Calculus Multivariable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 21

Parameters, Coordinates, and Integrals - all with Video Answers

Educators

Section 1

Coordinates and Parameterized Surfaces

Problem 1

Decide if the parameterization describes a curve or a surface.
$$\vec{r}(s)=s \vec{i}+(3-s) \vec{j}+s^{2} \vec{k}$$

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Decide if the parameterization describes a curve or a surface.
$$\vec{r}(s, t)=(s+t) \vec{i}+(3-s) \vec{j}$$

Lucas Finney

Numerade Educator

Problem 3

Decide if the parameterization describes a curve or a surface.
$$\vec{r}(s, t)=\cos s \vec{i}+\sin s \vec{j}+t^{2} \vec{k}$$

Lucas Finney

Numerade Educator

Problem 4

Decide if the parameterization describes a curve or a surface.
$$\vec{r}(s)=\cos s \vec{i}+\sin s \vec{j}+s^{2} \vec{k}$$

Lucas Finney

Numerade Educator

Problem 5

Describe in words the objects parameterized by the equations. (Note: $r$ and $\theta$ are cylindrical coordinates.)
$$\begin{array}{lll}
x=r \cos \theta & y=r \sin \theta & z=7 \\
0 \leq r \leq 5 & 0 \leq \theta \leq 2 \pi
\end{array}$$

Lucas Finney

Numerade Educator

Problem 6

Describe in words the objects parameterized by the equations. (Note: $r$ and $\theta$ are cylindrical coordinates.)
$$\begin{array}{lll}
x=5 \cos \theta & y=5 \sin \theta & z=z \\
0 \leq \theta \leq 2 \pi & 0 \leq z \leq 7
\end{array}$$

Lucas Finney

Numerade Educator

Problem 7

Describe in words the objects parameterized by the equations. (Note: $r$ and $\theta$ are cylindrical coordinates.)
$$\begin{array}{lll}
x=5 \cos \theta & y=5 \sin \theta & z=5 \theta \\
0 \leq \theta \leq 2 \pi &
\end{array}$$

Lucas Finney

Numerade Educator

Problem 8

Describe in words the objects parameterized by the equations. (Note: $r$ and $\theta$ are cylindrical coordinates.)
$$\begin{aligned}
&x=r \cos \theta \quad y=r \sin \theta \quad z=r\\
&0 \leq r \leq 5 \quad 0 \leq \theta \leq 2 \pi
\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 9

For a sphere parameterized using the spherical coordinates $\theta$ and $\phi,$ describe in words the part of the sphere given by the restrictions.
$$0 \leq \theta<2 \pi, \quad 0 \leq \phi \leq \pi / 2$$

Lucas Finney

Numerade Educator

Problem 10

For a sphere parameterized using the spherical coordinates $\theta$ and $\phi,$ describe in words the part of the sphere given by the restrictions.
$$\pi \leq \theta<2 \pi, \quad 0 \leq \phi \leq \pi$$

Lucas Finney

Numerade Educator

Problem 11

For a sphere parameterized using the spherical coordinates $\theta$ and $\phi,$ describe in words the part of the sphere given by the restrictions.
$$\pi / 4 \leq \theta<\pi / 3, \quad 0 \leq \phi \leq \pi$$

Lucas Finney

Numerade Educator

Problem 12

For a sphere parameterized using the spherical coordinates $\theta$ and $\phi,$ describe in words the part of the sphere given by the restrictions.
$$0 \leq \theta \leq \pi, \quad \pi / 4 \leq \phi<\pi / 3$$

Lucas Finney

Numerade Educator

Problem 13

Give parametric equations for the plane through the point with position vector $\vec{r}_{0}$ and containing the vectors $\vec{v}_{1}$ and $\vec{v}_{2}$.
$$\vec{r}_{0}=\vec{i}, \vec{v}_{1}=\vec{j}, \vec{v}_{2}=\vec{k}$$

Lucas Finney

Numerade Educator

Problem 14

Give parametric equations for the plane through the point with position vector $\vec{r}_{0}$ and containing the vectors $\vec{v}_{1}$ and $\vec{v}_{2}$.
$$\vec{r}_{0}=\vec{j}, \vec{v}_{1}=\vec{k}, \vec{v}_{2}=\vec{i}$$

Lucas Finney

Numerade Educator

Problem 15

Give parametric equations for the plane through the point with position vector $\vec{r}_{0}$ and containing the vectors $\vec{v}_{1}$ and $\vec{v}_{2}$.
$$\vec{r}_{0}=\vec{i}+\vec{j}, \vec{v}_{1}=\vec{j}+\vec{k}, \vec{v}_{2}=\vec{i}+\vec{k}$$

Lucas Finney

Numerade Educator

Problem 16

Give parametric equations for the plane through the point with position vector $\vec{r}_{0}$ and containing the vectors $\vec{v}_{1}$ and $\vec{v}_{2}$.
$$\vec{r}_{0}=\vec{i}+\vec{j}+\vec{k}, \vec{v}_{1}=\vec{i}-\vec{k}, \vec{v}_{2}=-\vec{j}+\vec{k}$$

Lucas Finney

Numerade Educator

Problem 17

Parameterize the plane that contains the three points.
$$(0,0,0),(1,2,3),(2,1,0)$$

Lucas Finney

Numerade Educator

Problem 18

Parameterize the plane that contains the three points.
$$(1,2,3),(2,5,8),(5,2,0)$$

Lucas Finney

Numerade Educator

Problem 19

Give two nonparallel vectors and the coordinates of a point in the plane with given parametric equations
$$x=2 s+3 t, \quad y=s-5 t, \quad z=-s+2 t$$

Lucas Finney

Numerade Educator

Problem 20

Give two nonparallel vectors and the coordinates of a point in the plane with given parametric equations
$$x=2+s+t, \quad y=s-t, \quad z=-1+s+t$$

Lucas Finney

Numerade Educator

Problem 21

Parameterize the plane through the point with the given normal vector.
$$(3,5,7), \vec{i}+\vec{j}+\vec{k}$$

Lucas Finney

Numerade Educator

Problem 22

Parameterize the plane through the point with the given normal vector.
$$(5,1,4), \vec{i}+2 \vec{j}+3 \vec{k}$$

Lucas Finney

Numerade Educator

Problem 23

Does the plane $\vec{r}(s, t)=(2+s) \vec{i}+(3+s+t) \vec{j}+4 t \vec{k}$
contain the following points?
(a) $\quad(4,8,12)$
(b) $\quad(1,2,3)$

Lucas Finney

Numerade Educator

Problem 24

Are the following two planes parallel?
$$\begin{array}{c}x=2+s+t, \quad y=4+s-t, \quad z=1+2 s, \quad \text { and } \\
x=2+s+2 t, \quad y=t, \quad z=s-t\end{array}$$.

Lucas Finney

Numerade Educator

Problem 25

Describe the families of parameter curves with $s=s_{0}$ and $t=t_{0}$ for the parameterized surface.
$$\begin{aligned}&x=s, \quad y=t, \quad z=1 \text { for }-\infty<s<\infty,-\infty<t<\\&\infty\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 26

Describe the families of parameter curves with $s=s_{0}$ and $t=t_{0}$ for the parameterized surface.
$$\begin{aligned}&x=s, \quad y=\cos t, \quad z=\sin t \text { for }-\infty<s<\infty\\&0 \leq t \leq 2 \pi\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 27

Describe the families of parameter curves with $s=s_{0}$ and $t=t_{0}$ for the parameterized surface.
$$\begin{aligned}&x=s \quad y=t, \quad z=s^{2}+t^{2} \text { for }-\infty<s<\infty\\&-\infty<t<\infty\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 28

Describe the families of parameter curves with $s=s_{0}$ and $t=t_{0}$ for the parameterized surface.
$$\begin{aligned}&x=\cos s \sin t, \quad y=\sin s \sin t, \quad z=\cos t \text { for } 0 \leq\\&s \leq 2 \pi, 0 \leq t \leq \pi\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 29

A city is described parametrically by the equation
$$\vec{r}=\left(x_{0} \vec{i}+y_{0} \vec{j}+z_{0} \vec{k}\right)+s \vec{v}_{1}+t \vec{v}_{2}$$
where $\vec{v}_{1}=2 \vec{i}-3 \vec{j}+2 \vec{k}$ and $\vec{v}_{2}=\vec{i}+4 \vec{j}+5 \vec{k} . \mathrm{A}$
city block is a rectangle determined by $\vec{v}_{1}$ and $\vec{v}_{2} .$ East is in the direction of $\vec{v}_{1}$ and north is in the direction of $\vec{v}_{2} .$ Starting at the point $\left(x_{0}, y_{0}, z_{0}\right),$ you walk 5 blocks east, 4 blocks north, 1 block west and 2 blocks south. What are the parameters of the point where you end up? What are your $x, y$ and $z$ coordinates at that point?

Lucas Finney

Numerade Educator

Problem 30

You are at a point on the earth with longitude $80^{\circ}$ West of Greenwich, England, and latitude $40^{\circ}$ North of the equator.
(a) If your latitude decreases, have you moved nearer to or farther from the equator?
(b) If your latitude decreases, have you moved nearer to or farther from the north pole?
(c) If your longitude increases (say, to $90^{\circ}$ West), have you moved nearer to or farther from Greenwich?

Lucas Finney

Numerade Educator

Problem 31

Describe in words the curve $\phi=\pi / 4$ on the surface of the globe.

Lucas Finney

Numerade Educator

Problem 32

Describe in words the curve $\theta=\pi / 4$ on the surface of the globe.

Lucas Finney

Numerade Educator

Problem 33

A decorative oak post is $48^{\prime \prime}$ long and is turned on a lathe so that its profile is sinusoidal, as shown in Figure 21.17.
(a) Describe the surface of the post parametrically using cylindrical coordinates.
(b) Find the volume of the post.
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 34

Find parametric equations for the sphere $(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=d^{2}$.

Lucas Finney

Numerade Educator

Problem 35

You are standing at a point on the equator of a sphere parameterized by spherical coordinates $\theta_{0}$ and $\phi_{0} .$ If you go halfway around the equator and halfway up toward the north pole along a longitude, what are your new $\theta$ and $\phi$ coordinates?

Lucas Finney

Numerade Educator

Problem 36

Find parametric equations for the cone $x^{2}+y^{2}=z^{2}$.

Lucas Finney

Numerade Educator

Problem 37

Parameterize the cone in Example 6 on page 1028 in terms of $r$ and $\theta$.

Lucas Finney

Numerade Educator

Problem 38

Give a parameterization of the circle of radius $a$ centered at the point $\left(x_{0}, y_{0}, z_{0}\right)$ and in the plane parallel to two given unit vectors $\vec{u}$ and $\vec{v}$ such that $\vec{u} \cdot \vec{v}=0$.

Bradley Duda

Numerade Educator

Problem 39

(a) Write an equation in $x, y, z$ and identify the parametric surface.
(b) Draw a picture of the surface.
$$\begin{array}{ll}
x=2 s & y=s+t \\
0 \leq s \leq 1 & 0 \leq t \leq 1
\end{array} z=1+s-t$$

Lucas Finney

Numerade Educator

Problem 40

(a) Write an equation in $x, y, z$ and identify the parametric surface.
(b) Draw a picture of the surface.
$$\begin{aligned}
&x=s \quad y=t \quad z=\sqrt{1-s^{2}-t^{2}}\\
&s^{2}+t^{2} \leq 1 \quad s, t \geq 0
\end{aligned}$$

Lucas Finney

Numerade Educator

Problem 41

(a) Write an equation in $x, y, z$ and identify the parametric surface.
(b) Draw a picture of the surface.
$$\begin{array}{lll}
x=s+t & y=s-t & z=s^{2}+t^{2} \\
0 \leq s \leq 1 & 0 \leq t \leq 1
\end{array}$$

Lucas Finney

Numerade Educator

Problem 42

Explain what is wrong with the statement.
The parameter curves of a parameterized surface intersect at right angles.

Lucas Finney

Numerade Educator

Problem 43

Explain what is wrong with the statement.
The parameter curves for constant $\phi$ on the sphere $\vec{r}(\theta, \phi)=R \sin \phi \cos \theta \vec{i}+R \sin \phi \sin \theta \vec{j}+R \cos \phi \vec{k}$
are circles of radius $R$.

Lucas Finney

Numerade Educator

Problem 44

Give an example of:
A parameterization $\vec{r}(s, t)$ of the plane tangent to the unit sphere at the point where $\theta=\pi / 4$ and $\phi=\pi / 4$.

Fuzail Shakir

Numerade Educator

Problem 45

Give an example of:
An equation of the form $f(x, y, z)=0$ for the plane
$$\vec{r}(s, t)=(s+1) \vec{i}+(t+2) \vec{j}+(s+t) \vec{k}$$.

Lucas Finney

Numerade Educator

Problem 46

Give an example of:
A parameterized curve on the sphere $\vec{r}(\theta, \phi)=$ $\sin \phi \cos \theta \vec{i}+\sin \phi \sin \theta \vec{j}+\cos \phi \vec{k}$ that is not a parameter curve.

Carson Merrill

Numerade Educator

Problem 47

Are the statements true or false? Give reasons for your answer.
The equations $x=s+1, y=t-2, z=3$ parameterize a plane.

Lucas Finney

Numerade Educator

Problem 48

Are the statements true or false? Give reasons for your answer.
The equations $x=2 s-1, y=-s+3, z=4+s$ parameterize a plane.

Lucas Finney

Numerade Educator

Problem 49

Are the statements true or false? Give reasons for your answer.
If $\vec{r}=\vec{r}(s, t)$ parameterizes the upper hemisphere $x^{2}+y^{2}+z^{2}=1, z \geq 0,$ then $\vec{r}=-\vec{r}(s, t)$ parameterizes the lower hemisphere $x^{2}+y^{2}+z^{2}=1, z \leq 0$.

Lucas Finney

Numerade Educator

Problem 50

Are the statements true or false? Give reasons for your answer.
If $\vec{r}=\vec{r}(s, t)$ parameterizes the upper hemisphere $x^{2}+y^{2}+z^{2}=1, z \geq 0,$ then $\vec{r}=\vec{r}(-s,-t)$ parameterizes the lower hemisphere $x^{2}+y^{2}+z^{2}=1, z \leq 0$.

Lucas Finney

Numerade Educator

Problem 51

Are the statements true or false? Give reasons for your answer.
If $\vec{r}_{1}(s, t)$ parameterizes a plane then $\vec{r}_{2}(s, t)=$ $\vec{r}_{1}(s, t)+2 \vec{i}-3 \vec{j}+\vec{k}$ parameterizes a parallel plane.

Lucas Finney

Numerade Educator

Problem 52

Are the statements true or false? Give reasons for your answer.
Every point on a parameterized surface has a parameter curve passing through it.

Lucas Finney

Numerade Educator

Problem 53

Every point on a parameterized surface has a parameter curve passing through it.
If $s_{0} \neq s_{1},$ then the parameter curves $\vec{r}\left(s_{0}, t\right)$ and $\vec{r}\left(s_{1}, t\right)$ do not intersect.

Lucas Finney

Numerade Educator

Problem 54

Match the parameterizations (I)-(IV) with the surfaces
(a)-(d). In all cases $0 \leq s \leq \pi / 2,0 \leq t \leq \pi / 2 .$ Note that only part of the surface may be described by the given parameterization.
(a) Cylinder
(b) Plane
(c) Sphere
(d) Cone
I. $x=\cos s, \quad y=\sin t, \quad z=\cos s+\sin t$
II. $x=\cos s, \quad y=\sin s, \quad z=\cos t$
III. $\quad x=\sin s \cos t, \quad y=\sin s \sin t, \quad z=\cos s$
IV. $x=\cos s, \quad y=\sin t, \quad z=\sqrt{\cos ^{2} s+\sin ^{2} t}$

Harshita Goel

Numerade Educator