Explanation Given: ( 2 cos t − cos 2 t , 2 sin t − sin 2 t ) , 0 ≤ t ≤ π 2 Formula used: (i) s = ∫ a b x ' ( t ) 2 + y ' ( t ) 2 d t , where s is the length of the path over the interval [ a , b ] . (ii) sin 2 θ + cos 2 θ = 1 (iii) 1 − cos t 2 = sin 2 t 2 ⇒ 1 − cos t = 2 sin t 2 (iv) sin 2 t = 2 sin t cos t Calculation: Given path is ( 2 cos t − cos 2 t , 2 sin t − sin 2 t ) , 0 ≤ t ≤ π 2 . Thus, x ( t ) = 2 cos t − cos 2 t , y ( t ) = 2 sin t − sin 2 t , 0 ≤ t ≤ π 2 . Hence, x ' ( t ) = − 2 sin t + 2 sin 2 t , y ' ( t ) = 2 cos t − 2 cos 2 t Therefore, s = ∫ a b x ' ( t ) 2 + y ' ( t ) 2 d t = ∫ 0 π / 2 ( − 2 sin t + 2 sin 2 t ) 2 + ( 2 cos t − 2 cos 2 t ) 2 d t = ∫ 0 π / 2 4 sin 2 t + 4 sin 2 2 t − 8 sin t sin 2 t + 4 cos 2 t + 4 cos 2 2 t

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ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 12.2, Problem 9E

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To find the length of the path

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Chapter 12 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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To find the length of the path

Solution Summary: The author explains the formula used to determine the length of the path.

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