Explanation Given: polar curve r = 4 cos 2 θ Calculation: Area of the polar coordinates- Assume that r = f ( θ ) define a polar curve, where f is continuous and r = 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 curve r = 4 cos 2 θ . This polar curve is equivalent to the equation of four petals with no rotation i.e. equivalent to r = 4 cos 2 θ . First, find the area of the top half of the right loop to make sue f ( θ ) ≥ 0 . It can be seen that this corresponds to angles with measures from θ = 0 to θ = π 4 . By symmetry, the entire area enclosed by the four-leaved rose is 8 times the area of the top half of the right loop. 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

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 6, Problem 17SP

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To find:The area of the region bounded by the given curve

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Chapter 6, Problem 16SP

Chapter 6, Problem 18SP

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The area of the region bounded by the given curve

Solution Summary: The objective is to find the area bounded by the polar curve r=4mathrmcos2theta .

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Chapter 6 Solutions