Explanation Given: The complete cardioid r = 2 − 2 sin θ . The function f ( θ ) = 2 − 2 sin θ . Formula used: Suppose the polar curve is r = f ( θ ) and the derivative f ′ is continuous on [ a , b ] . The arc length of the polar curve is L = ∫ a b f ( θ ) 2 + f ′ ( θ ) 2 d θ . Calculation: Compute f ′ ( θ ) as follows, f ′ ( θ ) = ( 2 − 2 sin θ ) ′ = − 2 cos θ It is observed that, f ′ ( θ ) is continuous. Since the function is a complete cardioid, the interval is 0 ≤ θ ≤ 2 π . Use arc length formula to obtain the arc length of the complete cardioid. L = ∫ 0 2 π f ( θ ) 2 + f ′ ( θ ) 2 d θ = ∫ 0 2 π ( 2 − 2 sin θ ) 2 + ( − 2 cos θ ) 2 d θ = ∫ 0 2 π 4 + 4 sin 2 θ − 8 sin θ + 4 cos 2 θ d θ = ∫ 0 2 π 4 + 4 ( sin 2 θ + cos 2 θ ) − 8 sin θ d θ On further simplification, L = ∫ 0

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 11.8, Problem 32E

To determine

To find: The arc length of the complete cardioid r=2−2sinθ.

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Chapter 11.8, Problem 31E

Chapter 11.8, Problem 33E

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Chapter 11 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 11.1 - Interpret the following statement: Points have a...Ch. 11.1 - What is a position vector?Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Given a position vector v, why are there...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - If u = u1, u2 and v = v1, v2, how do you find u +...Ch. 11.1 - Prob. 10E

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The arc length of the complete cardioid r = 2 − 2 sin θ .

To find: The arc length of the complete cardioid r=2−2sinθ.

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Chapter 11 Solutions