Explanation Given: The region inside both circles r = 1 and r = 2 sin θ Calculation: Method: Area in polar coordinates- Assume that r = f ( θ ) define a polar curve, where f is continuous and f ( θ ) ≥ 0 on the closed interval α ≤ θ ≤ β , where 0 ≤ β − α ≤ 2 π . Then the region bounded by the curve r = f ( θ ) and the rays θ = α and θ = β has area A = 1 2 ∫ α β r 2 d θ = 1 2 ∫ α β [ f ( θ ) ] 2 d θ The objective is to find the area bounded by the polar curves r = 2 sin θ and r = 1 . These are the curves of circles. First, find the point of intersection of the two curves as follows: Equate both equations; 2 sin θ = 1 Or sin θ = 1 2 which implies that the curves intersect each other at θ = π 6 , 7 π 6 . Here, the curve r = 1 is on the top in the interval [ π 6 , 7 π 6 ] and f ( θ ) ≥ 0 . It can be seen that this corresponds to angles with measures from θ = π 6 to θ = 7 π 4 . Thus, according to the method, the required area is provided by: A = 1 2 ∫ α β r 2 d θ = 1 2 ∫ α β [ f ( θ ) ] 2 d θ A = 1 2 ∫ π 6 7 π 6 ( 1 − 2 sin θ ) 2 d θ A = 1 2 ∫ π 6 7 π 6 ( 1 + 4 sin 2 θ − 4 sin θ ) d θ A = 1</

6th Edition

ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 6, Problem 18SP

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To find:The area of the regions bounded by the curves and lines

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Chapter 6, Problem 17SP

Chapter 6, Problem 19SP

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Chapter 6 Solutions

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 polar curves r=1 and R=2mathrmsintheta .

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