Explanation Given: The equation is C ( θ ) = ( R ( c o s θ + θ sin θ ) , R ( sin θ + θ cos θ ) ) Calculation: The arc length parameterization of C ( θ ) = ( R ( c o s θ + θ sin θ ) , R ( sin θ + θ cos θ ) ) If r (t) is a differentiable vector valued function, then the arc length function is defined by S ( t ) = ∫ 0 t | | r ′ ( t ) | | d t (I) So, for the equation is given by C ( θ ) = ( R ( c o s θ + θ sin θ ) , R ( sin θ + θ cos θ ) ) In the figure, suppose QT corresponding to θ is unwound, then the length of the segment QP= RQ Therefore, the x and y coordinates can be calculated as x = O A + A B = O A + E P = R cos θ + Q P sin θ x = R cos θ + R θ sin θ x = R ( c o s θ + θ sin θ ) y = Q A − Q E = R S i n θ − Q P cos θ y = R sin θ + R θ cos θ y = R ( sin θ + θ cos θ ) So the coordinates of point with respect to parameter θ is given by C ( θ ) = ( R ( c o s θ + θ sin θ ) , R ( sin θ + θ cos θ ) ) , where 0 ≤ θ ≤ 2 π (II) Differentiating equation (II) with respect to θ x ′ ( θ ) = d d θ ( R ( c o s θ + θ sin θ ) ) = R

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Chapter 14.3, Problem 42E

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Arc length parameterization of $C (θ) = (R (c o s θ + θ \sin θ), R (\sin θ + θ \cos θ))$

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Chapter 14.3, Problem 41E

Chapter 14.3, Problem 43E

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Chapter 14 Solutions

CALCULUS (CLOTH)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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Arc length parameterization of C ( θ ) = ( R ( c o s θ + θ sin θ ) , R ( sin θ + θ cos θ ) )

Solution Summary: The author explains the arc length parametrization of C ( ) = ( R c o

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Arc length parameterization of $C (θ) = (R (c o s θ + θ \sin θ), R (\sin θ + θ \cos θ))$

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Chapter 14 Solutions

Arc length parameterization of C ( θ ) = ( R ( c o s θ + θ sin θ ) , R ( sin θ + θ cos θ ) )

Solution Summary: The author explains the arc length parametrization of C ( ) = ( R c o

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Arc length parameterization of C(θ)=(R(cosθ+θsinθ),R(sinθ+θcosθ))

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Arc length parameterization of $C (θ) = (R (c o s θ + θ \sin θ), R (\sin θ + θ \cos θ))$