Explanation Given information: The region that is inside the circle r = a and outside the cardioid r = a ( 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 = a and r = a ( 1 − cos θ ) .(Given) To find the point of intersection of r = a and r = a ( 1 − cos θ ) , a = a ( 1 − cos θ ) . a = a ( 1 − cos θ ) cos θ = 0 θ = cos − 1 ( 0 ) θ = π 2 The area of the region that is inside the circle r = a and outside the cardioid r = a ( 1 − cos θ ) is given by A = 2 [ ( A r e a o f r = a ) − ( A r e a o f r = a ( 1 − c o s θ ) ) ] A = 2 ( 1 2 ∫ 0 π 2 ( a ) 2 d θ − 1 2 ∫ 0 π 2 ( a ( 1 − c o s θ ) ) 2 d θ ) = 2 2 ( a 2 ∫ 0 π 2 d θ − a 2 ∫ 0 π 2 ( 1 − c o s θ ) 2 d θ ) = a 2 (

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 45PS

To determine

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

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Chapter 6.3, Problem 44PS

Chapter 6.3, Problem 46PS

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Chapter 6 Solutions

Calculus

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

Solution Summary: The author calculates the area of the polar region enclosed by f(theta ) and outside the cardioid.

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

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Chapter 6 Solutions