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CalculusCalculus10th Edition

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Chapter 12.5, Problem 14E
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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 Space In Exercises 11-16, sketch the space curve and find its length over the given interval.

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

To determine

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

Explanation

Given: The provided function and the interval are,

r→(t)=〈cost+t.sint, sint - tcost,t2〉,     [0, π2]

Graph: Consider that the parametric equation of the vector equation as shown below,

x=cost+tsint,y=sint - tcost, z=t2 

On simplifying, then,

cost+tsint−x=0−tcost+sint−y=0

Now, By cross multiplication as shown below,

cost−ty+x=sintxt+y=11+t2∴cost=x−ty1+t2,sint=xt+y1+t2

Further simplification as shown below,

∴cos2t+sin2t=1or,(x−ty1+t2)2+(xt+y1+t2)2=1or,x2−2xyt+t2y2+t2x2+2xyt+y2(1+z)2=1or,(x2+y2)+t2(x2+y2)(1+z)2=1or,(x2+y2)(1+z)(1+z)2=1or,(x2+y2)(1+z)=1or,x2+y2=1+zor,x2+y2−z=1

Now, draw the graph,

The graph is shown below,

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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 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