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Spring 2022 - Calculus III (MATH-2630-LR), Spring 2022
INSTRUCTOR
Final Exam Practice Packet (Homework)
Jiayin Jin
Auburn University
Current Score
QUESTION
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
POINTS
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TOTAL SCORE
0/0
0.0%
Due Date
THU, MAY 5, 2022
10:59 PM CDT
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Assignment Submission & Scoring
Assignment Submission
For this assignment, you submit answers by question parts. The number of submissions remaining for each question part
only changes if you submit or change the answer.
Assignment Scoring
Your last submission is used for your score.
1.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.1.015.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the sphere that passes through the point (7, 3, −3) and has center (5, 6, 3).
2.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.1.017.
ASK YOUR TEACHER
Write the equation of the sphere in standard form.
x2 + y2 + z2 + 4x − 4y − 6z = 19
Find its center and radius.
center
radius
(x, y, z) =
PRACTICE ANOTHER
3.
[0/0 Points]
MY NOTES
DETAILS
PREVIOUS ANSWERS
ASK YOUR TEACHER
PRACTICE ANOTHER
Write the equation of the sphere in standard form.
x2 + y2 + z2 + 8x − 8y + 6z + 32 = 0
Find its center and radius.
center
radius
SCALCET8 12.1.018.
(x, y, z) =
6
4.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.1.023.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find equations of the spheres with center (1, −2, 6) that touch the following planes.
(a) xy-plane
(b) yz-plane
(c) xz-plane
5.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.1.024.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the largest sphere with center (8, 5, 9) that is contained in the first octant.
6.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.2.026.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the vector that has the same direction as 9, 2, −6 but has length 4.
7.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.2.024.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find a unit vector that has the same direction as the given vector.
−4i + 2j − k
8.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.3.041.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the scalar and vector projections of b onto a.
a = 4, 7, −4
b = 4, −1, 1
scalar projection of b onto a
vector projection of b onto a
9.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.3.047.
ASK YOUR TEACHER
PRACTICE ANOTHER
If a = 3, 0, −1 , find a vector b such that compab = 3.
b=
10.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.3.505.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)
a = 3, −1, 7 , b = −2, 4, 2
exact
approximate
11.
[–/0 Points]
MY NOTES
°
DETAILS
SCALCET8 12.3.510.XP.
ASK YOUR TEACHER
Find the scalar and vector projections of b onto a.
a = 3, −6, 2 , b = 2, 1, 2
compab =
projab =
PRACTICE ANOTHER
12.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.4.521.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the volume of the parallelepiped determined by the vectors a, b, and c.
a = 3i + 3j − 2k, b = 4i − 4j + 2k, c = −3i + 3j + 3k
cubic units
13.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.4.520.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the volume of the parallelepiped determined by the vectors a, b, and c.
a = 4, 3, −3 ,
b = 0, 2, 3 ,
c = 6, −2, 5
cubic units
14.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.4.516.XP.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find two unit vectors orthogonal to both given vectors.
1, −1, 1 ,
0, 8, 8
(smaller i-value)
(larger i-value)
15.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.4.506.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the parallelogram with vertices A(−2, 2), B(0, 5), C(4, 3), and D(2, 0).
16.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.4.507.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the parallelogram with vertices K(2, 1, 2), L(2, 3, 4), M(4, 8, 4), and N(4, 6, 2).
17.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.5.061.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation for the plane consisting of all points that are equidistant from the points (2, 0, −2) and (4, 14, 0).
18.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.5.062.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation for the plane consisting of all points that are equidistant from the points (−5, 1, 1) and (1, 3, 5).
19.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.5.065.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find parametric equations for the line through the point (0, 3, 2) that is parallel to the plane x + y + z = 5 and perpendicular to the line
x = 1 + t, y = 3 − t, z = 2t. (Use the parameter t.)
(x(t), y(t), z(t)) =
20.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.5.040.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the plane.
The plane that passes through the line of intersection of the planes x − z = 2 and y + 3z = 1 and is perpendicular to the plane
x + y − 3z = 3
21.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.5.026.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the plane.
The plane through the point (7, 0, 1) and perpendicular to the line x = 6t, y = 8 − t, z = 1 + 2t
22.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 12.5.020.
ASK YOUR TEACHER
PRACTICE ANOTHER
Determine whether the lines L1 and L2 are parallel, skew, or intersecting.
L1: x
L2: x
= 8 − 9t, y = 3 + 6t, z = 1 − 3t
= 3 + 6s, y = −4s, z = 1 + 2s
parallel
skew
intersecting
If they intersect, find the point of intersection. (If an answer does not exist, enter DNE.)
(x, y, z) =
23.
[–/0 Points]
DETAILS
Match the equation with its graph.
x2 + 4y2 + 9z2 = 1
SCALCET8 12.6.021.
MY NOTES
ASK YOUR TEACHER
24.
[–/0 Points]
DETAILS
SCALCET8 12.6.023.
MY NOTES
ASK YOUR TEACHER
SCALCET8 12.6.024.
MY NOTES
ASK YOUR TEACHER
Match the equation with its graph.
x2 − y2 + z2 = 1
25.
[–/0 Points]
DETAILS
Match the equation with its graph.
−x2 + y2 − z2 = 1
26.
[–/0 Points]
DETAILS
Match the equation with its graph.
y = 2x2 + z2
SCALCET8 12.6.025.
MY NOTES
ASK YOUR TEACHER
27.
[–/0 Points]
DETAILS
Match the equation with its graph.
y2 = x2 + 2z2
SCALCET8 12.6.026.
MY NOTES
ASK YOUR TEACHER
28.
[–/0 Points]
DETAILS
Match the equation with its graph.
x2 + 2z2 = 1
SCALCET8 12.6.027.
MY NOTES
ASK YOUR TEACHER
29.
[–/0 Points]
DETAILS
Match the equation with its graph.
y = x2 − z2
SCALCET8 12.6.028.
MY NOTES
ASK YOUR TEACHER
30.
[0/0 Points]
MY NOTES
DETAILS
PREVIOUS ANSWERS
ASK YOUR TEACHER
SCALCET8 13.1.012.
PRACTICE ANOTHER
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.
r(t) = 2 cos(t)i + 2 sin(t)j + k
31.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.1.019.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find a vector equation and parametric equations for the line segment that joins P to Q.
P(0, −2, 2),
Q
1 1 1
, ,
2 3 4
vector equation
r(t) =
parametric equations
32.
[–/0 Points]
MY NOTES
(x(t), y(t), z(t)) =
DETAILS
SCALCET8 13.1.031.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
At what points does the curve r(t) = ti + (4t − t2)k intersect the paraboloid z = x2 + y2? (If an answer does not exist, enter DNE.)
(x, y, z) =
(smaller t-value)
(x, y, z) =
(larger t-value)
33.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.1.032.
ASK YOUR TEACHER
PRACTICE ANOTHER
At what points does the helix r(t) = sin(t), cos(t), t intersect the sphere x2 + y2 + z2 = 65? (Round your answers to three decimal places. If
an answer does not exist, enter DNE.)
(x, y, z) =
(smaller t-value)
(x, y, z) =
(larger t-value)
34.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.1.044.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find a vector function, r(t), that represents the curve of intersection of the two surfaces.
The paraboloid z = 3x2 + y2 and the parabolic cylinder y = 6x2
r(t) =
35.
[–/0 Points]
DETAILS
MY NOTES
SCALCET8 13.1.043.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find a vector function, r(t), that represents the curve of intersection of the two surfaces.
x2 + y2
The cone z =
and the plane z = 9 + y
r(t) =
36.
[–/0 Points]
DETAILS
MY NOTES
SCALCET8 13.1.504.XP.MI.
ASK YOUR TEACHER
Find the limit.
lim
t→0
3et − 3 ,
t
1+t −1
t
,
4
1+t
PRACTICE ANOTHER
37.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.1.AE.006.
ASK YOUR TEACHER
PRACTICE ANOTHER
EXAMPLE 6 Find a vector function that represents the curve of intersection of the
cylinder x2 + y2 = 9 and the plane 4y + z = 13.
SOLUTION The top figure shows how the plane and the cylinder intersect, and the
bottom figure shows the curve of intersection C, which is an ellipse. The projection
of C onto the xy-plane is the circle x2 + y2 =
, z = 0. So we know
from this example that we can write
x = 3 cos(t)
y=
0 ≤ t ≤ 2𝜋.
From the equation of the plane, we have
z = 13 − 4y = 13 − 12 sin(t).
So we can write parametric equations for C as
x
= 3 cos(t)
y
=
z = 13 − 12 sin(t)
0 ≤ t ≤ 2𝜋.
The corresponding vector equation is
r(t) =
0 ≤ t ≤ 2𝜋.
This equation is called a parametrization of the curve C. The arrows in the bottom
figure indicate the direction in which C is traced as the parameter t increases.
Video Example
38.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.2.019.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the unit tangent vector T(t) at the point with the given value of the parameter t.
T(0) =
39.
r(t) = cos(t)i + 6tj + 4 sin(2t)k, t = 0
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.2.025.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
x = e−5t cos(5t), y = e−5t sin(5t), z = e−5t; (1, 0, 1)
x(t), y(t), z(t)
=
40.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.2.034.
ASK YOUR TEACHER
PRACTICE ANOTHER
At what point do the curves r1(t) = t, 2 − t, 35 + t2 and r2(s) = 7 − s, s − 5, s2 intersect?
(x, y, z) =
Find their angle of intersection, 𝜃, correct to the nearest degree.
𝜃=
°
41.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.003.
ASK YOUR TEACHER
Find the length of the curve.
2 t i + et j + e−t k,
0≤t≤9
PRACTICE ANOTHER
42.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.005.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the length of the curve.
r(t) = 2 i + t2 j + t3 k, 0 ≤ t ≤ 1
43.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.017.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider the vector function given below.
r(t) = 4t, 5 cos(t), 5 sin(t)
(a) Find the unit tangent and unit normal vectors T(t) and N(t).
T(t) =
N(t) =
(b) Use this formula to find the curvature.
𝜅(t) =
44.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.019.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider the following vector function.
r(t) = 2 2 t, e2t, e−2t
(a) Find the unit tangent and unit normal vectors T(t) and N(t).
T(t) =
N(t) =
(b) Use this formula to find the curvature.
𝜅(t) =
45.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.025.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the curvature of r(t) = 3t, t2, t3 at the point (3, 1, 1).
𝜅=
46.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.030.
ASK YOUR TEACHER
PRACTICE ANOTHER
At what point does the curve have maximum curvature?
y = 5 ln(x)
(x, y) =
What happens to the curvature as x → ∞?
𝜅(x) approaches
as x → ∞.
47.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.031.
ASK YOUR TEACHER
PRACTICE ANOTHER
At what point does the curve have maximum curvature?
y = 8ex
(x, y) =
What happens to the curvature as x → ∞?
𝜅(x) approaches
as x → ∞.
48.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.049.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider the following.
x = sin(2t), y = −cos(2t), z = 6t; (0, 1, 3𝜋)
Find the equation of the normal plane of the curve at the given point.
Find the equation of the osculating plane of the curve at the given point.
49.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.3.052.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find equations for the osculating circles of the parabola y =
1 2
1
x at the points (0, 0) and 1,
.
2
2
(0, 0)
1,
1
2
Graph both osculating circles and the parabola on the same screen.
50.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.4.010.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the velocity, acceleration, and speed of a particle with the given position function.
r(t) = 3 cos(t), 4t, 3 sin(t)
v(t)
=
a(t)
=
|v(t)| =
51.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.4.011.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the velocity, acceleration, and speed of a particle with the given position function.
r(t) = 6 2 t i + e6t j + e−6tk
v(t)
=
a(t)
=
|v(t)| =
52.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.4.018.
ASK YOUR TEACHER
PRACTICE ANOTHER
(a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position.
a(t) = 11t i + et j + e−t k, v(0) = k, r(0) = j + k
r(t) =
(b) On your own using a computer, graph the path of the particle.
53.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.4.024.
ASK YOUR TEACHER
PRACTICE ANOTHER
A projectile is fired with an initial speed of 240 m/s and angle of elevation 60°. The projectile is fired from a position 110 m above the
ground. (Recall g ≈ 9.8 m/s2. Round your answers to the nearest whole number.)
(a) Find the range of the projectile.
m
(b) Find the maximum height reached.
m
(c) Find the speed at impact.
m/s
54.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.4.023.
ASK YOUR TEACHER
PRACTICE ANOTHER
A projectile is fired with an initial speed of 160 m/s and angle of elevation 60°. (Recall g ≈ 9.8 m/s2. Round your answers to the nearest whole
number.)
(a) Find the range of the projectile.
m
(b) Find the maximum height reached.
m
(c) Find the speed at impact.
m/s
55.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 13.4.501.XP.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.
a(t) = 7 i + 4 j, v(0) = k, r(0) = i
v(t) =
r(t) =
56.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.2.504.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim
(x, y) → (2, 0)
ln
4 + y2
x2 + xy
57.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.2.505.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
x6 − y6
(x, y) → (0, 0) x3 + y3
lim
58.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.2.506.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
x2yey
4
(x, y) → (0, 0) x + 8y2
lim
59.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.2.508.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
yz
lim
(x, y, z) → (0, 0, 0) x2 + 2y2 + 8z2
60.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.3.047.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.
x2 + 2y2 + 3z2 = 2
∂z
=
∂x
∂z
=
∂y
61.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.3.049.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.
e4z = xyz
∂z
=
∂x
∂z
=
∂y
62.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.4.502.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the tangent plane to the given surface at the specified point.
z=
xy ,
(5, 5, 5)
63.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.4.507.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the tangent plane to the given surface at the specified point.
z = y ln(x), (1, 3, 0)
64.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.4.508.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find an equation of the tangent plane to the given surface at the specified point.
(2, −2, 1)
65.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.5.021.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the Chain Rule to find the indicated partial derivatives.
z = x4 + x2y, x = s + 2t − u, y = stu2;
∂z ∂z ∂z
, ,
when s = 3, t = 1, u = 4
∂s ∂t ∂u
∂z
=
∂s
∂z
=
∂t
∂z
=
∂u
66.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.5.024.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the Chain Rule to find the indicated partial derivatives.
P=
u2 + v2 + w2 ,
∂P ∂P
,
∂x ∂y
∂P
=
∂x
∂P
=
∂y
u = xey,
when x = 0, y = 2
v = yex,
w = exy;
67.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.5.027.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use this equation to find dy/dx.
3y cos(x) = x2 + y2
dy
dx
68.
=
[–/0 Points]
MY NOTES
DETAILS
ASK YOUR TEACHER
Use this equation to find dy/dx.
2 tan−1(x2y) = x + xy2
dy
dx
=
SCALCET8 14.5.029.
PRACTICE ANOTHER
69.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.5.031.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the equations to find ∂z/∂x and ∂z/∂y.
x2 + 4y2 + 9z2 = 1
∂z
=
∂x
∂z
=
∂y
70.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.5.033.
ASK YOUR TEACHER
Use the equations to find ∂z/∂x and ∂z/∂y.
ez = 3xyz
∂z
=
∂x
∂z
=
∂y
PRACTICE ANOTHER
71.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.6.017.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the directional derivative of the function at the given point in the direction of the vector v.
h(r, s, t) = ln(3r + 6s + 9t),
Dvh(1, 1, 1) =
72.
[–/0 Points]
MY NOTES
DETAILS
(1, 1, 1), v = 12i + 36j + 18k
SCALCET8 14.6.023.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the maximum rate of change of f at the given point and the direction in which it occurs.
f(x, y) = 9 sin(xy),
maximum rate of change
direction vector
(0, 3)
73.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.6.032.
ASK YOUR TEACHER
PRACTICE ANOTHER
The temperature at a point (x, y, z) is given by
2
2
2
T(x, y, z) = 300e−x − 3y − 9z
where T is measured in °C and x, y, z in meters.
(a) Find the rate of change of temperature at the point P(4, −1, 4) in the direction towards the point (5, −4, 5).
°C/m
(b) In which direction does the temperature increase fastest at P?
(c) Find the maximum rate of increase at P.
74.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.6.033.
ASK YOUR TEACHER
PRACTICE ANOTHER
Suppose that over a certain region of space the electrical potential V is given by the following equation.
V(x, y, z) = 5x2 − 4xy + xyz
(a) Find the rate of change of the potential at P(5, 6, 6) in the direction of the vector v = i + j − k.
(b) In which direction does V change most rapidly at P?
(c) What is the maximum rate of change at P?
75.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.048.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the dimensions of the box with volume 4096 cm3 that has minimal surface area. (Let x, y, and z be the dimensions of the box.)
(x, y, z) =
76.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.049.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane
x + 2y + 3z = 12.
77.
[–/0 Points]
DETAILS
SCALCET8 14.7.051.
MY NOTES
ASK YOUR TEACHER
Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c. (Let x, y, and z
be the dimensions of the rectangular box.)
(x, y, z) =
78.
[–/0 Points]
DETAILS
SCALCET8 14.7.055.
MY NOTES
If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?
cubic units
ASK YOUR TEACHER
79.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.053.
ASK YOUR TEACHER
PRACTICE ANOTHER
A cardboard box without a lid is to have a volume of 10,976 cm3. Find the dimensions that minimize the amount of cardboard used. (Let x, y,
and z be the dimensions of the cardboard box.)
(x, y, z) =
80.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.505.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the absolute maximum and minimum values of f on the set D.
f(x, y) = 4x + 6y − x2 − y2 + 9,
D = {(x, y) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 5}
absolute maximum value
absolute minimum value
81.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.506.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the absolute maximum and minimum values of f on the set D.
f(x, y) = x4 + y4 − 4xy + 6,
D = {(x, y) | 0 ≤ x ≤ 3, 0 ≤ y ≤ 2}
absolute maximum value
absolute minimum value
82.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.504.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph
the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated
list. If an answer does not exist, enter DNE.)
f(x, y) = 5ey(y2 − x2)
local maximum value(s)
local minimum value(s)
saddle point(s)
(x, y, f) =
83.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.503.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph
the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated
list. If an answer does not exist, enter DNE.)
2 2
f(x, y) = 9(x2 + y2)ey − x
local maximum value(s)
local minimum value(s)
saddle point(s)
(x, y, f) =
84.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.7.501.XP.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph
the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated
list. If an answer does not exist, enter DNE.)
f(x, y) = x3 − 27xy + 27y3
local maximum value(s)
local minimum value(s)
saddle point(s)
85.
[–/0 Points]
MY NOTES
(x, y, f) =
DETAILS
SCALCET8 14.8.009.
ASK YOUR TEACHER
PRACTICE ANOTHER
This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme
values of the function subject to the given constraint.
f(x, y, z) = xy2z;
maximum value
minimum value
x2 + y2 + z2 = 16
86.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.8.011.
ASK YOUR TEACHER
PRACTICE ANOTHER
This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme
values of the function subject to the given constraint.
f(x, y, z) = x2 + y2 + z2;
x4 + y4 + z4 = 7
maximum value
minimum value
87.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 14.8.013.
ASK YOUR TEACHER
PRACTICE ANOTHER
This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme
values of the function subject to the given constraint.
f(x, y, z, t) = x + y + z + t;
maximum value
minimum value
x2 + y2 + z2 + t2 = 49
88.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.2.017.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the double integral.
D
89.
5x cos(y) dA, D is bounded by y = 0, y = x2, x = 1
[–/0 Points]
DETAILS
SCALCET8 15.2.018.
MY NOTES
ASK YOUR TEACHER
MY NOTES
ASK YOUR TEACHER
Evaluate the double integral.
(x2 + 2y) dA, D is bounded by y = x, y = x3, x ≥ 0
D
90.
[–/0 Points]
DETAILS
SCALCET8 15.2.024.
Find the volume of the given solid.
Under the surface z = 1 + x2y2 and above the region enclosed by x = y2 and x = 4.
91.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.2.025.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the volume of the given solid.
Under the surface z = 4xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)
92.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.2.030.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the volume of the given solid.
Bounded by the cylinder y2 + z2 = 64 and the planes x = 2y, x = 0, z = 0 in the first octant
93.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.2.031.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the volume of the given solid.
Bounded by the cylinder x2 + y2 = 4 and the planes y = 3z, x = 0, z = 0 in the first octant
94.
[–/0 Points]
DETAILS
MY NOTES
SCALCET8 15.2.051.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the integral by reversing the order of integration.
1
0
95.
3
3y
2
13ex dx dy
[–/0 Points]
DETAILS
MY NOTES
SCALCET8 15.2.056.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the integral by reversing the order of integration.
3
27
0
3
y
4
6ex dx dy
96.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.3.007.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the given integral by changing to polar coordinates.
x2y dA,
where D is the top half of the disk with center the origin and radius 5.
D
97.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.3.008.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the given integral by changing to polar coordinates.
(4x − y) dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 16 and the lines x = 0 and y = x
R
98.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.3.011.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the given integral by changing to polar coordinates.
D
99.
2 2
e−x − y dA , where D is the region bounded by the semicircle x =
[–/0 Points]
MY NOTES
DETAILS
16 − y2 and the y−axis
SCALCET8 15.3.014.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the given integral by changing to polar coordinates.
x dA , where D is the region in the first quadrant that lies between the circles x2 + y2 = 16 and x2 + y2 = 4x
D
100.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.3.019.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use polar coordinates to find the volume of the given solid.
Under the paraboloid z = x2 + y2 and above the disk x2 + y2 ≤ 9
101.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.3.021.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use polar coordinates to find the volume of the given solid.
Below the plane 6x + y + z = 8 and above the disk x2 + y2 ≤ 1
102.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.4.017.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the moments of inertia Ix, Iy, I0 for a lamina that occupies the region D and has the given density function 𝜌.
D = {(x, y) | 1 ≤ x ≤ 3, 1 ≤ y ≤ 4}; 𝜌(x, y) = ky2
Ix
=
Iy
=
I0
=
103.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.4.029.
ASK YOUR TEACHER
PRACTICE ANOTHER
Suppose X and Y are random variables with joint density function.
f(x, y) =
0.1e−(0.5x + 0.2y)
0
if x ≥ 0, y ≥ 0
otherwise
(a) Is f a joint density function?
Yes
No
(b) Find P(Y ≥ 6). (Round your answer to four decimal places.)
Find P(X ≤ 5, Y ≤ 3). (Round your answer to four decimal places.)
(c) Find the expected value of X.
Find the expected value of Y.
104.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.4.501.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the mass and center of mass of the lamina that occupies the region D and has the given density function 𝜌.
D is the triangular region enclosed by the lines x = 0, y = x,
m
and 2x + y = 6; 𝜌(x, y) = 4x2
=
(x, y) =
105.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.4.503.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the mass and center of mass of the lamina that occupies the region D and has the given density function 𝜌.
D is bounded by the parabolas y = x2 and x = y2;
m
=
(x, y) =
𝜌(x, y) = 13
x
106.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.5.005.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the surface.
The part of the paraboloid z = 1 − x2 − y2 that lies above the plane z = −2
107.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.5.007.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the surface.
The part of the hyperbolic paraboloid z = y2 − x2 that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 16.
108.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.5.010.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the surface.
The part of the sphere x2 + y2 + z2 = 36 that lies above the plane z = 5.
109.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.6.020.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use a triple integral to find the volume of the given solid.
The solid enclosed by the paraboloids y = x2 + z2 and y = 8 − x2 − z2.
110.
[–/0 Points]
MY NOTES
DETAILS
ASK YOUR TEACHER
Express the integral
E
y = x2,
SCALCET8 15.6.031.
z = 0,
f(x, y, z) dV
PRACTICE ANOTHER
as an iterated integral in six different ways, where E is the solid bounded by the given surfaces.
y + 4z = 16
f(x, y, z) dz dy dx
−4
x2
0
f(x, y, z) dz dx dy
0
−
y
0
f(x, y, z) dx dz dy
0
−
0
y
f(x, y, z) dx dy dz
0
−
0
y
f(x, y, z) dy dz dx
−4
x2
0
f(x, y, z) dy dx dz
0
−
16−4z
x2
111.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.6.034.
ASK YOUR TEACHER
PRACTICE ANOTHER
The figure shows the region of integration for the integral.
25−x2
5
0
0
5−x
0
f(x, y, z) dy dz dx
Rewrite this integral as an equivalent iterated integral in the five other orders. (Assume y(x) = 5 − x and z(x) = 25 − x2. )
f(x, y, z) dy dx dz
0
0
0
f(x, y, z) dz dx dy
0
0
0
f(x, y, z) dz dy dx
0
0
0
f(x, y, z) dx dy dz
0
0
0
+
f(x, y, z) dx dy dz
5−
0
25−z
0
f(x, y, z) dx dz dy
0
0
0
+
f(x, y, z) dx dz dy
10y−y2
0
112.
[–/0 Points]
MY NOTES
DETAILS
0
SCALCET8 15.6.041.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the mass and center of mass of the solid E with the given density 𝜌.
E is the cube
0 ≤ x ≤ a, 0 ≤ y ≤ a, 0 ≤ z ≤ a; 𝜌(x, y, z) = 2x2 + 2y2 + 2z2.
m=
x, y, z
113.
=
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.7.023.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use cylindrical coordinates.
Find the volume of the solid that is enclosed by the cone z =
x2 + y2
and the sphere x2 + y2 + z2 = 18.
114.
[–/0 Points]
DETAILS
SCALCET8 15.7.024.
MY NOTES
Use cylindrical coordinates.
Find the volume of the solid that lies between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z2 = 2.
115.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.7.030.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the integral by changing to cylindrical coordinates.
5
25 − x2
−5 0
116.
[–/0 Points]
MY NOTES
25 − x2 − y2
0
DETAILS
x2 + y2 dz dy dx
SCALCET8 15.7.029.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the integral by changing to cylindrical coordinates.
7
49 − y2
−7 −
49 − y2
8
x2 + y2
xz dz dx dy
ASK YOUR TEACHER
117.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 15.8.019.
ASK YOUR TEACHER
PRACTICE ANOTHER
Set up the triple integral of an arbitrary continuous function f(x, y, z) in cylindrical coordinates over the solid shown.
f
,
E
f(x, y, z) dV
=
,
0
0
0
dz dr d𝜃
118.
[–/0 Points]
DETAILS
SCALCET8 15.8.022.
MY NOTES
Use spherical coordinates.
Evaluate
119.
E
y2z2 dV,
[–/0 Points]
MY NOTES
where E lies above the cone 𝜑 = 𝜋/3 and below the sphere 𝜌 = 1.
DETAILS
SCALCET8 15.8.023.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use spherical coordinates.
Evaluate
E
2
2
2
2
2
2
(x2 + y2) dV , where E lies between the spheres x + y + z = 1 and x + y + z = 9.
ASK YOUR TEACHER
120.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.2.007.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the line integral, where C is the given curve.
C
121.
(x + 7y) dx + x2 dy, C consists of line segments from (0, 0) to (7, 1) and from (7, 1) to (8, 0)
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.2.013.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the line integral, where C is the given curve.
C
xyeyz dy, C: x = 4t,
y = 4t2,
z = 3t3,
0≤t≤1
122.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.2.021.
ASK YOUR TEACHER
Evaluate the line integral
PRACTICE ANOTHER
F · dr, where C is given by the vector function r(t).
C
F(x, y, z) = sin(x) i + cos(y) j + xz k
r(t) = t5 i − t3 j + t k, 0 ≤ t ≤ 1
123.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.2.041.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the work done by the force field F(x, y, z) = x − y2, y − z2, z − x2 on a particle that moves along the line segment from (0, 0, 1) to
(4, 1, 0).
124.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.3.505.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is not conservative,
enter DNE.)
F(x, y) = (ln(y) + 18xy3)i + (27x2y2 + x/y)j
f(x, y) =
125.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.3.507.XP.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider F and C below.
F(x, y) = x2 i + y2 j
C is the arc of the parabola y = 2x2 from (−1, 2) to (1, 2)
(a) Find a function f such that F = ∇f.
f(x, y) =
(b) Use part (a) to evaluate
∇f · dr along the given curve C.
C
126.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.3.508.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider F and C below.
F(x, y) = 4xy2 i + 4x2y j
C: r(t) = t + sin
1
1
𝜋t , t + cos 𝜋t
2
2
, 0≤t≤1
(a) Find a function f such that F = ∇f.
f(x, y) =
(b) Use part (a) to evaluate
∇f · dr along the given curve C.
C
127.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.3.513.XP.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider F and C below.
F(x, y, z) = 2xz + y2 i + 2xy j + x2 + 9z2 k
C: x = t2,
y = t + 1,
z = 4t − 1,
0≤t≤1
(a) Find a function f such that F = ∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
∇f · dr along the given curve C.
C
128.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.4.009.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
8y3 dx − 8x3 dy
C
C is the circle x2 + y2 = 4
129.
[–/0 Points]
DETAILS
SCALCET8 16.4.012.
Use Green's Theorem to evaluate
C
MY NOTES
F · dr. (Check the orientation of the curve before applying the theorem.)
F(x, y) = e−x + y2, e−y + x2 , C consists of the arc of the curve y = cos(x) from − 𝜋 , 0 to
2
𝜋 , 0 to − 𝜋 , 0
2
2
130.
[–/0 Points]
MY NOTES
DETAILS
𝜋,0
2
and the line segment from
SCALCET8 16.4.013.
ASK YOUR TEACHER
Use Green's Theorem to evaluate
ASK YOUR TEACHER
C
PRACTICE ANOTHER
F · dr. (Check the orientation of the curve before applying the theorem.)
F(x, y) = y − cos(y), x sin(y) , C is the circle (x − 4)2 + (y + 5)2 = 16 oriented clockwise
131.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.5.003.
ASK YOUR TEACHER
Consider the given vector field.
F(x, y, z) = 4xyezi + yzexk
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.
div F =
PRACTICE ANOTHER
132.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.5.007.
ASK YOUR TEACHER
PRACTICE ANOTHER
Consider the vector field.
F(x, y, z) = 8ex sin(y), 4ey sin(z), 5ez sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.
div F =
133.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.5.013.MI.SA.
ASK YOUR TEACHER
PRACTICE ANOTHER
This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the
skipped part, and you will not be able to come back to the skipped part.
Tutorial Exercise
Find a function f such that F = ∇f.
F(x, y, z) = 6y2z3i + 12xyz3j + 18xy2z2k
134.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.5.013.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f. (If the vector field is not
conservative, enter DNE.)
F(x, y, z) = 5y2z3 i + 10xyz3 j + 15xy2z2 k
f(x, y, z) =
135.
[–/0 Points]
DETAILS
SCALCET8 16.5.017.
MY NOTES
ASK YOUR TEACHER
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f. (If the vector field is not
conservative, enter DNE.)
F(x, y, z) = eyzi + xzeyzj + xyeyzk
f(x, y, z) =
136.
[–/0 Points]
DETAILS
SCALCET8 16.5.030.
MY NOTES
ASK YOUR TEACHER
Let r = x i + y j + z k and r = |r|. Find each of the following. (Give your answers in terms of r.)
(a) ∇ · r
(b) ∇ · (r r)
(c) ∇2r3
137.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.6.026.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find a parametric representation for the surface.
The part of the plane z = x + 2 that lies inside the cylinder x2 + y2 = 4. (Enter your answer as a comma-separated list of equations.
Let x, y, and z be in terms of s and/or 𝜃.)
138.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.6.039.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the part of the plane 5x + 3y + z = 15 that lies in the first octant.
139.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.6.041.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the surface.
The part of the plane x + 2y + 3z = 1 that lies inside the cylinder x2 + y2 = 7
140.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.6.045.
ASK YOUR TEACHER
PRACTICE ANOTHER
Find the area of the surface.
The part of the surface z = xy that lies within the cylinder x2 + y2 = 16.
141.
[–/0 Points]
MY NOTES
DETAILS
SCALCET8 16.7.031.
ASK YOUR TEACHER
Evaluate the surface integral
PRACTICE ANOTHER
F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S.
S
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = x2 i + y2 j + z2 k
S is the boundary of the solid half-cylinder
142.
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9 − y2 , 0 ≤ x ≤ 5
SCALCET8 16.7.032.
ASK YOUR TEACHER
Evaluate the surface integral
0≤z≤
PRACTICE ANOTHER
F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S.
S
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = y i + (z − y) j + x k
S is the surface of the tetrahedron with vertices (0, 0, 0), (6, 0, 0), (0, 6, 0), and (0, 0, 6)
143.
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SCALCET8 16.7.019.
ASK YOUR TEACHER
PRACTICE ANOTHER
Evaluate the surface integral.
xz dS
S
S is the boundary of the region enclosed by the cylinder y2 + z2 = 25
144.
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DETAILS
Evaluate the surface integral
SCALCET8 16.7.021.
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = zexyi − 3zexyj + xyk,
S is the parallelogram of this exercise with upward orientation.
F · dS =
MY NOTES
ASK YOUR TEACHER
F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S.
S
S
and the planes x = 0 and x + y = 9
145.
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DETAILS
SCALCET8 16.7.026.
ASK YOUR TEACHER
Evaluate the surface integral
PRACTICE ANOTHER
F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S.
S
For closed surfaces, use the positive (outward) orientation.
F(x, y, z) = yi − xj + 4zk, S is the hemisphere x2 + y2 + z2 = 4, z ≥ 0, oriented downward
146.
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DETAILS
SCALCET8 16.8.002.
ASK YOUR TEACHER
Use Stokes' Theorem to evaluate
PRACTICE ANOTHER
curl F · dS.
S
F(x, y, z) = x2 sin(z)i + y2j + xyk,
S is the part of the paraboloid z = 1 − x2 − y2 that lies above the xy-plane, oriented upward.
147.
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SCALCET8 16.8.003.
ASK YOUR TEACHER
Use Stokes' Theorem to evaluate
PRACTICE ANOTHER
curl F · dS.
S
F(x, y, z) = zeyi + x cos(y)j + xz sin(y)k,
S is the hemisphere x2 + y2 + z2 = 4, y ≥ 0, oriented in the direction of the positive y-axis.
148.
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DETAILS
SCALCET8 16.8.004.
ASK YOUR TEACHER
Use Stokes' Theorem to evaluate
PRACTICE ANOTHER
curl F · dS.
S
F(x, y, z) = tan−1(x2yz2)i + x2yj + x2z2k, S is the cone x =
149.
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y2 + z2 , 0 ≤ x ≤ 3,
oriented in the direction of the positive x-axis.
SCALCET8 16.8.007.MI.
ASK YOUR TEACHER
Use Stokes' Theorem to evaluate
PRACTICE ANOTHER
F · dr where C is oriented counterclockwise as viewed from above.
C
F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k,
C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3).
150.
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SCALCET8 16.8.008.
ASK YOUR TEACHER
Use Stokes' Theorem to evaluate
PRACTICE ANOTHER
F · dr where C is oriented counterclockwise as viewed from above.
C
F(x, y, z) = i + (x + yz)j + (xy −
z )k,
C is the boundary of the part of the plane
151.
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DETAILS
5x + 4y + z = 1 in the first octant.
SCALCET8 16.8.009.
Use Stokes' Theorem to evaluate
MY NOTES
ASK YOUR TEACHER
F · dr where C is oriented counterclockwise as viewed from above.
C
F(x, y, z) = xyi + yzj + zxk,
C is the boundary of the part of the paraboloid z = 1 − x2 − y2
152.
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in the first octant.
SCALCET8 16.9.005.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the Divergence Theorem to calculate the surface integral
F · dS; that is, calculate the flux of F across S.
S
F(x, y, z) = xyezi + xy2z3j − yezk,
S is the surface of the box bounded by the coordinate plane and the planes x = 3, y = 4, and z = 1.
153.
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SCALCET8 16.9.006.
MY NOTES
Use the Divergence Theorem to calculate the surface integral
ASK YOUR TEACHER
F · dS; that is, calculate the flux of F across S.
S
F(x, y, z) = x2yzi + xy2zj + xyz2k,
S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, where a, b, and c are positive
numbers.
154.
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SCALCET8 16.9.007.MI.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the Divergence Theorem to calculate the surface integral
S
F · dS; that is, calculate the flux of F across S.
F(x, y, z) = 3xy2i + xezj + z3k,
S is the surface of the solid bounded by the cylinder y2 + z2 = 9
and the planes x = −3 and x = 1.
155.
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DETAILS
SCALCET8 16.9.008.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the Divergence Theorem to calculate the surface integral
F · dS; that is, calculate the flux of F across S.
S
F(x, y, z) = (x3 + y3)i + (y3 + z3)j + (z3 + x3)k,
S is the sphere with center the origin and radius 3.
156.
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SCALCET8 16.9.011.
ASK YOUR TEACHER
PRACTICE ANOTHER
Use the Divergence Theorem to calculate the surface integral
F · dS; that is, calculate the flux of F across S.
S
F(x, y, z) = (2x3 + y3)i + (y3 + z3)j + 3y2zk,
S is the surface of the solid bounded by the paraboloid z = 1 − x2 − y2 and the xy-plane.
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