 2 t + 5 s , y = 2 + 5 t + s , z = - 1 - t + s$

Question 11

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?

Question 12

Let and S be parametric surface oriented upward.
Use the formula for a flux integral over a parametric surface to find .

Question 13

Let and Find a vector which is perpendicular to and to find an equation of the plane through the point (1, 2, -1)and with normal vector perpendicular to both and .Express your answer in the form

Question 14

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)?

Question 15

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

A) $x = 3 \cos \theta , y = 3 \sin \theta , z = z , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
B) $x = 3 \cos \theta , y = 3 \sin \theta , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
C) $x = 9 \cos \theta , y = 9 \sin \theta , z = z , \quad 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
D) $x = 3 \sin \theta , y = 3 \cos \theta , z = z , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$
E) $x = 9 \cos \theta , y = 9 \sin \theta , 0 \leq \theta \leq 2 \pi , - \infty < z < \infty$

Question 16

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).

Question 17

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

A) $x = 5 \cos t , y = 7 \sin t , z = s , 0 \leq t \leq 2 \pi , - 7 \leq s \leq 5$
B) $x = \sqrt { 5 } \cos t _ { 1 } y = 7 \sin t _ { 1 } \quad z = t _ { 1 } , 0 \leq t \leq 2 \pi , - 7 \leq s \leq 5$
C) $x = \sqrt { 7 } \cos t _ { 1 } y = 5 \sin t _ { 1 } \quad z = s , \quad 0 \leq t \leq 2 \pi , - 7 \leq s \leq 5$
D) $x = \sqrt { 5 } \cos t _ { 1 } y = 7 \sin t , \quad z = s , \quad 0 \leq t \leq \pi , - 7 \leq s \leq 5 .$
E) $x = \sqrt { 5 } \cos t _ { 1 } y = 7 \sin t _ { 1 } \quad z = s , \quad 0 \leq t \leq 2 \pi , - 7 \leq s \leq 5$

Question 18

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.

Question 19

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.

A) $\frac { 1 } { \sqrt { 2 } } ( \vec { i } + \vec { k } )$
B) $\frac { 1 } { \sqrt { 3 } } ( \vec { i } + \vec { j } - \vec { k } )$
C) $\frac { 1 } { \sqrt { 2 } } ( \vec { i } + \vec { j } )$
D) $\frac { 1 } { \sqrt { 3 } } ( \vec { i } + \vec { j } + \vec { k } )$
E) $\frac { 1 } { \sqrt { 2 } } ( \vec { i } - \vec { j } )$

Question 20

Consider the change of variables x = s + 4t, y = s - 5t.
Find the absolute value of the Jacobian .

Question 21

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.

Question 22

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 $x = s + t ^ { 2 } , y = s ^ { 2 } , z = 4 t$

Question 23

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.

A) $\int _ { T } \frac { \left( x ^ { 2 } + 7 y ^ { 2 } \left( 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } \right) \right. } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y$
B) $\int _ { 0 } ^ { 2 \pi } \int _ { 1 } ^ { 2 } r ^ { 2 } ( 1 + r ) d r d \theta$
C) $\int _ { 0 } ^ { 2 \pi } \int _ { 1 } ^ { 2 } r ^ { 2 } ( 1 + r ) n d \theta d r$
D) $\int _ { 0 } ^ { 2 \pi } \int _ { 1 } ^ { 2 } r ^ { 2 } ( 1 + r ) \left( \cos ^ { 2 } \theta + b \sin ^ { 2 } \theta \right) d r d \theta$
E) $- \int _ { T } \frac { \left( x ^ { 2 } + y ^ { 2 } \right) \left( 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } \right) } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y$