Math

CalculusMultivariable CalculusEvaluating a Line Integral In exercises 23–32, evaluate ∫ C F . d r along each path. ( Hint: If F is conservative, the integration may be easier on an alternative path.) F (x, y, z) = − y i + x j + 3 x z 2 k (a). C 1 : r 1 ( t ) = cos t i + sin t j + t k , 0 ≤ t ≤ π (b). C 2 : r 2 ( t ) = ( 1 − 2 t ) i + π t k , 0 ≤ t ≤ 1BuyFind*launch*

11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275378

Chapter 15.3, Problem 30E

Textbook Problem

**Evaluating a Line Integral** In exercises 23–32, evaluate
*Hint:* If **F** is conservative, the integration may be easier on an alternative path.)

**F**(x, y, z) =

(a).

(b).

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

Evaluate the integral. x3+1x31dx

Calculus (MindTap Course List)

Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

Finite Mathematics and Applied Calculus (MindTap Course List)

Prove that the function f(x)=x101+x51+x+1 has neither a local maximum nor a local minimum.

Single Variable Calculus: Early Transcendentals

In Exercises 16-18, find the vertical and horizontal asymptotes of the graph of the function.
17.

Calculus: An Applied Approach (MindTap Course List)

Match the differential equations with the solution graphs labeled IIV. Give reasons for your choices. (a) y = 1...

Single Variable Calculus

The graph of is:
a)
b)
c)
d)

Study Guide for Stewart's Multivariable Calculus, 8th

Sketch the region enclosed by the given curves and find its area. y = cos x, y = 4x2 1

Single Variable Calculus: Early Transcendentals, Volume I

Critical Numbers Consider the function f(x)=x4x+2. Is x=2 a critical number of f? Why or why not?

Calculus: Early Transcendental Functions

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. ...

Calculus: Early Transcendentals

Show that a conic with focus at the origin, eccentricity e, and directrix y = d has polar equation r=ed1esin

Multivariable Calculus

Solving Trigonometric Equations by Using Identities Solve the given equation. 3. 2 cos2 + sin = 1

Precalculus: Mathematics for Calculus (Standalone Book)

Determining Convergence or Divergence In Exercises 37-40, use the polynomial test given in Exercise 36 to deter...

Calculus of a Single Variable

PHOTOSYNTHESIS The rate of production R in photosynthesis is related to the light intensity I by the function R...

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

Air is being pumped into a chamber so that after t seconds the pressure in the chamber is 3 + 3x + x2 pounds/in...

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

Horizontal and Vertical Tangency In Exercises 27 and 28, find all points (if any) of horizontal and vertical ta...

Calculus: Early Transcendental Functions (MindTap Course List)

Calculate each expression in Exercises 124, giving the answer as a whole number or a fraction in lowest terms. ...

Finite Mathematics

Combination locks How many permutations does a combination lock have if each combination has 3 numbers, no two ...

College Algebra (MindTap Course List)

In Exercises 13 to 18, use only the given information to state the reason why ABCDBC a'. Redraw the figure and ...

Elementary Geometry For College Students, 7e

Find the exact value of each of the following. sin210

Trigonometry (MindTap Course List)

The SAT is standardized so that the average score on the verbal test is 500 each year. If you select a group of...

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

Game Shows The spinner for the television game show Wheel of Fortune is shown below. On a single spin of the wh...

Mathematical Excursions (MindTap Course List)

Solve, using the substitution method, and check; 2xy=4y=x

Elementary Technical Mathematics

Subtract each of the fractions in Exercises 7 through 9. Reduce to lowest terms where necessary. 9.a. 7458 b. 3...

Mathematics For Machine Technology

Let X have a standard beta density with parameters and . a. Verify the formula for E(X) given in the section. ...

Probability and Statistics for Engineering and the Sciences

SOC A random sample of 100 cable TV movies contained an average of 3.38 acts of physical violence per program w...

Essentials Of Statistics

In Exercises 23-28, characterize the solutions to the following equations by evaluating the discriminant.

Elements Of Modern Algebra

Marketable Accounts Quick Acid Test Company Cash Securities Receivable Assets Ratio 6. Super-saver, Inc. $39,35...

Contemporary Mathematics for Business & Consumers

For Problems 27-36, simplify each of the algebraic expressions by combining similar terms. 3(2x3y)4(3x+5y)x

Intermediate Algebra

In Problems 1-16, find each function’s relative maxima, relative minima, and saddle points, if they exist.
5.
...

Mathematical Applications for the Management, Life, and Social Sciences

A cereal box measures 2 in. by 8 in. by 10 in. What is the volume of the box? How many square inches of cardboa...

Elementary Geometry for College Students

In Exercise 15-24, compute the steady-state matrix of each stochastic matrix. [350251]

Finite Mathematics for the Managerial, Life, and Social Sciences

Critical Thinking You are interested in the weights of backpacks students carry to class and decide to conduct ...

Understanding Basic Statistics

Refer to the study description and sample statistics given in the previous exercise. a. Suppose that the sample...

Introduction To Statistics And Data Analysis

15. In Example 11.2 (p. 343) we presented a repeated measures research study demonstrating that swearing can he...

Statistics for The Behavioral Sciences (MindTap Course List)

Refraction of Light When light waves traveling in a transparent medium strike the surface of a second transpare...

Calculus (MindTap Course List)

Consider the following data for two variables, x and y. x 2 3 4 5 7 7 7 8 9 y 4 5 4 6 4 6 9 5 11 a. Does there ...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Explain the general advantages and disadvantages of within-subjects designs compared to between-subjects design...

Research Methods for the Behavioral Sciences (MindTap Course List)

The following crosstabulation shows household income by educational level of the head of household (Statistical...

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

In Exercises 27-56, use a calculator to find each value. Give the entire display of your calculator. a. 2e5.6e3...

Mathematics: A Practical Odyssey

Compare and contrast the four scales of measurement (nominal, ordinal, interval, and ratio) and identify exampl...

Research Methods for the Behavioral Sciences (MindTap Course List)

Solving an Equation In Exercises 61-64, solve the equation. (Round your solution to three decimal places.) 2.76...

College Algebra

Prove each statement in 8-23 by mathematical induction. 32x -1 is divible by 8, for each integer n0.

Discrete Mathematics With Applications

Cost Estimation Conclusion. Refer to exercise 21, where data on production volume and cost were used to develop...

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

Using Laptops on Flights. An airline promotion to business travelers is based on the assumption that two-thirds...

Essentials Of Statistics For Business & Economics

In Problems 3546 find the general solution of the given system. x=(4554)x

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

Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons...

Introductory Statistics

[T] The cost to remove a toxin from a lake is modeled by the function C(p)=75p/(85p) , where C is the cost (in ...

Calculus Volume 1

In the following exercises, express the limits as integrals. 61. limni=1n(5( x i * )23( x i * )3)x over [0, 2]

Calculus Volume 2

Reminder Round all answers to two decimal places unless otherwise indicated. Profit The total cost C for a manu...

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