Explanation Given information: The region is inside the circle r = 6 cos θ and outside the cardioid r = 2 ( 1 + cos θ ) . Formula used: The area of the polar region enclosed by f ( θ ) for a ≤ θ ≤ b is given by A = 1 2 ∫ a b [ f ( θ ) ] 2 d θ . Calculation: r = 6 cos θ and r = 2 ( 1 + cos θ ) .(Given) To find the point of intersection of r = 6 cos θ and r = 2 ( 1 + cos θ ) , 6 cos θ = 2 ( 1 + cos θ ) . 3 cos θ = ( 1 + cos θ ) 3 cos θ − cos θ = 1 cos θ = 1 2 θ = cos − 1 1 2 θ = π 3 The area of the region that is inside the circle r = 6 cos θ and outside the cardioid r = 2 ( 1 + cos θ ) is given by A = 2 [ ( A r e a o f r = 6 cos θ ) − ( A r e a o f r = 2 ( 1 + c o s θ ) ) ] A = 2 ( 1 2 ∫ 0 π 3 ( 6 cos θ ) 2 d θ − 1 2 ∫ 0 π 3 ( 2 ( 1 + c o s θ ) ) 2 d θ ) = 36 ∫ 0 π 3 cos 2 θ d θ − 4 ∫ 0 π 3 ( 1 + c o s θ ) 2 d θ = 36 ∫<

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Question

Chapter 6.3, Problem 47PS

To determine

To find:The area of the region that is inside the circle r=6cosθ and outside the cardioid r=2(1+cosθ) .

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Chapter 6.3, Problem 46PS

Chapter 6.3, Problem 48PS

Chapter 6 Solutions

Calculus

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The area of the region that is inside the circle r = 6 cos θ and outside the cardioid r = 2 ( 1 + cos θ ) .

Solution Summary: The area of the polar region enclosed by r=6mathrmcostheta is 4pi -9sqrt3 square units

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To find:The area of the region that is inside the circle r=6cosθ and outside the cardioid r=2(1+cosθ) .

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Chapter 6 Solutions