10th Edition

ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 10.3, Problem 85E

Areas of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of Exercise 77 to match the closed curse with its area. (These exercises were based on “The Surveyor’s Area Formula'’ by Bart Braden, College Mathematics Journal, September 1986, pp. 335-337, by permission of the author.)

(a) 8 3 a b

(b) 3 8 π a 2

(c) 2 π a 2

(d) π a b

(e) 2 π a b

(f) 6 π a 2

Hourglass: ( 0 ≤ t ≤ 2 π )

x = a sin 2 t y = b sin t

Chapter 10.3, Problem 85E, Areas of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of

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Chapter 10.3, Problem 84E

Chapter 10.3, Problem 86E

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Chapter 10 Solutions

Calculus

Ch. 10.1 - Matching In Exercises 5-10, match the equation...Ch. 10.1 - Matching In Exercises 5-10, match the equation...Ch. 10.1 - Matching In Exercises 5-10, match the equation...Ch. 10.1 - Matching In Exercises 5-10, match the equation...Ch. 10.1 - Matching In Exercises 5-10, match the equation...Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Sketching a Parabola In Exercises 1116, find the...

Ch. 10.1 - Sketching a Parabola In Exercises 1116, find the...Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Finding the Standard Equation of an Ellipse In...Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.1 - Prob. 41E Ch. 10.1 - Prob. 42E Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Prob. 45E Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 53E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.1 - Prob. 57E Ch. 10.1 - Classifying the Graph of an Equation In Exercises...Ch. 10.1 - Prob. 55E Ch. 10.1 - Prob. 54E Ch. 10.1 - Prob. 63E Ch. 10.1 - Prob. 69E Ch. 10.1 - Prob. 56E Ch. 10.1 - Prob. 59E Ch. 10.1 - Prob. 60E Ch. 10.1 - Prob. 61E Ch. 10.1 - Eccentricity Define the eccentricity of an...Ch. 10.1 - Prob. 64E Ch. 10.1 - Solar Collector A solar collector for heating...Ch. 10.1 - Beam Deflection A simply supported beam that is 16...Ch. 10.1 - Prob. 67E Ch. 10.1 - Proof (a) Prove that if any two tangent lines to a...Ch. 10.1 - Bridge Design A cable of a suspension bridge is...Ch. 10.1 - Prob. 71E Ch. 10.1 - Surface Area A satellite signal receiving dish is...Ch. 10.1 - Prob. 73E Ch. 10.1 - Prob. 74E Ch. 10.1 - Prob. 75E Ch. 10.1 - Prob. 76E Ch. 10.1 - Prob. 77E Ch. 10.1 - Prob. 78E Ch. 10.1 - Prob. 79E Ch. 10.1 - Prob. 80E Ch. 10.1 - Prob. 81E Ch. 10.1 - Conjecture Show that the equation of an ellipse...Ch. 10.1 - Prob. 83E Ch. 10.1 - Prob. 84E Ch. 10.1 - Prob. 85E Ch. 10.1 - Prob. 86E Ch. 10.1 - Prob. 87E Ch. 10.1 - Hyperbolic Mirror A hyperbolic mirror (used in...Ch. 10.1 - Prob. 89E Ch. 10.1 - Prob. 90E Ch. 10.1 - True or False? In Exercises 8792, determine...Ch. 10.1 - Prob. 92E Ch. 10.1 - Prob. 93E Ch. 10.1 - True or False? In Exercises 87-92, determine...Ch. 10.1 - True or False? In Exercises 8792, determine...Ch. 10.1 - Prob. 96E Ch. 10.1 - For a point P on an ellipse, let d be the distance...Ch. 10.1 - Prob. 98E Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Prob. 15E Ch. 10.2 - Using Parametric Equations In Exercises 5-22,...Ch. 10.2 - Using Parametric Equations In Exercises 5-22,...Ch. 10.2 - Using Parametric Equations In Exercises 5-22,...Ch. 10.2 - Using Parametric Equations In Exercises 23-34, use...Ch. 10.2 - Using Parametric Equations In Exercises 23-34, use...Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Using Parametric Equations In Exercises 23-34, use...Ch. 10.2 - Prob. 26E Ch. 10.2 - Using Parametric Equations In Exercises 23-34, use...Ch. 10.2 - Using Parametric Equations In Exercises 23-34, use...Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - Prob. 32E Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - Prob. 35E Ch. 10.2 - Prob. 36E Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Writing a Set of Parametric Equations In Exercises...Ch. 10.2 - Prob. 42E Ch. 10.2 - Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.2 - Prob. 47E Ch. 10.2 - Prob. 48E Ch. 10.2 - Prob. 49E Ch. 10.2 - Prob. 50E Ch. 10.2 - Prob. 51E Ch. 10.2 - Prob. 52E Ch. 10.2 - Prob. 53E Ch. 10.2 - Prob. 54E Ch. 10.2 - Prob. 55E Ch. 10.2 - Prob. 56E Ch. 10.2 - Prob. 57E Ch. 10.2 - Prob. 58E Ch. 10.2 - Prob. 59E Ch. 10.2 - Prob. 60E Ch. 10.2 - Prob. 61E Ch. 10.2 - Prob. 62E Ch. 10.2 - Prob. 63E Ch. 10.2 - Prob. 64E Ch. 10.2 - Prob. 65E Ch. 10.2 - Plane Curve Explain the process of sketching a...Ch. 10.2 - Prob. 67E Ch. 10.2 - Prob. 68E Ch. 10.2 - Prob. 69E Ch. 10.2 - Prob. 70E Ch. 10.2 - Prob. 71E Ch. 10.2 - Prob. 72E Ch. 10.2 - Prob. 73E Ch. 10.2 - Epicycloid A circle of radius 1 rolls around the...Ch. 10.2 - Prob. 75E Ch. 10.2 - Prob. 76E Ch. 10.2 - Prob. 77E Ch. 10.2 - Prob. 78E Ch. 10.2 - Projectile Motion In Exercises 81 and 82, consider...Ch. 10.2 - Projectile Motion In Exercises 81 and 82, consider...Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - Prob. 6E Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Finding an Equation of a Tangent Line In Exercises...Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Finding Equations of Tangent Lines In Exercises...Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Horizontal and Vertical Tangency In Exercises 27...Ch. 10.3 - Horizontal and Vertical Tangency In Exercises 31...Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 37E Ch. 10.3 - Prob. 36E Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.3 - Prob. 40E Ch. 10.3 - Prob. 41E Ch. 10.3 - Prob. 42E Ch. 10.3 - Prob. 43E Ch. 10.3 - Prob. 44E Ch. 10.3 - Prob. 45E Ch. 10.3 - Prob. 46E Ch. 10.3 - Prob. 47E Ch. 10.3 - Prob. 48E Ch. 10.3 - Arc Length In Exercises 49-54, find the arc length...Ch. 10.3 - Arc Length In Exercises 49-54, find the arc length...Ch. 10.3 - Arc Length In Exercises 55-58, find the arc length...Ch. 10.3 - Arc Length In Exercises 55-58, find the arc length...Ch. 10.3 - Prob. 53E Ch. 10.3 - Arc Length In Exercises 5154, find the are length...Ch. 10.3 - Prob. 55E Ch. 10.3 - Prob. 56E Ch. 10.3 - Prob. 57E Ch. 10.3 - Prob. 58E Ch. 10.3 - Prob. 59E Ch. 10.3 - Prob. 60E Ch. 10.3 - Surface Area In Exercises 63-68, find the area of...Ch. 10.3 - Prob. 66E Ch. 10.3 - Surface Area In Exercises 63-68, find the area of...Ch. 10.3 - Prob. 68E Ch. 10.3 - Prob. 69E Ch. 10.3 - Prob. 70E Ch. 10.3 - Prob. 61E Ch. 10.3 - Surface Area In Exercises 6164, write an integral...Ch. 10.3 - Prob. 63E Ch. 10.3 - Prob. 64E Ch. 10.3 - Prob. 71E Ch. 10.3 - Prob. 72E Ch. 10.3 - Prob. 73E Ch. 10.3 - Prob. 74E Ch. 10.3 - Prob. 75E Ch. 10.3 - Prob. 76E Ch. 10.3 - Prob. 77E Ch. 10.3 - Prob. 78E Ch. 10.3 - Area In Exercises 79 and 80, find the area of the...Ch. 10.3 - Prob. 80E Ch. 10.3 - Prob. 81E Ch. 10.3 - Prob. 82E Ch. 10.3 - Prob. 83E Ch. 10.3 - Prob. 84E Ch. 10.3 - Areas of Simple Closed Curves In Exercises 81-86,...Ch. 10.3 - Prob. 86E Ch. 10.3 - Prob. 87E Ch. 10.3 - Prob. 88E Ch. 10.3 - Prob. 89E Ch. 10.3 - Volume In Exercises 89 and 90, find the volume of...Ch. 10.3 - Prob. 91E Ch. 10.3 - Prob. 92E Ch. 10.3 - Prob. 93E Ch. 10.3 - Prob. 94E Ch. 10.3 - Prob. 95E Ch. 10.3 - Prob. 96E Ch. 10.3 - True or False? In Exercises 97-100, determine...Ch. 10.3 - Prob. 98E Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Prob. 10E Ch. 10.4 - Prob. 11E Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Prob. 24E Ch. 10.4 - Rectangular-to Polar Conversion In Exercises 2534,...Ch. 10.4 - Prob. 26E Ch. 10.4 - Prob. 27E Ch. 10.4 - Prob. 28E Ch. 10.4 - Prob. 29E Ch. 10.4 - Prob. 30E Ch. 10.4 - Prob. 31E Ch. 10.4 - Rectangular-to-Polar Conversion In Exercises...Ch. 10.4 - Prob. 33E Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E Ch. 10.4 - Prob. 36E Ch. 10.4 - Prob. 37E Ch. 10.4 - Prob. 38E Ch. 10.4 - Prob. 39E Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Polar-to-Rectangular Conversion In Exercises...Ch. 10.4 - Prob. 43E Ch. 10.4 - Prob. 44E Ch. 10.4 - Prob. 45E Ch. 10.4 - Prob. 46E Ch. 10.4 - Prob. 47E Ch. 10.4 - Prob. 48E Ch. 10.4 - Prob. 49E Ch. 10.4 - Prob. 50E Ch. 10.4 - Prob. 51E Ch. 10.4 - Prob. 52E Ch. 10.4 - Prob. 53E Ch. 10.4 - Prob. 96E Ch. 10.4 - Prob. 97E Ch. 10.4 - Prob. 54E Ch. 10.4 - Prob. 55E Ch. 10.4 - Prob. 56E Ch. 10.4 - Prob. 57E Ch. 10.4 - Prob. 58E Ch. 10.4 - Prob. 60E Ch. 10.4 - Prob. 59E Ch. 10.4 - Prob. 61E Ch. 10.4 - Prob. 62E Ch. 10.4 - Prob. 63E Ch. 10.4 - Prob. 64E Ch. 10.4 - Prob. 65E Ch. 10.4 - Prob. 66E Ch. 10.4 - Prob. 67E Ch. 10.4 - Prob. 68E Ch. 10.4 - Prob. 69E Ch. 10.4 - Prob. 70E Ch. 10.4 - Prob. 71E Ch. 10.4 - Prob. 72E Ch. 10.4 - Prob. 73E Ch. 10.4 - Prob. 74E Ch. 10.4 - Prob. 75E Ch. 10.4 - Prob. 76E Ch. 10.4 - Prob. 77E Ch. 10.4 - Prob. 78E Ch. 10.4 - Prob. 79E Ch. 10.4 - Prob. 80E Ch. 10.4 - Prob. 81E Ch. 10.4 - Prob. 82E Ch. 10.4 - Prob. 83E Ch. 10.4 - Prob. 84E Ch. 10.4 - Prob. 85E Ch. 10.4 - Prob. 86E Ch. 10.4 - Prob. 87E Ch. 10.4 - Prob. 88E Ch. 10.4 - Prob. 89E Ch. 10.4 - Prob. 90E Ch. 10.4 - Prob. 91E Ch. 10.4 - Prob. 92E Ch. 10.4 - Prob. 93E Ch. 10.4 - Prob. 94E Ch. 10.4 - Prob. 95E Ch. 10.4 - Prob. 98E Ch. 10.4 - Prob. 99E Ch. 10.4 - Prob. 100E Ch. 10.4 - Rotated Curve In Exercises 101104, use the results...Ch. 10.4 - Rotated Curve In Exercises 103-105, use the...Ch. 10.4 - Prob. 103E Ch. 10.4 - Proof Prove that the tangent of the angle (0/2)...Ch. 10.4 - Prob. 105E Ch. 10.4 - Prob. 106E Ch. 10.4 - Prob. 107E Ch. 10.4 - Prob. 108E Ch. 10.4 - Prob. 109E Ch. 10.4 - Prob. 110E Ch. 10.4 - Prob. 111E Ch. 10.4 - Prob. 112E Ch. 10.4 - Prob. 113E Ch. 10.4 - Prob. 114E Ch. 10.5 - Points of Intersection Explain why finding points...Ch. 10.5 - Area of a Polar Region In Exercises 3-6, write an...Ch. 10.5 - Area of a Polar Region In Exercises 3-6, write an...Ch. 10.5 - Prob. 3E Ch. 10.5 - Prob. 4E Ch. 10.5 - Finding the Area of a Polar Region In Exercises...Ch. 10.5 - Prob. 6E Ch. 10.5 - Finding the Area of a Polar Region In Exercises...Ch. 10.5 - Prob. 8E Ch. 10.5 - Prob. 9E Ch. 10.5 - Finding the Area of a Polar Region In Exercises...Ch. 10.5 - Prob. 11E Ch. 10.5 - Prob. 12E Ch. 10.5 - Prob. 13E Ch. 10.5 - Prob. 14E Ch. 10.5 - Prob. 15E Ch. 10.5 - Prob. 16E Ch. 10.5 - Prob. 17E Ch. 10.5 - Prob. 18E Ch. 10.5 - Prob. 19E Ch. 10.5 - Prob. 20E Ch. 10.5 - Prob. 21E Ch. 10.5 - Prob. 22E Ch. 10.5 - Prob. 23E Ch. 10.5 - Prob. 24E Ch. 10.5 - Prob. 25E Ch. 10.5 - Finding Points of Intersection In Exercises 27-34,...Ch. 10.5 - Prob. 27E Ch. 10.5 - Prob. 28E Ch. 10.5 - Prob. 29E Ch. 10.5 - Prob. 30E Ch. 10.5 - Finding Points of Intersection In Exercises 27-34,...Ch. 10.5 - Prob. 32E Ch. 10.5 - Prob. 33E Ch. 10.5 - Prob. 34E Ch. 10.5 - Prob. 35E Ch. 10.5 - Prob. 36E Ch. 10.5 - Prob. 37E Ch. 10.5 - Finding the Area of a Polar Region Between Two...Ch. 10.5 - Prob. 39E Ch. 10.5 - Prob. 40E Ch. 10.5 - Prob. 41E Ch. 10.5 - Prob. 42E Ch. 10.5 - Prob. 43E Ch. 10.5 - Prob. 44E Ch. 10.5 - Prob. 45E Ch. 10.5 - Finding the Area of a Polar Region Between Two...Ch. 10.5 - 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Prob. 51E Ch. 10.6 - Prob. 52E Ch. 10.6 - Prob. 53E Ch. 10.6 - Prob. 54E Ch. 10.6 - Prob. 55E Ch. 10.6 - Prob. 56E Ch. 10.6 - Area of a Region In Exercises 55-58, use the...Ch. 10.6 - Prob. 58E Ch. 10.6 - Prob. 59E Ch. 10.6 - Prob. 60E Ch. 10.6 - Prob. 61E Ch. 10.6 - Prob. 62E Ch. 10.6 - Prob. 63E Ch. 10.6 - Prob. 64E Ch. 10.6 - Prob. 65E Ch. 10.6 - Prob. 66E Ch. 10.6 - Prob. 67E Ch. 10.6 - Comet Hale-Bopp The comet Hale-Bopp has an...Ch. 10.6 - Eccentricity In Exercises 67 and 68, let r0...Ch. 10.6 - Prob. 70E Ch. 10 - Matching In Exercises 1-6, match the equation with...Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Prob. 12RE Ch. 10 - Prob. 13RE Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - Prob. 25RE Ch. 10 - 26. 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