To determine Calculate the length of the curve. Explanation Given information: The curve function is y = x 3 3 + x 2 + x + 1 4 x + 4 . The curve lies from x = 0 to x = 2 . Calculation: Show the curve function as follows: y = x 3 3 + x 2 + x + 1 4 x + 4 (1) Differentiate Equation (1) with respect to x . d d x ( y ) = d d x ( x 3 3 + x 2 + x + 1 4 x + 4 ) d y d x = 1 3 3 x 2 + 2 x + 1 − 1 ( 4 x + 4 ) 2 = x 2 + 2 x + 1 − 1 4 1 ( 1 + x ) 2 = ( 1 + x ) 2 − 1 4 1 ( 1 + x ) 2 The expression to find the length of the curve y = f ( x ) on the interval [ a , b ] is shown as follows: L = ∫ a b 1 + ( d y d x ) 2 d x Substitute ( 1 + x ) 2 − 1 4 1 ( 1 + x ) 2 for d y d x , 0 for a , and 2 for b . L = ∫ 0 2 1 + ( ( 1 + x ) 2 − 1 4 1 ( 1 + x ) 2 ) 2 d x = ∫ 0 2 1 + ( 1 + x ) 4 − 2 ( 1 + x ) 2 ( 1 4 1 ( 1 + x ) 2 ) + ( 1 + x ) − 4 16 d x = ∫ 0 2 1 + ( 1 + x ) 4 − 1 2 + ( 1 + x ) − 4 16 d x

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 6.3, Problem 8E

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Calculate the length of the curve.

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University Calculus: Early Transcendentals (4th Edition)

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