Explanation Given: The region inside one loop of the Lemniscate r 2 = cos 2 θ Calculation: 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 curve r 2 = cos 2 θ . This polar curve is equivalent to the equation of Lemniscate with no rotation i.e. equivalent to r 2 = a cos 2 θ . First, find the area of the top half of the right loop to make sure f ( θ ) ≥ 0 . It can be seen that this corresponds to angles with measures from θ = 0 to θ = π 4 . By symmetry, the area enclosed by one loop of Lemniscate is 2 times the area of the top half of the right loop

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 19SP

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

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Chapter 6, Problem 18SP

Chapter 6, Problem 20SP

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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 objective is to find the area bounded by the polar curve r2=mathrm

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Chapter 6 Solutions