Explanation Given: r → ( t ) = 〈 2 ( sin t − t cos t ) , 2 ( cos t + t sin t ) , t 〉 , [ 0 , π 2 ] Graph: Let us consider the parametric equation of the vector equation as x = 2 ( sin t - t cos t ) , y = 2 ( cos t + t sin t ) , z= t On simplifying, we get 2 cos t + 2 t sin t − y = 0 − 2 t cos t + 2 sin t − x = 0 By cross multiplication, we get cos t − 2 t x + 2 y = sin t 2 y t + 2 x = 1 4 + 4 t 2 ∴ cos t = y − t x 2 ( 1 + t 2 ) , sin t = x + y t 2 ( 1 + t 2 ) Further simplification gives ∴ cos 2 t + sin 2 t = 1 o r , ( y − t x 2 ( 1 + t 2 ) ) 2 + ( x + y t 2 ( 1 + t 2 ) ) 2 = 1 o r , y 2 − 2 x y t + t 2 x 2 + x 2 + 2 x y t + y 2 t 2 ( 1 + z 2 ) 2 = 4 o r , ( x 2 + y 2 ) + t 2 ( x 2 + y 2 ) ( 1 + z ) 2 = 4 o r , ( x 2 + y 2 ) ( 1 + z 2 ) ( 1 + z ) 2 = 4 o r , ( x 2 + y 2 ) ( 1 + z 2 ) = 4 o r , x 2 + y 2 = 4 + 4 z 2 o r , x 2 + y 2 − 4 z 2 = 4 The sketch of the given equation is as follows

Calculus: Early Transcendental Functions

7th Edition

ISBN: 9781337552516

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 12, Problem 62RE

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To Graph: The plane curve and find the length over the interval [0,π2].

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Chapter 12, Problem 61RE

Chapter 12, Problem 63RE

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Calculus: Early Transcendental Functions

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To Graph: The plane curve and find the length over the interval [ 0 , π 2 ] .

Solution Summary: The author explains how to Graph the plane curve and find the length over the interval left[0,pi2right].

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To Graph: The plane curve and find the length over the interval [0,π2].

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Chapter 12 Solutions