Calculus
10th Edition
ISBN: 9781285057095
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 15.5, Problem 58E
To determine
To prove: The Surface area given by
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Chapter 15 Solutions
Calculus
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 - Prob. 7ECh. 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 - Prob. 12ECh. 15.1 - Prob. 13ECh. 15.1 - Prob. 14ECh. 15.1 - Prob. 15ECh. 15.1 - Prob. 16ECh. 15.1 - Prob. 17ECh. 15.1 - Prob. 18ECh. 15.1 - Prob. 19ECh. 15.1 - Prob. 20ECh. 15.1 - Prob. 21ECh. 15.1 - Prob. 22ECh. 15.1 - Prob. 23ECh. 15.1 - Prob. 24ECh. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Testing for a Conservative Vector Field In...Ch. 15.1 - Prob. 27ECh. 15.1 - Prob. 28ECh. 15.1 - Prob. 29ECh. 15.1 - Prob. 30ECh. 15.1 - Prob. 31ECh. 15.1 - Prob. 32ECh. 15.1 - Finding a Potential Function In Exercises 3342,...Ch. 15.1 - Prob. 34ECh. 15.1 - Prob. 35ECh. 15.1 - Finding a Potential Function In Exercises 3744,...Ch. 15.1 - Prob. 37ECh. 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 3342,...Ch. 15.1 - Prob. 42ECh. 15.1 - Prob. 43ECh. 15.1 - Finding the Curl of a Vector Field In Exercises...Ch. 15.1 - Prob. 45ECh. 15.1 - Prob. 46ECh. 15.1 - Prob. 47ECh. 15.1 - Prob. 48ECh. 15.1 - Prob. 49ECh. 15.1 - Prob. 50ECh. 15.1 - Prob. 51ECh. 15.1 - Prob. 52ECh. 15.1 - Prob. 53ECh. 15.1 - Prob. 54ECh. 15.1 - Prob. 55ECh. 15.1 - Finding a Potential Function In Exercises 51-56,...Ch. 15.1 - Finding the Divergence of a Vector Field In...Ch. 15.1 - Prob. 58ECh. 15.1 - Prob. 59ECh. 15.1 - Prob. 60ECh. 15.1 - Prob. 61ECh. 15.1 - Prob. 62ECh. 15.1 - Prob. 63ECh. 15.1 - Prob. 64ECh. 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 - 57095-15.1-67E-Question-Digital.docx Curl Define...Ch. 15.1 - Prob. 68ECh. 15.1 - Curl of a Cross Product In Exercises 69 and 70,...Ch. 15.1 - Prob. 70ECh. 15.1 - Prob. 71ECh. 15.1 - Prob. 72ECh. 15.1 - Prob. 73ECh. 15.1 - Prob. 74ECh. 15.1 - Divergence of the Curl of a Vector Field In...Ch. 15.1 - Prob. 76ECh. 15.1 - Proof In parts (a) - (h), prove the property for...Ch. 15.1 - Prob. 78ECh. 15.1 - Prob. 79ECh. 15.1 - Prob. 80ECh. 15.1 - Prob. 81ECh. 15.1 - True or False? In Exercises 7982, determine...Ch. 15.1 - Earths magnetic field A cross section of Earths...Ch. 15.2 - Finding a Piecewise Smooth Parametrization In...Ch. 15.2 - Prob. 2ECh. 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 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 - 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 - Prob. 14ECh. 15.2 - Evaluating a Line Integral In Exercises 15–18, (a)...Ch. 15.2 - Prob. 16ECh. 15.2 - Prob. 17ECh. 15.2 - Prob. 18ECh. 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 - 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 - Prob. 23ECh. 15.2 - Prob. 24ECh. 15.2 - Prob. 25ECh. 15.2 - Prob. 26ECh. 15.2 - Evaluating a Line Integral of a Vector Field In...Ch. 15.2 - Prob. 28ECh. 15.2 - Prob. 29ECh. 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 - Prob. 32ECh. 15.2 - Evaluating a Line Integral of a Vector Field Using...Ch. 15.2 - Prob. 34ECh. 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 - Prob. 39ECh. 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 - Prob. 46ECh. 15.2 - Prob. 47ECh. 15.2 - Prob. 48ECh. 15.2 - Prob. 49ECh. 15.2 - Prob. 50ECh. 15.2 - Evaluating a Line Integral in Differential Form In...Ch. 15.2 - Prob. 52ECh. 15.2 - Prob. 53ECh. 15.2 - Prob. 54ECh. 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 - Prob. 60ECh. 15.2 - Prob. 61ECh. 15.2 - Prob. 62ECh. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Prob. 64ECh. 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 - Prob. 67ECh. 15.2 - Prob. 68ECh. 15.2 - Prob. 69ECh. 15.2 - Lateral Surface Area In Exercises 65-72, find the...Ch. 15.2 - Prob. 71ECh. 15.2 - Prob. 72ECh. 15.2 - Prob. 73ECh. 15.2 - Prob. 74ECh. 15.2 - Prob. 75ECh. 15.2 - Prob. 76ECh. 15.2 - Prob. 77ECh. 15.2 - Prob. 78ECh. 15.2 - Prob. 79ECh. 15.2 - Prob. 80ECh. 15.2 - Prob. 81ECh. 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 - Prob. 85ECh. 15.2 - True or False? In Exercises 8386, determine...Ch. 15.2 - Prob. 87ECh. 15.3 - Line Integral of a Conservative Vector Field In...Ch. 15.3 - 57095-15.3-2E-Question-Digital.docx Evaluating a...Ch. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Testing for Conservative Vector Fields In...Ch. 15.3 - Prob. 6ECh. 15.3 - Prob. 7ECh. 15.3 - Prob. 8ECh. 15.3 - Prob. 9ECh. 15.3 - Prob. 10ECh. 15.3 - Evaluating a Line Integral of a Vector Field In...Ch. 15.3 - Prob. 12ECh. 15.3 - Prob. 13ECh. 15.3 - Prob. 14ECh. 15.3 - Prob. 15ECh. 15.3 - Evaluating a Line Integral In Exercises 23-32,...Ch. 15.3 - Prob. 17ECh. 15.3 - Prob. 18ECh. 15.3 - Evaluating a Line Integral of a Vector Field In...Ch. 15.3 - Prob. 20ECh. 15.3 - Prob. 21ECh. 15.3 - Prob. 22ECh. 15.3 - Prob. 23ECh. 15.3 - Prob. 24ECh. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Prob. 26ECh. 15.3 - Prob. 27ECh. 15.3 - Prob. 28ECh. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Prob. 30ECh. 15.3 - Prob. 31ECh. 15.3 - Prob. 32ECh. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Using the Fundamental Theorem of Line Integrals In...Ch. 15.3 - Work In Exercises 35 and 36, find the work done by...Ch. 15.3 - Prob. 36ECh. 15.3 - Prob. 37ECh. 15.3 - Prob. 38ECh. 15.3 - Work A zip line is installed 50 meters above...Ch. 15.3 - Prob. 40ECh. 15.3 - CONCEPT CHECK Fundamental Theorem of Line...Ch. 15.3 - Independence of Path What does it mean for a line...Ch. 15.3 - 43. Thing About It Let Find the value of the line...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 - Prob. 47ECh. 15.3 - Prob. 48ECh. 15.3 - Prob. 49ECh. 15.3 - Prob. 50ECh. 15.3 - Prob. 51ECh. 15.3 - Kinetic and Potential Energy The kinetic energy of...Ch. 15.3 - Prob. 53ECh. 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 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.4 - Prob. 7ECh. 15.4 - Prob. 8ECh. 15.4 - Prob. 9ECh. 15.4 - Evaluating a Line Integral Using Greens TheoremIn...Ch. 15.4 - Prob. 11ECh. 15.4 - Prob. 12ECh. 15.4 - Prob. 13ECh. 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 - Prob. 17ECh. 15.4 - Prob. 18ECh. 15.4 - Prob. 19ECh. 15.4 - Prob. 20ECh. 15.4 - Prob. 21ECh. 15.4 - Work In Exercises 25-28, use Greens Theorem to...Ch. 15.4 - Prob. 23ECh. 15.4 - Prob. 24ECh. 15.4 - Prob. 25ECh. 15.4 - Area In Exercises 25–28, use a line integral to...Ch. 15.4 - Prob. 27ECh. 15.4 - Prob. 28ECh. 15.4 - Prob. 29ECh. 15.4 - Prob. 30ECh. 15.4 - Using Green's Theorem to Verify a Formula In...Ch. 15.4 - Prob. 32ECh. 15.4 - Prob. 33ECh. 15.4 - Prob. 34ECh. 15.4 - Prob. 35ECh. 15.4 - Prob. 36ECh. 15.4 - Prob. 37ECh. 15.4 - Prob. 38ECh. 15.4 - Prob. 39ECh. 15.4 - Prob. 40ECh. 15.4 - Prob. 41ECh. 15.4 - HOW DO YOU SEE IT? The figure shows a region R...Ch. 15.4 - Prob. 43ECh. 15.4 - Greens Theorem: Region with a Hole Let R be the...Ch. 15.4 - Prob. 45ECh. 15.4 - Prob. 46ECh. 15.4 - Prob. 47ECh. 15.4 - Prob. 48ECh. 15.4 - Prob. 49ECh. 15.4 - Prob. 50ECh. 15.4 - Prob. 51ECh. 15.4 - Prob. 52ECh. 15.5 - 57095-15.5-47E-Question-Digital.docx Parametric...Ch. 15.5 - Prob. 48ECh. 15.5 - Prob. 1ECh. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Prob. 3ECh. 15.5 - Matching In Exercises 3-8, match the vector-valued...Ch. 15.5 - Prob. 5ECh. 15.5 - Matching In Exercises 1–6, match the vector-valued...Ch. 15.5 - Prob. 7ECh. 15.5 - Prob. 8ECh. 15.5 - Prob. 9ECh. 15.5 - Sketching a Parametric Surface In Exercises 9-12,...Ch. 15.5 - Prob. 11ECh. 15.5 - Prob. 12ECh. 15.5 - Prob. 13ECh. 15.5 - Prob. 14ECh. 15.5 - Prob. 15ECh. 15.5 - Prob. 16ECh. 15.5 - Prob. 17ECh. 15.5 - Prob. 18ECh. 15.5 - Prob. 19ECh. 15.5 - Prob. 20ECh. 15.5 - Prob. 21ECh. 15.5 - Prob. 22ECh. 15.5 - Prob. 23ECh. 15.5 - Prob. 24ECh. 15.5 - Prob. 25ECh. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Prob. 27ECh. 15.5 - Representing a Surface Parametrically In Exercises...Ch. 15.5 - Prob. 29ECh. 15.5 - Prob. 30ECh. 15.5 - Prob. 31ECh. 15.5 - Prob. 32ECh. 15.5 - Prob. 33ECh. 15.5 - Prob. 34ECh. 15.5 - Prob. 35ECh. 15.5 - Prob. 36ECh. 15.5 - Prob. 37ECh. 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 - Prob. 40ECh. 15.5 - Prob. 41ECh. 15.5 - Prob. 42ECh. 15.5 - Prob. 43ECh. 15.5 - 57095-15.5-44E-Question-Digital.docx Area In...Ch. 15.5 - Prob. 45ECh. 15.5 - Prob. 46ECh. 15.5 - Representing a Cone Parametrically Show that the...Ch. 15.5 - Prob. 50ECh. 15.5 - Prob. 51ECh. 15.5 - Different Views of a Surface Use a computer...Ch. 15.5 - Prob. 53ECh. 15.5 - Prob. 54ECh. 15.5 - Prob. 55ECh. 15.5 - Prob. 56ECh. 15.5 - Area Use a computer algebra system to graph one...Ch. 15.5 - Prob. 58ECh. 15.5 - Prob. 59ECh. 15.5 - Prob. 60ECh. 15.6 - 57095-15.6-39E-Question-Digital.docx WRITING ABOUT...Ch. 15.6 - Prob. 40ECh. 15.6 - Prob. 41ECh. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Prob. 2ECh. 15.6 - Evaluating a surface Integral In Exercise 58,...Ch. 15.6 - Prob. 4ECh. 15.6 - Evaluating a Surface Integral In Exercises 9 and...Ch. 15.6 - Prob. 6ECh. 15.6 - Prob. 9ECh. 15.6 - Prob. 10ECh. 15.6 - Prob. 7ECh. 15.6 - Prob. 8ECh. 15.6 - Prob. 11ECh. 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 - Prob. 14ECh. 15.6 - Prob. 15ECh. 15.6 - Evaluating a Surface Integral In Exercises 15-18,...Ch. 15.6 - Evaluating a Surface Integral In Exercises 19-24,...Ch. 15.6 - Prob. 18ECh. 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 - Prob. 24ECh. 15.6 - Evaluating a Flux Integral In Exercises 25-30,...Ch. 15.6 - Prob. 26ECh. 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 - Prob. 37ECh. 15.6 - Prob. 38ECh. 15.6 - Prob. 31ECh. 15.6 - Prob. 32ECh. 15.6 - Prob. 33ECh. 15.6 - Prob. 34ECh. 15.6 - Prob. 35ECh. 15.6 - Prob. 36ECh. 15.6 - HOW DO YOU SEE IT? Is the surface shown in the...Ch. 15.6 - Prob. 43ECh. 15.7 - Classifying a Point in a Vector Field How do you...Ch. 15.7 - Prob. 1ECh. 15.7 - Verifying the Divergence Theorem In Exercises 1–6,...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 - Prob. 12ECh. 15.7 - Prob. 13ECh. 15.7 - Prob. 14ECh. 15.7 - Prob. 15ECh. 15.7 - Prob. 16ECh. 15.7 - Prob. 17ECh. 15.7 - Prob. 18ECh. 15.7 - Prob. 19ECh. 15.7 - Prob. 21ECh. 15.7 - HOW DO YOU SEE IT? The graph of a vector field F...Ch. 15.7 - Prob. 23ECh. 15.7 - Prob. 24ECh. 15.7 - Prob. 25ECh. 15.7 - Prob. 26ECh. 15.7 - Prob. 27ECh. 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 - Prob. 1ECh. 15.8 - Prob. 2ECh. 15.8 - Prob. 3ECh. 15.8 - Prob. 4ECh. 15.8 - Prob. 5ECh. 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 - Prob. 9ECh. 15.8 - Prob. 10ECh. 15.8 - Prob. 11ECh. 15.8 - Prob. 12ECh. 15.8 - Prob. 13ECh. 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 - Prob. 19ECh. 15.8 - Prob. 20ECh. 15.8 - Prob. 23ECh. 15.8 - HOW DO YOU SEE IT? Let S1 be the portion of the...Ch. 15.8 - Prob. 25ECh. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Sketching a Vector Field In Exercises 1 and 2,...Ch. 15 - Prob. 3RECh. 15 - Prob. 4RECh. 15 - Prob. 5RECh. 15 - Prob. 6RECh. 15 - Prob. 7RECh. 15 - Prob. 8RECh. 15 - Prob. 9RECh. 15 - Prob. 10RECh. 15 - Prob. 11RECh. 15 - Prob. 12RECh. 15 - Prob. 13RECh. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Divergence and Curl In Exercises 19-26, find (a)...Ch. 15 - Prob. 16RECh. 15 - Prob. 17RECh. 15 - Prob. 18RECh. 15 - Prob. 19RECh. 15 - Prob. 20RECh. 15 - Prob. 21RECh. 15 - Prob. 22RECh. 15 - Evaluating a Line Integral In Exercises 27-30,...Ch. 15 - Prob. 24RECh. 15 - Prob. 25RECh. 15 - Prob. 26RECh. 15 - Prob. 27RECh. 15 - Prob. 28RECh. 15 - Prob. 29RECh. 15 - Prob. 30RECh. 15 - Prob. 31RECh. 15 - Prob. 32RECh. 15 - Prob. 33RECh. 15 - Prob. 34RECh. 15 - Prob. 35RECh. 15 - Prob. 36RECh. 15 - Prob. 37RECh. 15 - Prob. 38RECh. 15 - Prob. 39RECh. 15 - Prob. 40RECh. 15 - Prob. 41RECh. 15 - Prob. 42RECh. 15 - Prob. 43RECh. 15 - Prob. 44RECh. 15 - Prob. 45RECh. 15 - Prob. 46RECh. 15 - 57095-15-47RE-Question-Digital.docx Evaluating a...Ch. 15 - Prob. 48RECh. 15 - Prob. 49RECh. 15 - Prob. 50RECh. 15 - Prob. 51RECh. 15 - Prob. 52RECh. 15 - Prob. 53RECh. 15 - Prob. 54RECh. 15 - Prob. 55RECh. 15 - Prob. 56RECh. 15 - Prob. 57RECh. 15 - Prob. 58RECh. 15 - Prob. 59RECh. 15 - Prob. 60RECh. 15 - Prob. 61RECh. 15 - Prob. 1PSCh. 15 - Heat Flux Consider a single heat source located at...Ch. 15 - Prob. 3PSCh. 15 - Prob. 4PSCh. 15 - Prob. 5PSCh. 15 - Prob. 6PSCh. 15 - Prob. 7PSCh. 15 - Prob. 8PSCh. 15 - Prob. 9PSCh. 15 - Prob. 10PSCh. 15 - Prob. 11PSCh. 15 - Prob. 12PSCh. 15 - Prob. 13PS
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- Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = x2 + 4y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forwardWalking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a.In each case, find z’(t). b.Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = 4x2 - y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forwarda. Set up an integral for the area of the surface generated byrevolving the given curve about the indicated axis.b. Graph the curve to see what it looks like. If you can, graphthe surface too.c. Use your utility’s integral evaluator to find the surface’s areanumerically. x = sin y, 0 ≤ y ≤ π; y-axisarrow_forward
- Stokes’ Theorem for evaluating line integrals Evaluate theline integral ∮C F ⋅ dr by evaluating the surface integral in Stokes’Theorem with an appropriate choice of S. Assume C has a counterclockwiseorientation. F = ⟨y2, -z2, x⟩; C is the circle r(t) = ⟨3 cos t, 4 cos t, 5 sin t⟩, for 0 ≤ t ≤ 2π.arrow_forwardFine the volume generated by revolving the area bounded by the given curves and lines revolved about the given axis of rotation. Y = 1-x^2 y=0 is revolved about the x=1arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluatethe line integral in Stokes’ Theorem to determine the value of thesurface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upwarddirection. F = ⟨2y, -z, x - y - z⟩; S is the cap of the spherex2 + y2 + z2 = 25, for 3 ≤ x ≤ 5 (excluding its base).arrow_forward
- Surface integrals using an explicit description Evaluate the surface integral ∫∫S ƒ(x, y, z) dS using an explicit representation of the surface. ƒ(x, y, z) = 25 - x2 - y2; S is the hemisphere centered at theorigin with radius 5, for z ≥ 0.arrow_forwardWork Let R be the region in the first quadrant bounded by the curve y = sec-1 x and the line y = π/3. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.arrow_forwarda. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. b. Graph the curve to see what it looks like. If you can, graph the surface too. c. Use your utility’s integral evaluator to find the surface’s area numerically. y = tan x, 0 … x … pi/4; x-axisarrow_forward
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