Explanation Given: the region inside the circle, r = 2 a sin θ The region outside the circle, r = a Calculation: Area in polar coordinates- Let’s consider that r = f ( θ ) defines a polar curve, Here f is continuous f ( θ ) ≥ 0 on the closed interval α ≤ θ ≤ β , where 0 ≤ β − α ≤ 2 π Therefore the region bounded by the curve, r 1 = f 1 ( θ ) , r 2 = f 2 ( θ ) and the rays θ = α and θ = β has area A = 1 2 ∫ α β ( r 1 2 − r 2 2 ) d θ = 1 2 ∫ α β ( f 1 2 ( θ ) − f 2 2 ( θ ) ) d θ The objective is to find the region inside the circle r = 2 a sin θ and outside the circle r = a Suppose, f 1 ( x ) = 2 a sin θ and f 2 ( x ) = a are the curves of circles. First, to find the point to intersection of the two curves; Equate both equation; 2 a sin θ = a or sin θ = 1 2 which implies that the curves intersect each other at θ = π 6 , 5 π 6 Here, the curve r = 2 a sin θ is on the top in the interval [ π 6 , 5 π 6 ] and f ( θ ) ≥ 0 . It can be seen that this corresponds to angles with measures from θ = π 6 to θ = 5 π 6 . Thus, according to the method, the required area is; A = 1 2 ∫ α β ( r 1 2 − r 2 2 ) d θ = 1 2 ∫ α β ( f 1 2 ( θ ) − f 2 2 ( θ ) ) d θ A = 1 2 ∫ π 6 5 π 6 [ ( 2 a sin θ ) 2 − a 2 ] d θ

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

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

Publisher: Kendall Hunt Publishing

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Chapter 6, Problem 21SP

To determine

To calculate: the area of the regions bounded by the curves and lines

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Chapter 6, Problem 20SP

Chapter 6, Problem 22SP

Chapter 6 Solutions

Student's Solution and Survival Manual for Calculus

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the area of the regions bounded by the curves and lines

Solution Summary: The author calculates the area of the regions bounded by the curves and lines. The objective is to find the region inside and outside the circle.

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To calculate: the area of the regions bounded by the curves and lines

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Chapter 6 Solutions