Explanation If the derivative of the function is continuous on [ a , b ] , then the length (arc length) of the curve y = f ( x ) from the point A = ( a , f ( a ) ) to the point B = ( b , f ( b ) ) is the value of the integral L = ∫ a b 1 + [ f ′ ( x ) ] 2 d x = ∫ a b 1 + ( d y d x ) 2 d x Example: Consider the curve function is y = x 3 / 2 . The curve lies from x = 0 to x = 4 . Calculation: Show the curve function as follows: y = x 3 / 2 (1) Differentiate Equation (1) with respect to x . d d x ( y ) = d d x ( x 3 / 2 ) d y d x = 3 2 x 1 / 2 = 3 2 x 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 3 2 x for d y d x , 0 for a , and 4 for b . L = ∫ 0 4 1 + ( 3 2 x ) 2 d x = ∫ 0 4 ( 1 + 9 4 x ) d x (2) Let consider u = 1 + 9 4 x . Substitute 0 for x . u = 1 + 0 = 1 Substitute 4 for x . u = 1 + 9 4 ( 4 ) = 10 Differentiate u = 1 + 9 4 x as follows: d u = 9 4 d x 4 9 d u = d x Substitute u for 1 + 9 4 x , 4 9 d u for dx , and limit 1 to 10 for limit 0 to 4

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, Problem 4GYR

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Provide notes about the functions that do not have continuous first derivatives.

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Provide notes about the calculation of the length of the graph of a smooth function with example. Provide notes about the functions that do not have continuous first derivatives.