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CalculusCalculus10th Edition

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Chapter 12, Problem 60RE
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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Finding the Arc Length of a Curve in SpaceIn Exercises 59–62, sketch the space curve and find its length over the given interval.

r ( t ) = 〈 2 ( sin t − t cos t ) , 2 ( cos t + t sin t ) , t 〉 ,         [ 0 , π 2 ]

To determine

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

Explanation

Given: r→(t)=〈2(sint−tcost),2(cost+tsint), t〉,   [0,π2]

Graph: Let us consider the parametric equation,

x=2(sint - tcost),y=2(cost+tsint), z = t 

simplify the above equation and to get,

2cost+2tsint−y=0−2tcost+2sint−x=0

By cross multiplication of the above equation,

cost−2tx+2y=sint2yt+2x=14+4t2∴cost=y−tx2(1+t2),sint=x+yt2(1+t2)

Simplify the above equation,

∴cos2t+sin2t=1or,(y−tx2(1+t2))2+(x+yt2(1+t2))2=1or,y2−2xyt+t2x2+x2+2xyt+y2t2(1+z2)2=4or,(x2+y2)+t2(x2+y2)(1+z)2=4

or,(x2+y2)(1+z2)(1+z)2=4or,(x2+y2)(1+z2)=4or,x2+y2=4+4z2or,x2+y2−4z2=4

The sketch of the provided equation is as follows,

By the use of 3D graphing calculator, graph has been drawn

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Sect-12.1 P-1ESect-12.1 P-2ESect-12.1 P-3ESect-12.1 P-4ESect-12.1 P-5ESect-12.1 P-6ESect-12.1 P-7ESect-12.1 P-8ESect-12.1 P-9ESect-12.1 P-10E
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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 P-10PSCh-12 P-11PSCh-12 P-12PSCh-12 P-13PSCh-12 P-14PS

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