   Chapter 10.3, Problem 58E

Chapter
Section
Textbook Problem

Arc Length In Exercises 55-58, find the arc length of the curve on the interval [ 0 , 2 π ] .Nephroid perimater: x = a ( 3 cos t − cos 3 t ) y = a ( 3 sin t − sin 3 t )

To determine

To calculate: The arc length of curve x=acos3θ,y=asin3θ on the interval [0,2π].

Explanation

Given:

Parametric equations,

x=a(3costcos3t)y=a(3sintsin3t)

Formula used:

Arc length of curve is given by:

s=02π((dxdθ)2+(dydθ)2)dθ

And,

cos2t+sin2t=1cos23t+sin23t=1

Cosine formula:

cos(a±b)=cosacosbsinasinb

Cosine double angle formula,

cos2a=12sin2a

Calculation:

Consider the given equations,

x=a(3costcos3t)y=a(3sintsin3t)

Differentiate x=a(3costcos3t) with respect to t, to get

dxdt=a(3sint+3sin3t)

Differentiate y=a(3sintsin3t) with respect to t, to get

dydt=a(3cost3cos3t)

Arc length of Curve is given by:

s=02π((dxdθ)2+(dydθ)2)dθ

Substitute the values of dxdt and dydt in above equation, to get,

s=02π(a(3sint+3sin3t)2+(a(3cost3cos3t))2dt=3a

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