Explanation Definition used: Arc length for y = f ( x ) : Let y = f ( x ) have a continuous first derivative on the interval [ a , b ] . The length of the curve from ( a , f ( a ) ) to ( b , f ( b ) ) is L = ∫ a b 1 + f ′ ( x ) 2 d x . Here, the function y = ln x . Find the first derivative of the function y = ln x in order to obtain the length of the curve. f ′ ( x ) = d d x ( ln x ) = 1 x Thus, the first derivative of the function is 1 x . The function y = 1 x is continuous on [ 1 , b ] where b > 0 , so the function y = 1 x is continuous first derivative on the interval [ 1 , b ] where b > 0 . Use the definition of the arc length for y = f ( x ) to find the length of the curve y = 1 x from x = 0 and x = b . Substitute 1 x for f ′ ( x ) , 1 for a and b for b in L = ∫ a b 1 + f ′ ( x ) 2 d x , L = ∫ 1 b 1 + ( 1 x ) 2 d x = ∫ 1 b 1 + 1 x 2 d x = ∫ 1 b x 2 + 1 x 2 d x = ∫ 1 b x 2 + 1 2 x d x Use the value of integral ∫ x 2 + a 2 x d x = x 2 + a 2 − a ln ( a + x 2 + a 2 x ) + C

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

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

Publisher: PEARSON

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Chapter 6, Problem 65RE

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To find: The value of b when length of curve y=lnx between x=1 and x=b>1 is 2.

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Chapter 6, Problem 64RE

Chapter 6, Problem 66RE

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