Explanation Formula used: The length of the curve is L = ∫ 0 π r 2 + ( d r d θ ) 2 d θ where the function r = f ( θ ) . Calculation: Consider the polar equation r = sin 3 θ 3 . Differentiate the equation r = sin 3 θ 3 with respect to θ . d r d θ = [ 3 sin 2 ( θ 3 ) ] cos ( θ 3 ) ( 1 3 ) d r d θ = sin 2 ( θ 3 ) cos ( θ 3 ) Length is determined for the limits of 0 to π . Use above mentioned formula and find the length of curve as follows. L = ∫ 0 π r 2 + ( d r d θ ) 2 d θ = ∫ 0 π ( sin 3 θ 3 ) 2 + ( sin 2 ( θ 3 ) cos ( θ 3 ) ) 2 d θ = ∫ 0 π sin 6 θ 3 + sin 4 ( θ 3 ) cos 2 ( θ 3 ) d θ = ∫ 0 π sin 4 θ 3 [ sin 2 ( θ 3 ) + cos 2 ( θ 3 ) ] d θ On further simplification,

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10, Problem 40RE

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The length of the curve r=sin3θ3 for 0≤θ≤π.

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Chapter 10, Problem 39RE

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The length of the curve r = sin 3 θ 3 for 0 ≤ θ ≤ π .

Solution Summary: The author calculates the length of the curve r=mathrmsin3theta.

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The length of the curve r=sin3θ3 for 0≤θ≤π.

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