   Chapter 12, Problem 60RE

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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(sinttcost),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+2tsinty=02tcost+2sintx=0

By cross multiplication of the above equation,

cost2tx+2y=sint2yt+2x=14+4t2cost=ytx2(1+t2),sint=x+yt2(1+t2)

Simplify the above equation,

cos2t+sin2t=1or,(ytx2(1+t2))2+(x+yt2(1+t2))2=1or,y22xyt+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+y24z2=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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