BuyFind*launch*

11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275378

Chapter 15.5, Problem 29E

Textbook Problem

Representing a Surface of Revolution Parametrically

In Exercises 27-32, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis.

Function Axis of Revolution

Multivariable Calculus

Ch. 15.1 - Vector Field Define a vector field in the plane...Ch. 15.1 - CONCEPT CHECK Conservative Vector Field What is a...Ch. 15.1 - Potential Function Describe how to find a...Ch. 15.1 - CONCEPT CHECK Vector Field A vector field in space...Ch. 15.1 - Matching In Exercise 5-8, match the vector field...Ch. 15.1 - Matching In Exercise 5-8, match the vector field...Ch. 15.1 - Matching In Exercise 5-8, match the vector field...Ch. 15.1 - Matching In Exercise 5-8, match the vector field...Ch. 15.1 - Sketching a Vector Field In Exercises 914, find F...Ch. 15.1 - Sketching a Vector Field In Exercises 914, find F...

Ch. 15.1 - Sketching a Vector Field In Exercises 914, find F...Ch. 15.1 - Sketching a Vector Field In Exercises 914, find F...Ch. 15.1 - Sketching a Vector Field In Exercises 914, find F...Ch. 15.1 - Sketching a Vector Field In Exercises 914, find F...Ch. 15.1 - Graphing a Vector Field Using Technology In...Ch. 15.1 - Sketching a Vector Field In Exercises 1518, use a...Ch. 15.1 - Graphing a Vector Field Using Technology In...Ch. 15.1 - Graphing a Vector Field Using Technology In...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - In Exercises 1928, find the conservative vector...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - In Exercises 1928, find the conservative vector...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Finding a Conservative Vector Field In Exercises...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Finding a Potential Function In Exercises 37-44,...Ch. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Finding a Potential Function In Exercises 37-44,...Ch. 15.1 - Finding the Curl of a Vector Field In Exercises...Ch. 15.1 - Finding the Curl of a Vector Field In Exercises...Ch. 15.1 - Finding the Curl of a Vector Field In Exercises...Ch. 15.1 - Finding the Curl of the Vector Field In Exercises...Ch. 15.1 - Finding the Curl of a Vector Field Using...Ch. 15.1 - Finding the Curl of a Vector Field Using...Ch. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - Finding the Divergence of the Vector Field In...Ch. 15.1 - Finding the Divergence of the Vector Field In...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.1 - EXPLORING CONCEPTS Think About It In Exercise...Ch. 15.1 - EXPLORING CONCEPTS Think About It In Exercise...Ch. 15.1 - HOW DO YOU SEE IT? Several representative vectors...Ch. 15.1 - Curl of a Cross Product In Exercises 69 and 70,...Ch. 15.1 - Curl of a Cross Product In Exercises 69 and 70,...Ch. 15.1 - Curl of the Curl of a Vector Field In Exercises 71...Ch. 15.1 - Curl of the Curl of a Vector Field In Exercises 71...Ch. 15.1 - Divergence of a Cross Product In Exercises 73 and...Ch. 15.1 - Divergence of a Cross Product In Exercises 73 and...Ch. 15.1 - Divergence of the Curl of a Vector Field In...Ch. 15.1 - Divergence of the Curl of a Vector Field In...Ch. 15.1 - Proof In parts (a) - (h), prove the property for...Ch. 15.1 - Earths magnetic field A cross section of Earths...Ch. 15.2 - CONCEPT CHECK Line integral What is the physical...Ch. 15.2 - CONCEPT CHECK Orientation of a Curve Describe how...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 9-12, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line IntegralIn Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 1316, (a)...Ch. 15.2 - Evaluating a Line Integral In Exercises 17 and 18,...Ch. 15.2 - Evaluating a Line Integral In Exercises 17 and 18,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Evaluating a Line Integral In Exercises 19-22,...Ch. 15.2 - Mass In Exercises 23 and 24, find the total mass...Ch. 15.2 - Mass In Exercises 23 and 24, find the total mass...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Mass In Exercises 25-28, find the total mass of...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field Using...Ch. 15.2 - Evaluating a Line Integral of a Vector Field Using...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 3742, find the work done by the...Ch. 15.2 - Work In Exercises 37-42, find the work done by the...Ch. 15.2 - Work In Exercises 3742, find the work done by the...Ch. 15.2 - Work In Exercises 43-46, determine whether the...Ch. 15.2 - Work In Exercises 43-46, determine whether the...Ch. 15.2 - Work In Exercises 43-46, determine whether the...Ch. 15.2 - Work In Exercises 43-46, determine whether the...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Demonstrating a Property In Exercises 49-52,...Ch. 15.2 - Demonstrating a Property In Exercises 49-52,...Ch. 15.2 - Demonstrating a Property In Exercises 49-52,...Ch. 15.2 - Demonstrating a Property In Exercises 49-52,...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Engine Design A tractor engine has a steel...Ch. 15.2 - Building Design The ceiling of a building has a...Ch. 15.2 - Moment of Inertia Consider a wire of density (x,y)...Ch. 15.2 - Moment of Inertia Consider a wire of density (x,y)...Ch. 15.2 - Investigation The top outer edge of a solid with...Ch. 15.2 - Work A particle moves along the path y=x2 from the...Ch. 15.2 - Work Find the work done by a person weighing 175...Ch. 15.2 - Investigation Determine the value of c such that...Ch. 15.2 - Think About It A path C is given by x=t,y=2t where...Ch. 15.2 - Line Integrals Let F(x,y)=2xi+xy2j and consider...Ch. 15.2 - Ordering Surfaces Order the surfaces in ascending...Ch. 15.2 - HOW DO YOU SEE IT? For each of the following,...Ch. 15.2 - True or False? In Exercises 85 and 86, determine...Ch. 15.2 - True or False? In Exercises 85 and 86, determine...Ch. 15.2 - Work Consider a particle that moves through the...Ch. 15.3 - CONCEPT CHECK Fundamental Theorem of Line...Ch. 15.3 - Independence of Path What does it mean for a line...Ch. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - In Exercises 918, Using the Fundamental Theorem of...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals....Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Finding Work in a Conservative Force Field In...Ch. 15.3 - Finding Work in a Conservative Force Field In...Ch. 15.3 - Finding Work in a Conservative Force Field In...Ch. 15.3 - Finding Work in a Conservative Force Field In...Ch. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Evaluating a Line IntegralIn Exercises 2332,...Ch. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Evaluating a Line Integral In exercises 2332,...Ch. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Evaluating a Line Integral In exercises 2332,...Ch. 15.3 - Evaluating a Line Integral In exercises 2332,...Ch. 15.3 - Evaluating a Line Integral In exercises 2332,...Ch. 15.3 - Evaluating a Line Integral In exercises 2332,...Ch. 15.3 - Work A stone weighing 1 pound is attached to the...Ch. 15.3 - Work A grappling hook weighing 1 kilogram is...Ch. 15.3 - Work A zip line is installed 50 meters above...Ch. 15.3 - Can you find a path for the zip line in Exercise...Ch. 15.3 - EXPLORING CONCEPTS Think About It Consider...Ch. 15.3 - Work Let F(x, y, z) = a1i+a2j+a3k be a constant...Ch. 15.3 - Using Different Methods Use two different methods...Ch. 15.3 - HOW DO YOU SEE IT? Consider the force field shown...Ch. 15.3 - Graphical Reasoning In Exercises 41 and 42,...Ch. 15.3 - Graphical Reasoning In Exercises 41 and 42,...Ch. 15.3 - True or False? In Exercises 43-46, determine...Ch. 15.3 - True or False? In Exercises 43-46, determine...Ch. 15.3 - True or False? In Exercises 43-46, determine...Ch. 15.3 - True or False? In Exercises 4346, determine...Ch. 15.3 - Harmonic Function A function f is called harmonic...Ch. 15.3 - Kinetic and Potential Energy The kinetic energy of...Ch. 15.3 - Let F(x,y)=yx2+y2ixx2+y2j Show that Nx=My. Let...Ch. 15.4 - CONCEPT CHECK WritingWhat does it mean for a curve...Ch. 15.4 - Green's Theorem Explain the usefulness of Green's...Ch. 15.4 - Integral SignWhat information do you learn from...Ch. 15.4 - AreaDescribe how to find the area of a plane...Ch. 15.4 - Verifying Greens TheoremIn Exercises 58, verify...Ch. 15.4 - Verifying Greens TheoremIn Exercises 58, verify...Ch. 15.4 - Verifying Greens TheoremIn Exercises 58, verify...Ch. 15.4 - Verifying Green's Theorem In Exercises 5-8, verify...Ch. 15.4 - Verifying Greens TheoremIn Exercises 9 and 10,...Ch. 15.4 - Verifying Greens TheoremIn Exercises 9 and 10,...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens Theorem In...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - WorkIn Exercises 2528, use Greens Theorem to...Ch. 15.4 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15.4 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15.4 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15.4 - Area In Exercises 29-32, use a line integral to...Ch. 15.4 - Area In Exercises 29-32, use a line integral to...Ch. 15.4 - Area In Exercises 29-32, use a line integral to...Ch. 15.4 - Area In Exercises 2932, use a line integral to...Ch. 15.4 - Using Green's Theorem to Verify a Formula In...Ch. 15.4 - Using Green's Theorem to Verify a Formula In...Ch. 15.4 - Centroid In Exercises 35-38, use the results of...Ch. 15.4 - Centroid In Exercises 35-38, use the results of...Ch. 15.4 - Centroid In Exercises 35-38, use the results of...Ch. 15.4 - Centroid In Exercises 35-38, use the results of...Ch. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Area In Exercises 39-42, use the result of...Ch. 15.4 - Maximum Value (a) Evaluate C1y3dx+(27xx3)dy, where...Ch. 15.4 - HOW DO YOU SEE IT? The figure shows a region R...Ch. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Think About It Let I=Cydxxdyx2+y2 Where C is a...Ch. 15.4 - Think About It For each given path, verify Greens...Ch. 15.4 - Proff (a) Let C be the line segment joining...Ch. 15.4 - Area Use the result of Exercise 49(b) to find the...Ch. 15.4 - Proof In Exercises 51 and 52, prove the identity,...Ch. 15.4 - Proof In Exercises 51 and 52, prove the identity,...Ch. 15.4 - Proof Let F=Mi+Nj, where M and N have continuous...Ch. 15.4 - Find the least possible area of a convex set in...Ch. 15.5 - CONCEPT CHECK Parametric Surface Explain how a...Ch. 15.5 - CONCEPT CHECK Surface Area A surface S is...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Matching In Exercises 38, match the vector-valued...Ch. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Graphing a Parametric Surface In Exercises 1316,...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Representing a Surface of Revolution...Ch. 15.5 - Representing a Surface of Revolution...Ch. 15.5 - Representing a Surface of Revolution...Ch. 15.5 - Representing a Surface Revolution ParametricallyIn...Ch. 15.5 - Representing a Surface Revolution ParametricallyIn...Ch. 15.5 - Representing a Surface Revolution ParametricallyIn...Ch. 15.5 - Finding a Tangent Plane In Exercises 33-36, find...Ch. 15.5 - Finding a Tangent Plane In Exercises 3336, find an...Ch. 15.5 - Finding a Tangent Plane In Exercises 33-36, find...Ch. 15.5 - Finding a Tangent Plane In Exercises 33-36, find...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 3742, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - Finding Surface Area In Exercises 37-42, find the...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - EXPLORING CONCEPTS Think About It In Exercises...Ch. 15.5 - Representing a Cone Parametrically Show that the...Ch. 15.5 - HOW DO YOU SEE IT? The figures below are graphs of...Ch. 15.5 - Astroidal Sphere An equation of an astroidal...Ch. 15.5 - Different Views of a Surface Use a computer...Ch. 15.5 - Investigation Use a computer algebra system to...Ch. 15.5 - Investigation Consider the function in Exercise...Ch. 15.5 - Surface Area The surface of the dome on a new...Ch. 15.5 - Hyperboloid Find a vector-valued function for the...Ch. 15.5 - Area Use a computer algebra system to graph one...Ch. 15.5 - Surface Area Lei f be a nonnegative function such...Ch. 15.5 - Open-Ended Project The parametric equations...Ch. 15.5 - Mobius Strip The surface shown in the figure is...Ch. 15.6 - CONCEPT CHECK Surface Integral Explain how to set...Ch. 15.6 - CONCEPT CHECK Surface Integral For what conditions...Ch. 15.6 - CONCEPT CHECK Orientation of a Surface Integral...Ch. 15.6 - CONCEPT CHECK Flux What is the physical...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 9 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 9 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 11 and...Ch. 15.6 - Evaluating a Surface Integral In Exercises 11 and...Ch. 15.6 - Mass In Exercises 13 and 14, find the mass of the...Ch. 15.6 - Mass In Exercises 13 and 14, find the mass of the...Ch. 15.6 - Evaluating a Surface Integral In Exercises15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 31 and 32,...Ch. 15.6 - Evaluating a Flux Integral In Exercises 31 and 32,...Ch. 15.6 - Flow Rate In Exercises 33 and 34, use a computer...Ch. 15.6 - Flow Rate In Exercises 33-34, use a computer...Ch. 15.6 - Gauss's Law In Exercises 35 and 36, evaluate SENdS...Ch. 15.6 - Gauss's Law In Exercises 35 and 36, evaluate SENdS...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - Moments of Inertia In Exercises 37-40, use the...Ch. 15.6 - EXPLORING CONCEPTS Using Different Methods...Ch. 15.6 - HOW DO YOU SEE IT? Is the surface shown in the...Ch. 15.6 - Investigation (a) Use a computer algebra system to...Ch. 15.7 - CONCEPT CHECK Using Different Methods Suppose that...Ch. 15.7 - Classifying a Point in a Vector Field How do you...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Verifying the Divergence Theorem In Exercises 38,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Using the Divergence Theorem In Exercises 9-18,...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - Classifying a Point In Exercises 19-22, a vector...Ch. 15.7 - EXPLORING CONCEPTS Closed Surface What is the...Ch. 15.7 - HOW DO YOU SEE IT? The graph of a vector field F...Ch. 15.7 - Volume (a) Use the Divergence Theorem to verify...Ch. 15.7 - Constant Vector Field For the constant vector...Ch. 15.7 - Volume For the vector field F(x,y,z)=xi+yj+zk,...Ch. 15.7 - Verifying an Identity For the vector field...Ch. 15.7 - Proof In Exercises 31 and 32, prove the identity,...Ch. 15.7 - Proof In Exercises 31 and 32, prove the identity,...Ch. 15.8 - CONCEPT CHECK Stokess Theorem Explain the benefit...Ch. 15.8 - Curl What is the physical interpretation of curl?Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokess Theorem In Exercises 3-6, verify...Ch. 15.8 - Verifying Stokes Theorem In Exercises 3-6, verify...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess Theorem In Exercises 716, use...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess TheoremIn Exercises 716, use Stokess...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokess Theorem In Exercises 7-16, use...Ch. 15.8 - Using Stokes Theorem In Exercises 7-16, use Stokes...Ch. 15.8 - Using Stokes Theorem In Exercises 7-16, use Stokes...Ch. 15.8 - Motion of a Liquid In Exercises 17 and 18, the...Ch. 15.8 - Motion of a Liquid In Exercises 17 and 18, the...Ch. 15.8 - EXPLORING CONCEPTS Think About It Let K be a...Ch. 15.8 - HOW DO YOU SEE IT? Let S1 be the portion of the...Ch. 15.8 - Let G(x,y)=(yx2+4y2,xx2+4y2,0). Prove or disprove...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Finding a Conservative Vector Field In Exercises...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Testing for a Conservative Vector Field In...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Finding a Potential Function In Exercises 11-18,...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Evaluating a Line Integral Using Technology In...Ch. 15 - Mass In Exercises 33 and 34, find the total mass...Ch. 15 - Mass In Exercises 33 and 34, find the total mass...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Evaluating a Line Integral of a Vector Field In...Ch. 15 - Work In Exercises 39 and 40, find the work done by...Ch. 15 - Work In Exercises 39 and 40, find the work done by...Ch. 15 - Evaluating a Line Integral in Differential Form In...Ch. 15 - Evaluating a Line Integral in Differential Form...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Lateral Surface Area In Exercises 43 and44, find...Ch. 15 - Line Integral of a Conservative Vector Field In...Ch. 15 - Line Integral of a Conservative Vector Field In...Ch. 15 - Using the Fundamental Theorem of Line Integrals In...Ch. 15 - Using the Fundamental Theorem of Line Integrals In...Ch. 15 - Using the Fundamental Theorem of Line Integrals in...Ch. 15 - Using the Fundamental Theorem of Line Integrals in...Ch. 15 - Finding Work in a Conservative Force Field In...Ch. 15 - Finding Work in a Conservative Force Field In...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Evaluating a Line Integral Using Green's Theorem...Ch. 15 - Work In Exercises 59 and 60, use Greens Theorem to...Ch. 15 - Work In Exercises 59 and 60, use Greens Theorem to...Ch. 15 - Area In Exercises 61 and 61, use a line integral...Ch. 15 - Area In Exercises 61 and 62, use a line integral...Ch. 15 - Sketching a Parametric Surface In Exercise 63 and...Ch. 15 - Sketching a Parametric Surface In Exercise 63 and...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - Graphing a Parametric SurfaceIn Exercise 65 and...Ch. 15 - Representing a Surface Parametrically In Exercises...Ch. 15 - Representing a Surface Parametrically In Exercises...Ch. 15 - Representing a Surface of Revolution...Ch. 15 - Representing a Surface of Revolution...Ch. 15 - Finding a Surface Area In Exercises 71 and 72,...Ch. 15 - Finding a Surface Area In Exercises 71 and 72,...Ch. 15 - Evaluating a Surface Integral In Exercises 73 and...Ch. 15 - Evaluating a Surface Integral In Exercises 73 and...Ch. 15 - Mass In Exercises 75 and 76, find the mass of the...Ch. 15 - Mass In Exercises 75 and 76, find the mass of the...Ch. 15 - Evaluating a Surface Integral In Exercises 77 and...Ch. 15 - Evaluating a Surface Integral In Exercises 77 and...Ch. 15 - Evaluating a Flux Integral In Exercises 79 and 80,...Ch. 15 - Evaluating a Flux IntegralIn Exercises 79 and 80,...Ch. 15 - Using the Divergence TheoremIn Exercises 81 and...Ch. 15 - Using the Divergence TheoremIn Exercises 81 and...Ch. 15 - Using Stokess Theorem In Exercises 83 and 84, use...Ch. 15 - Using Stokess Theorem In Exercises 83 and 84, use...Ch. 15 - Motion of a Liquid In Exercises 85 and 86, the...Ch. 15 - Motion of a Liquid In Exercises 85 and 86, the...Ch. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Moments of Inertia Consider a wire of density...Ch. 15 - Moments of Inertia Using the formulas from...Ch. 15 - Laplace's Equation Let F(x,y,z)=xi+yj+zk, and let...Ch. 15 - Greens Theorem Consider the line integral...Ch. 15 - Area Use a line integral to find the area bounded...Ch. 15 - Area Use a line integral to find the area bounded...Ch. 15 - Work The force field F(x,y)=(x+y)i+(x2+1)j acts on...Ch. 15 - Work The force field F(x,y)=3x2y2i+2x3yj is shown...Ch. 15 - Area and Work How does the area of the ellipse...Ch. 15 - Verifying Identities (a) Let f and g be scalar...

Find more solutions based on key concepts

List the first five terms of the sequence. 12. a1 = 2, a2 = 1, an+1 = an an1

Multivariable Calculus

A particle moves on a vertical line so that its coordinate at time t is y = t3 12t + 3, t 0. (a) Find the vel...

Single Variable Calculus: Early Transcendentals

Find the sum of the areas of approximating rectangles for the area under f (x) = 48 − x2, between x = 1 and x =...

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Find the limit or show that it does not exist. limxsin2xx2+1

Calculus: Early Transcendentals

Convert the expressions in Exercises 31-36 to positive exponent form. 13x4+0.1x23

Finite Mathematics and Applied Calculus (MindTap Course List)

Use series to evaluate the limit. 64. limx01+x112xx2

Single Variable Calculus

Proof Prove that for any real number y there exists x in (/2,/2) such that tanx=y.

Calculus: Early Transcendental Functions (MindTap Course List)

find the real roots of each equation by factoring. 133. 14x2x+1=0

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Find an iterated integral for where S is the surface r(u, v) = u2i + v2j + uvk, 0 ≤ u ≤ 1, 0 ≤ v ≤ 1, and F(x,...

Study Guide for Stewart's Multivariable Calculus, 8th

Finding Higher-Order Derivatives In Exercises 93100, find the higher-order derivative. GivenDerivativef(x)=8x5/...

Calculus: An Applied Approach (MindTap Course List)

Using the Second Derivative Test In Exercises 37-58, find all relative extrema of the function. Use the Second ...

Calculus: Early Transcendental Functions

Find dy/dx by implicit differentiation. x24xy+y2=4

Calculus (MindTap Course List)

Find the value of the sum. 23. j=163j+1

Single Variable Calculus: Early Transcendentals, Volume I

Illumination The amount of illumination of a surface is proportional to the intensity of the light source, inve...

Calculus of a Single Variable

Domain Determine the values of the variable for which the expression is defined as a real number. 101. x29

Precalculus: Mathematics for Calculus (Standalone Book)

For Problems 5-54, perform the following operations with real numbers. Objectives 3-6 -12-7

Intermediate Algebra

Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the hor...

Trigonometry (MindTap Course List)

Practice Solve each problem. Geometric problem Find the dimensions of a rectangle whose area is 180 cm2 and who...

College Algebra (MindTap Course List)

In the text, it is said that a statistical test is a package of five basic ingredients. List these ingredients,...

Understanding Basic Statistics

For each measurement find: a. the degree of precision. b. the value that is equal to or less than the range of ...

Mathematics For Machine Technology

In Exercises 9-14, decide whether the specified values come from a linear, quadratic, exponential, or absolute ...

Finite Mathematics

As noted on page 304, when the two population means are equal, the estimated standard error for the independent...

Statistics for The Behavioral Sciences (MindTap Course List)

Light The percent of light that will pass through a material is given by the formula log P=kd, where P is the p...

Mathematical Excursions (MindTap Course List)

Automatic identification of the boundaries of significant structures within a medical image is an area of ongoi...

Probability and Statistics for Engineering and the Sciences

Consider the study described in the previous exercise. a. Is the conclusionthat reasoning using information dis...

Introduction To Statistics And Data Analysis

CJ The town of Shinbone, Kansas, has a population of 211,732 and experienced 47 bank robberies, 13 murders, and...

Essentials Of Statistics

In Problems 53-56, use a change-of-base formula to rewrite each logarithm. Then use a graphing utility to grap...

Mathematical Applications for the Management, Life, and Social Sciences

What are the two requirements for a random sample?

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Answer true or false for review Exercises 1 to 7. The value of a ratio must be less than 1.

Elementary Geometry for College Students

PSYCHOLOGY EXPERIMENTS A psychologist has constructed the following maze for use in an experiment. The maze is ...

Finite Mathematics for the Managerial, Life, and Social Sciences

Let H1={ [ 0 ],[ 15 ] },H2={ [ 0 ],[ 10 ],[ 20 ] }, and H3={ [ 0 ],[ 6 ],[ 12 ],[ 18 ],[ 24 ] } be subgroups of...

Elements Of Modern Algebra

Subtract the following hexadecimal numbers. Check using decimals. F35

Elementary Technical Mathematics

In Exercises 9 to 24, find an equation of the line described. Leave the solution in the form Ax+By = C. The lin...

Elementary Geometry For College Students, 7e

Reduce the following fractions to lowest terms.
a. b.

Contemporary Mathematics for Business & Consumers

In Exercises 11 and 12, it was assumed that there was a single initial injection of the tracer drug into the co...

Calculus (MindTap Course List)

A technician services mailing machines at companies in the Phoenix area. Depending onthe type of malfunction, t...

Statistics for Business & Economics, Revised (MindTap Course List)

Refer to exercise 18. Use the Bonferroni adjustment to test for a significant difference between all pairs of m...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Describe the general characteristics of a within-subject experimental design and identify these designs when th...

Research Methods for the Behavioral Sciences (MindTap Course List)

Define the validity of measurement and explain why and how it is measured.

Research Methods for the Behavioral Sciences (MindTap Course List)

The data from exercise 2 follow.
The estimated regression equation for these data is .
Compute SSE, SST, and S...

Essentials Of Statistics For Business & Economics

Are the melanin hair color genes MBd, MBw and MBk dominant, recessive or co dominant? Are the redness hair colo...

Mathematics: A Practical Odyssey

Reminder: Round all answer to two decimal places unless otherwise indicated. a. Use a formula to express the wo...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

Extend the argument given in the proof of Lemma 10.4.1 to show that a tree with more than one vertex has at lea...

Discrete Mathematics With Applications

Refer to the Hudson Marine problem in exercise 53.
Compute the centered moving average values for this time ser...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Business In Exercise 77, there are a total number of 2206 stores at the end of 2010. Write the terms of a seque...

College Algebra

In Problems 914 use an appropriate infinite series method about x = 0 to find two solutions of the given differ...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

In the following exercises, solve for the antiderivative f of with C = 0, the given interval [a, b]. Approximat...

Calculus Volume 2