Explanation Given: The curve is y = f ( x ) and the interval is [ − b , b ] . The first derivative of f is continuous and the curve is symmetric about the origin as well as y -axis. Formula used: The arc length is L = ∫ a b 1 + f ′ ( x ) 2 d x . Definition used A function f ( x ) is said to be even function if f ( − x ) = f ( x ) . A function f ( x ) is said to be odd function if f ( − x ) = − f ( x ) . Proof: The given function is y = f ( x ) Calculate the derivative of f ( x ) . That is, f ′ ( x ) = d ( f ( x ) ) d x Since the given curve y = f ( x ) is symmetric about origin, this implies f ( x ) is odd. That is, f ( − x ) = − f ( x ) f ′ ( − x ) = − f ′ ( x ) Similarly, the curve is also symmetric about the y -axis, this implies f ( x ) is even function

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

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

Publisher: PEARSON

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Chapter 6.5, Problem 40E

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To show: The arc length of the curve y=f(x) from x=−b to x=b is twice the length of the curve from x=0 to x=b when the curve is symmetric about the origin or the curve is symmetric about the y-axis.

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Chapter 6.5, Problem 39E

Chapter 6.5, Problem 41E

Chapter 6 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Ch. 6.1 - A police officer leaves his station on a...Ch. 6.1 - Describe a possible motion of an object along a...Ch. 6.1 - Is the position s(t) a number or a function? For...Ch. 6.1 - Without doing further calculations, what are the...Ch. 6.1 - Prob. 5QC Ch. 6.1 - Prob. 6QC Ch. 6.1 - Explain the meaning of position, displacement, and...Ch. 6.1 - Suppose the velocity of an object moving along a...Ch. 6.1 - Given the velocity function v of an object moving...Ch. 6.1 - Prob. 4E

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The arc length of the curve y = f ( x ) from x = − b to x = b is twice the length of the curve from x = 0 to x = b when the curve is symmetric about the origin or the curve is symmetric about the y -axis.

Solution Summary: The author explains that the arc length of the curve y=f(x) is twice the length when it is symmetric about the origin.

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To show: The arc length of the curve y=f(x) from x=−b to x=b is twice the length of the curve from x=0 to x=b when the curve is symmetric about the origin or the curve is symmetric about the y-axis.

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Chapter 6 Solutions