University Calculus: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780321999580
Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.
Publisher: PEARSON
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Chapter 14.1, Problem 12E
To determine
Calculate the iterated
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(7) (12 points) Let F(x, y, z) = (y, x+z cos yz, y cos yz).
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F.ds;
as denotes the boundary of S. Explain your answer.
(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
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steps otherwise no credit.
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(x, y) = R².
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=
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So S√ ²
0
Chapter 14 Solutions
University Calculus: Early Transcendentals (3rd Edition)
Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2ECh. 14.1 - Prob. 3ECh. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E
Ch. 14.1 - In Exercises 1-14. evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - In Exercises 1–14, evaluate the iterated...Ch. 14.1 - In Exercises 1–14, evaluate the iterated...Ch. 14.1 - In Exercises 17-24, evaluate the double integral...Ch. 14.1 - In Exercises 17-24, evaluate the double integral...Ch. 14.1 - In Exercises 17-24, evaluate the double integral...Ch. 14.1 - Prob. 18ECh. 14.1 - In Exercises 17–24, evaluate the double integral...Ch. 14.1 - In Exercises 17–24, evaluate the double integral...Ch. 14.1 - In Exercises 17–24, evaluate the double integral...Ch. 14.1 - In Exercises 17–24, evaluate the double integral...Ch. 14.1 - In Exercises 25 and 26, integrate f over the given...Ch. 14.1 - In Exercises 25 and 26, integrate f over the given...Ch. 14.1 - Find the volume of the region hounded above by the...Ch. 14.1 - Find the volume of the region bounded above by the...Ch. 14.1 - Prob. 27ECh. 14.1 - Prob. 28ECh. 14.1 - Prob. 29ECh. 14.1 - Prob. 30ECh. 14.1 - Find a value of the constant k so that
Ch. 14.1 - Prob. 32ECh. 14.1 - Prob. 33ECh. 14.1 - Prob. 34ECh. 14.1 - Prob. 35ECh. 14.1 - Prob. 36ECh. 14.2 - In Exercises 1-8, sketch the described regions of...Ch. 14.2 - Prob. 2ECh. 14.2 - Prob. 3ECh. 14.2 - In Exercises 1-8, sketch the described regions of...Ch. 14.2 - In Exercises 1-8, sketch the described regions of...Ch. 14.2 - In Exercises 1-8, sketch the described regions of...Ch. 14.2 - In Exercises 1-8, sketch the described regions of...Ch. 14.2 - In Exercises 1-8, sketch the described regions of...Ch. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - Prob. 14ECh. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - In Exercises 9-18, write an iterated integral for...Ch. 14.2 - In Exercises 9-18, write an iterated integral for...Ch. 14.2 - In Exercises 9–18, write an iterated integral for ...Ch. 14.2 - Finding Regions of Integration and Double...Ch. 14.2 - Finding Regions of Integration and Double...Ch. 14.2 - In Exercises 19–24, sketch the region of...Ch. 14.2 - Prob. 22ECh. 14.2 - In Exercises 19–24, sketch the region of...Ch. 14.2 - Prob. 24ECh. 14.2 - In Exercises 25-28, integrate f over the given...Ch. 14.2 - Prob. 26ECh. 14.2 - Prob. 27ECh. 14.2 - In Exercises 25–28, integrate f over the given...Ch. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Each of Exercises 29–32 gives an integral over a...Ch. 14.2 - Prob. 32ECh. 14.2 - In Exercises 33–46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - Prob. 40ECh. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - Prob. 44ECh. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - Prob. 46ECh. 14.2 - In Exercises 33-46, sketch the region of...Ch. 14.2 - Prob. 48ECh. 14.2 - In Exercises 47-56, sketch the region of...Ch. 14.2 - Prob. 50ECh. 14.2 - In Exercises 47-56, sketch the region of...Ch. 14.2 - Prob. 52ECh. 14.2 - In Exercises 47-56, sketch the region of...Ch. 14.2 - Prob. 54ECh. 14.2 - In Exercises 47–56, sketch the region of...Ch. 14.2 - Prob. 56ECh. 14.2 - Find the volume of the region bounded above by the...Ch. 14.2 - Prob. 58ECh. 14.2 - Find the volume of the solid whose base is the...Ch. 14.2 - Prob. 60ECh. 14.2 - Find the volume of the solid in the first octant...Ch. 14.2 - Prob. 62ECh. 14.2 - Find the volume of the wedge cut from the first...Ch. 14.2 - Prob. 64ECh. 14.2 - Find the volume of the solid that is bounded on...Ch. 14.2 - Prob. 66ECh. 14.2 - In Exercises 67 and 68, sketch the region of...Ch. 14.2 - Prob. 68ECh. 14.2 - Prob. 69ECh. 14.2 - Prob. 70ECh. 14.2 - Prob. 71ECh. 14.2 - Prob. 72ECh. 14.2 - In Exercises 73 and 74, approximate the double...Ch. 14.2 - Prob. 74ECh. 14.2 - Circular sector Integrate over the smaller sector...Ch. 14.2 - Unbounded region Integrate f(x, y) = 1/ [(x2 –...Ch. 14.2 - Noncircular cylinder A solid right (noncircular)...Ch. 14.2 - Prob. 78ECh. 14.2 - Maximizing a double integral What region R in the...Ch. 14.2 - Minimizing a double integral What region R in the...Ch. 14.2 - Is it possible to evaluate the integral of a...Ch. 14.2 - How would you evaluate the double integral of a...Ch. 14.2 - Prob. 83ECh. 14.2 - Prob. 84ECh. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - Prob. 2ECh. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - Prob. 6ECh. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - Prob. 8ECh. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - Prob. 10ECh. 14.3 - Prob. 11ECh. 14.3 - In Exercises 1-12, sketch the region bounded by...Ch. 14.3 - The integrals and sums of integrals in Exercises...Ch. 14.3 - Prob. 14ECh. 14.3 - The integrals and sums of integrals in Exercises...Ch. 14.3 - The integrals and sums of integrals in Exercises...Ch. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Find the average value of f(x, y) = sin(x + y)...Ch. 14.3 - Which do you think will be larger, the average...Ch. 14.3 - Find the average height of the paraboloid z = x2 +...Ch. 14.3 - Find the average value of f(x, y) = 1/(xy) over...Ch. 14.3 - Geometric area Find the area of the region
using...Ch. 14.3 - Prob. 24ECh. 14.3 - Bacterium population If f(x, y) = (10,000ey)/ (1 +...Ch. 14.3 - Prob. 26ECh. 14.3 - Average temperature in Texas According to the...Ch. 14.3 - Prob. 28ECh. 14.3 - Suppose f(x, y) is continuous over a region R in...Ch. 14.3 - Prob. 30ECh. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 - In Exercises 1-8, describe the given region in...Ch. 14.4 -
In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - Prob. 18ECh. 14.4 - In Exercises 9-22, change the Cartesian integral...Ch. 14.4 - Prob. 20ECh. 14.4 - In Exercises 9–22, change the Cartesian integral...Ch. 14.4 - In Exercises 9–22, change the Cartesian integral...Ch. 14.4 - In Exercises 23-26, sketch the region of...Ch. 14.4 - In Exercises 23–26, sketch the region of...Ch. 14.4 - In Exercises 23–26, sketch the region of...Ch. 14.4 - In Exercises 23–26, sketch the region of...Ch. 14.4 - Find the area of the region cut from the first...Ch. 14.4 - Prob. 28ECh. 14.4 - One leaf of a rose Find the area enclosed by one...Ch. 14.4 - Prob. 30ECh. 14.4 - Prob. 31ECh. 14.4 - Overlapping cardioids Find the area of the region...Ch. 14.4 - In polar coordinates, the average value of a...Ch. 14.4 - Prob. 34ECh. 14.4 - In polar coordinates, the average value of a...Ch. 14.4 - Prob. 36ECh. 14.4 - Converting to a polar integral Integrate over the...Ch. 14.4 - Prob. 38ECh. 14.4 - Volume of noncircular right cylinder The region...Ch. 14.4 - Prob. 40ECh. 14.4 - Prob. 41ECh. 14.4 - Prob. 42ECh. 14.4 - Prob. 43ECh. 14.4 - Area formula in polar coordinates Use the double...Ch. 14.4 - Prob. 45ECh. 14.4 - Prob. 46ECh. 14.4 - Evaluate the integral , where R is the region...Ch. 14.4 - Prob. 48ECh. 14.5 - Evaluate the integral in Example 3, taking F(x, y,...Ch. 14.5 - Prob. 2ECh. 14.5 - Volume of tetrahedron Write six different iterated...Ch. 14.5 - Prob. 4ECh. 14.5 - Volume enclosed by paraboloids Let D be the region...Ch. 14.5 - Prob. 6ECh. 14.5 - Evaluate the integrals in Exercises 7–20.
7.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
8.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
9.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
10.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
11.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
12.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
13.
Ch. 14.5 - Prob. 14ECh. 14.5 - Evaluate the integrals in Exercises 7–20.
15.
Ch. 14.5 - Prob. 16ECh. 14.5 - Evaluate the integrals in Exercises 7–20.
17.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
18.
Ch. 14.5 - Evaluate the integrals in Exercises 7–20.
19.
Ch. 14.5 - Prob. 20ECh. 14.5 - Here is the region of integration of the integral...Ch. 14.5 - Here is the region of integration of the...Ch. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Find the volumes of the regions in Exercises 2336....Ch. 14.5 - Find the volumes of the regions in Exercises 2336....Ch. 14.5 - Prob. 28ECh. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Prob. 32ECh. 14.5 - Find the volumes of the regions in Exercises...Ch. 14.5 - Prob. 34ECh. 14.5 - The region cut from the solid elliptical cylinder...Ch. 14.5 - Prob. 36ECh. 14.5 - In Exercises 37–40, find the average value of F(x,...Ch. 14.5 - Prob. 38ECh. 14.5 - In Exercises 37–40, find the average value of F(x,...Ch. 14.5 - Prob. 40ECh. 14.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 14.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 14.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 14.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 14.5 - Finding an upper limit of an iterated integral...Ch. 14.5 - Prob. 46ECh. 14.5 - Minimizing a triple integral What domain D in...Ch. 14.5 - Maximizing a triple integral What domain D in...Ch. 14.6 - Finding a center of mass find the center of mass...Ch. 14.6 - Prob. 2ECh. 14.6 - Finding a centroid Find the centroid of the region...Ch. 14.6 - Prob. 4ECh. 14.6 - Prob. 5ECh. 14.6 - Finding a centroid Find the centroid of the region...Ch. 14.6 - Prob. 7ECh. 14.6 - Prob. 8ECh. 14.6 - The centroid of an infinite region Find the...Ch. 14.6 - Prob. 10ECh. 14.6 - Prob. 11ECh. 14.6 - Prob. 12ECh. 14.6 - Finding a center of mass Find the center of mass...Ch. 14.6 - Prob. 14ECh. 14.6 - Prob. 15ECh. 14.6 - Prob. 16ECh. 14.6 - Center of mass, moment of inertia Find the center...Ch. 14.6 - Prob. 18ECh. 14.6 - Prob. 19ECh. 14.6 - Prob. 20ECh. 14.6 - Moments of inertia Find the moments of inertia of...Ch. 14.6 - Prob. 22ECh. 14.6 - Center of mass and moments of inertia A solid...Ch. 14.6 - Prob. 24ECh. 14.6 - a. Center of mass Find the center of mass of a...Ch. 14.6 - Prob. 26ECh. 14.6 - Moment of inertia about a line A wedge like the...Ch. 14.6 - Prob. 28ECh. 14.6 - In Exercises 29 and 30, find
the mass of the...Ch. 14.6 - In Exercises 29 and 30, find
a. the mass of the...Ch. 14.6 - In Exercises 31 and 32, find
the mass of the...Ch. 14.6 - Prob. 32ECh. 14.6 - Mass Find the mass of the solid bounded by the...Ch. 14.6 - Prob. 34ECh. 14.7 - Evaluate the cylindrical coordinate integrals in...Ch. 14.7 - Evaluate the cylindrical coordinate integrals in...Ch. 14.7 - Evaluate the cylindrical coordinate integrals in...Ch. 14.7 - Prob. 4ECh. 14.7 - Evaluate the cylindrical coordinate integrals in...Ch. 14.7 - Prob. 6ECh. 14.7 - The integrals we have seen so far suggest that...Ch. 14.7 - Prob. 8ECh. 14.7 - Prob. 9ECh. 14.7 - Prob. 10ECh. 14.7 - Let D be the region bounded below by the plane z =...Ch. 14.7 - Let D be the region bounded below by the cone and...Ch. 14.7 - Give the limits of integration for evaluating the...Ch. 14.7 - Convert the integral
to an equivalent integral in...Ch. 14.7 - In Exercises 37–42, set up the iterated integral...Ch. 14.7 - In Exercises 37–42, set up the iterated integral...Ch. 14.7 - In Exercises 37–42, set up the iterated integral...Ch. 14.7 - In Exercises 37–42, set up the iterated integral...Ch. 14.7 - In Exercises 37–42, set up the iterated integral...Ch. 14.7 - Prob. 20ECh. 14.7 - Evaluate the spherical coordinate integrals in...Ch. 14.7 - Evaluate the spherical coordinate integrals in...Ch. 14.7 - Evaluate the spherical coordinate integrals in...Ch. 14.7 - Prob. 24ECh. 14.7 - Evaluate the spherical coordinate integrals in...Ch. 14.7 - Prob. 26ECh. 14.7 - The previous integrals suggest there are preferred...Ch. 14.7 - The previous integrals suggest there are preferred...Ch. 14.7 - The previous integrals suggest there are preferred...Ch. 14.7 - Prob. 30ECh. 14.7 - Let D be the region in Exercise 33. Set up the...Ch. 14.7 - Let D be the region bounded below by the cone and...Ch. 14.7 - In Exercises 55–60, (a) find the spherical...Ch. 14.7 - In Exercises 55–60, (a) find the spherical...Ch. 14.7 - In Exercises 55–60, (a) find the spherical...Ch. 14.7 - Prob. 36ECh. 14.7 - In Exercises 55–60, (a) find the spherical...Ch. 14.7 - In Exercises 55–60, (a) find the spherical...Ch. 14.7 - Set up triple integrals for the volume of the...Ch. 14.7 - Prob. 40ECh. 14.7 - Let D be the smaller cap cut from a solid ball of...Ch. 14.7 - Express the moment of inertia Iz of the solid...Ch. 14.7 - Find the volumes of the solids in Exercises...Ch. 14.7 - Find the volumes of the solids in Exercises...Ch. 14.7 - Find the volumes of the solids in Exercises...Ch. 14.7 - Prob. 46ECh. 14.7 - Find the volumes of the solids in Exercises...Ch. 14.7 - Prob. 48ECh. 14.7 - Sphere and cones Find the volume of the portion of...Ch. 14.7 - Prob. 50ECh. 14.7 - Prob. 51ECh. 14.7 - Prob. 52ECh. 14.7 - Cylinder and paraboloid Find the volume of the...Ch. 14.7 - Cylinder and paraboloids Find the volume of the...Ch. 14.7 - Prob. 55ECh. 14.7 - Prob. 56ECh. 14.7 - Prob. 57ECh. 14.7 - Prob. 58ECh. 14.7 - Region trapped by paraboloids Find the volume of...Ch. 14.7 - Paraboloid and cylinder Find the volume of the...Ch. 14.7 - Prob. 61ECh. 14.7 - Prob. 62ECh. 14.7 - Prob. 63ECh. 14.7 - Prob. 64ECh. 14.7 - Find the average value of the function f(, , ) = ...Ch. 14.7 - Find the average value of the function f(ρ, ϕ, θ)...Ch. 14.7 - Prob. 67ECh. 14.7 - Prob. 68ECh. 14.7 - Prob. 69ECh. 14.7 - Prob. 70ECh. 14.7 - Prob. 71ECh. 14.7 - Prob. 72ECh. 14.7 - Prob. 73ECh. 14.7 - Prob. 74ECh. 14.7 - Prob. 75ECh. 14.7 - Prob. 76ECh. 14.7 - Variable density A solid is bounded below by the...Ch. 14.7 - Variable density A solid ball is bounded by the...Ch. 14.7 - Prob. 79ECh. 14.7 - Prob. 80ECh. 14.7 - Prob. 81ECh. 14.7 - Mass of planet’s atmosphere A spherical planet of...Ch. 14.8 - Solve the system
for x and y in terms of u and v....Ch. 14.8 - Prob. 2ECh. 14.8 - Solve the system
for x and y in terms of u and v....Ch. 14.8 - Prob. 4ECh. 14.8 - Prob. 5ECh. 14.8 - Prob. 6ECh. 14.8 - Use the transformation in Exercise 3 to evaluate...Ch. 14.8 - Prob. 8ECh. 14.8 - Let R be the region in the first quadrant of the...Ch. 14.8 - Find the Jacobian of the transformation and...Ch. 14.8 - Prob. 11ECh. 14.8 - The area of an ellipse The area πab of the ellipse...Ch. 14.8 - Prob. 13ECh. 14.8 - Prob. 14ECh. 14.8 - Prob. 15ECh. 14.8 - Prob. 16ECh. 14.8 - Prob. 17ECh. 14.8 - Prob. 18ECh. 14.8 - Prob. 19ECh. 14.8 - Prob. 20ECh. 14.8 - Prob. 21ECh. 14.8 - Prob. 22ECh. 14.8 - Prob. 23ECh. 14.8 - Substitutions in single integrals How can...Ch. 14.8 - Prob. 25ECh. 14.8 - Prob. 26ECh. 14.8 - Prob. 27ECh. 14.8 - Prob. 28ECh. 14 - Prob. 1GYRCh. 14 - Prob. 2GYRCh. 14 - Prob. 3GYRCh. 14 - How can you change a double integral in...Ch. 14 - Prob. 5GYRCh. 14 - Prob. 6GYRCh. 14 - How are double and triple integrals in rectangular...Ch. 14 - Prob. 8GYRCh. 14 - How are triple integrals in cylindrical and...Ch. 14 - Prob. 10GYRCh. 14 - How are substitutions in triple integrals pictured...Ch. 14 - Prob. 1PECh. 14 - Prob. 2PECh. 14 - Prob. 3PECh. 14 - Prob. 4PECh. 14 - Prob. 5PECh. 14 - Prob. 6PECh. 14 - Prob. 7PECh. 14 - Prob. 8PECh. 14 - Prob. 9PECh. 14 - Prob. 10PECh. 14 - Prob. 11PECh. 14 - Prob. 12PECh. 14 - Prob. 13PECh. 14 - Prob. 14PECh. 14 - Prob. 15PECh. 14 - Prob. 16PECh. 14 - Prob. 17PECh. 14 - Prob. 18PECh. 14 - Prob. 19PECh. 14 - Prob. 20PECh. 14 - Prob. 21PECh. 14 - Prob. 22PECh. 14 - Prob. 23PECh. 14 - Prob. 24PECh. 14 - Prob. 25PECh. 14 - Prob. 26PECh. 14 - Prob. 27PECh. 14 - Prob. 28PECh. 14 - Prob. 29PECh. 14 - Prob. 30PECh. 14 - Prob. 31PECh. 14 - Prob. 32PECh. 14 - Prob. 33PECh. 14 - Prob. 34PECh. 14 - Prob. 35PECh. 14 - Prob. 36PECh. 14 - Prob. 37PECh. 14 - Prob. 38PECh. 14 - Prob. 39PECh. 14 - Prob. 40PECh. 14 - Prob. 41PECh. 14 - Prob. 42PECh. 14 - Prob. 43PECh. 14 - Prob. 44PECh. 14 - Prob. 45PECh. 14 - Prob. 46PECh. 14 - Prob. 47PECh. 14 - Prob. 48PECh. 14 - Prob. 49PECh. 14 - Prob. 50PECh. 14 - Prob. 51PECh. 14 - Centroid Find the centroid of the plane region...Ch. 14 - Prob. 53PECh. 14 - Prob. 54PECh. 14 - Prob. 1AAECh. 14 - Water in a hemispherical bowl A hemispherical bowl...Ch. 14 - Prob. 3AAECh. 14 - Prob. 4AAECh. 14 - Prob. 5AAECh. 14 - Prob. 6AAECh. 14 - Prob. 7AAECh. 14 - Prob. 8AAECh. 14 - Prob. 9AAECh. 14 - Prob. 10AAECh. 14 - Prob. 11AAECh. 14 - Prob. 12AAECh. 14 - Prob. 13AAECh. 14 - Prob. 14AAECh. 14 - Minimizing polar inertia A thin plate of constant...Ch. 14 - Prob. 16AAECh. 14 - Prob. 17AAECh. 14 - Centroid of a boomerang Find the centroid of the...Ch. 14 - Prob. 19AAECh. 14 - Prob. 20AAECh. 14 - Prob. 21AAECh. 14 - Prob. 22AAECh. 14 - Prob. 23AAECh. 14 - Prob. 24AAECh. 14 - Prob. 25AAECh. 14 - Prob. 26AAECh. 14 - Prob. 27AAECh. 14 - Prob. 28AAE
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- (10) (16 points) Let R>0. Consider the truncated sphere S given as x² + y² + (z = √15R)² = R², z ≥0. where F(x, y, z) = −yi + xj . (a) (8 points) Consider the vector field V (x, y, z) = (▼ × F)(x, y, z) Think of S as a hot-air balloon where the vector field V is the velocity vector field measuring the hot gasses escaping through the porous surface S. The flux of V across S gives the volume flow rate of the gasses through S. Calculate this flux. Hint: Parametrize the boundary OS. Then use Stokes' Theorem. (b) (8 points) Calculate the surface area of the balloon. To calculate the surface area, do the following: Translate the balloon surface S by the vector (-15)k. The translated surface, call it S+ is part of the sphere x² + y²+z² = R². Why do S and S+ have the same area? ⚫ Calculate the area of S+. What is the natural spherical parametrization of S+?arrow_forward(1) (8 points) Let c(t) = (et, et sint, et cost). Reparametrize c as a unit speed curve starting from the point (1,0,1).arrow_forward(9) (16 points) Let F(x, y, z) = (x² + y − 4)i + 3xyj + (2x2 +z²)k = - = (x²+y4,3xy, 2x2 + 2²). (a) (4 points) Calculate the divergence and curl of F. (b) (6 points) Find the flux of V x F across the surface S given by x² + y²+2² = 16, z ≥ 0. (c) (6 points) Find the flux of F across the boundary of the unit cube E = [0,1] × [0,1] x [0,1].arrow_forward
- (8) (12 points) (a) (8 points) Let C be the circle x² + y² = 4. Let F(x, y) = (2y + e²)i + (x + sin(y²))j. Evaluate the line integral JF. F.ds. Hint: First calculate V x F. (b) (4 points) Let S be the surface r² + y² + z² = 4, z ≤0. Calculate the flux integral √(V × F) F).dS. Justify your answer.arrow_forwardDetermine whether the Law of Sines or the Law of Cosines can be used to find another measure of the triangle. a = 13, b = 15, C = 68° Law of Sines Law of Cosines Then solve the triangle. (Round your answers to four decimal places.) C = 15.7449 A = 49.9288 B = 62.0712 × Need Help? Read It Watch Itarrow_forward(4) (10 points) Evaluate √(x² + y² + z²)¹⁄² exp[}(x² + y² + z²)²] dV where D is the region defined by 1< x² + y²+ z² ≤4 and √√3(x² + y²) ≤ z. Note: exp(x² + y²+ 2²)²] means el (x²+ y²+=²)²]¸arrow_forward
- (2) (12 points) Let f(x,y) = x²e¯. (a) (4 points) Calculate Vf. (b) (4 points) Given x directional derivative 0, find the line of vectors u = D₁f(x, y) = 0. (u1, 2) such that the - (c) (4 points) Let u= (1+3√3). Show that Duƒ(1, 0) = ¦|▼ƒ(1,0)| . What is the angle between Vf(1,0) and the vector u? Explain.arrow_forwardFind the missing values by solving the parallelogram shown in the figure. (The lengths of the diagonals are given by c and d. Round your answers to two decimal places.) a b 29 39 66.50 C 17.40 d 0 54.0 126° a Ꮎ b darrow_forward(5) (10 points) Let D be the parallelogram in the xy-plane with vertices (0, 0), (1, 1), (1, 1), (0, -2). Let f(x,y) = xy/2. Use the linear change of variables T(u, v)=(u,u2v) = (x, y) 1 to calculate the integral f(x,y) dA= 0 ↓ The domain of T is a rectangle R. What is R? |ǝ(x, y) du dv. |ð(u, v)|arrow_forward
- 2 Anot ined sove in peaper PV+96252 Q3// Find the volume of the region between the cylinder z = y2 and the xy- plane that is bounded by the planes x=1, x=2,y=-2,andy=2. vertical rect a Q4// Draw and Evaluate Soxy-2sin (ny2)dydx D Lake tarrow_forwardDetermine whether the Law of Sines or the Law of Cosines can be used to find another measure of the triangle. B 13 cm 97° Law of Sines Law of Cosines A 43° Then solve the triangle. (Round your answers to two decimal places.) b = x C = A = 40.00arrow_forwardFind the missing values by solving the parallelogram shown in the figure. (The lengths of the diagonals are given by c and d. Round your answers to two decimal places.) a 29 b 39 d Ꮎ 126° a Ꮎ b darrow_forward
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