Explanation Given: The parametric equation for the variable x is as below. x = cos t + ln ( tan 1 2 t ) The parametric equation for the variable y is as below. y = sin t The value t ranges from π 4 to 3 π 4 . Calculation: The length of the curve is obtained by the formula. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t Differentiate the variable x with respect to t . x = cos t + ln ( tan 1 2 t ) d x d t = − sin t + 1 2 sec 2 ( t 2 ) tan ( t 2 ) = − sin t + 1 2 sin ( t 2 ) cos ( t 2 ) = − sin t + 1 sin t Differentiate the variable y with respect to t . y = sin t d y d t = cos t Write the length of the curve formula as below. L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t Substitute ( − sin t + 1 sin t ) for d x d t , ( cos t ) for d y d t , π 4 for α , and 3 π 4 for β in the above equation, L = ∫ α β ( d x d t ) 2 + ( d y d t ) 2 d t = ∫ π 4 3 π 4 ( − sin t + 1 sin t ) 2 + ( cos t ) 2 d t = 2 ∫ π 4 π 2 ( − sin t + 1 sin t ) 2 + ( cos t ) 2 d t = 2 ∫ π 4 π 2 − sin 2 t + 1 sin 2 t + cos t d t Simplify further, L = 2 ∫ π 4 π 2 cot t d t = 2 ∫ π 4 π 2 1 − 2 + 1 sin 2 t d t = 2 ∫ π 4 π 2 csc 2

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

8th Edition

ISBN: 9781305779198

Author: James Stewart

Publisher: Cengage Learning

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Chapter 10.2, Problem 46E

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To find: The length of the curve and graph the curve obtained from the parametric equations x=cost+ln(tan12t) and y=sint.

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Chapter 10 Solutions

Bundle: Multivariable Calculus, 8th + WebAssign Printed Access Card for Stewart's Calculus, 8th Edition, Single-Term

Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Sketch the curve by using the parametric equations...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

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The length of the curve and graph the curve obtained from the parametric equations x = cos t + ln ( tan 1 2 t ) and y = sin t .

Solution Summary: The length of the curve is obtained by the formula.

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To find: The length of the curve and graph the curve obtained from the parametric equations x=cost+ln(tan12t) and y=sint.

Chapter 10 Solutions