   Chapter 12.5, Problem 14E

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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+tsintx=0tcost+sinty=0

Now, By cross multiplication as shown below,

costty+x=sintxt+y=11+t2cost=xty1+t2,sint=xt+y1+t2

Further simplification as shown below,

cos2t+sin2t=1or,(xty1+t2)2+(xt+y1+t2)2=1or,x22xyt+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+y2z=1

Now, draw the graph,

The graph is shown below,

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