Explanation Given information: The curve is f ( x ) = x 3 2 − 1 3 x 1 2 on [ 4 , 16 ] . Formula used: The length of the arc of the curve f ( x ) on [ a , b ] is given by s = ∫ a b 1 + [ f ' ( x ) ] 2 d x Calculation: f ( x ) = x 3 2 − 1 3 x 1 2 on [ 4 , 16 ] (Given) Differentiating f ( x ) = x 3 2 − 1 3 x 1 2 with respect to ‘x’, we get f ' ( x ) = 3 2 ( x 1 2 ) − 1 6 ( x − 1 2 ) = 3 2 ( x ) − 1 6 x = ( 9 x − 1 6 x ) ⇒ [ f ' ( x ) ] 2 = ( 9 x − 1 6 x ) 2 The length of the arc of the curve f ( x ) = x 3 2 − 1 3 x 1 2 on [ 4 , 16 ] is given by s = ∫ 4 16 1 + [ f ' ( x ) ] 2 d x = ∫ 4 16 1 + ( 9 x − 1 6 x <

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 6.4, Problem 18PS

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To find:The length of the arc of the curve f(x)=x32−13x12 on [4,16] .

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Chapter 6.4, Problem 17PS

Chapter 6.4, Problem 19PS

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Chapter 6 Solutions

Calculus

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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY

The length of the arc of the curve f ( x ) = x 3 2 − 1 3 x 1 2 on [ 4 , 16 ] .

Solution Summary: The formula used to find the length of the arc of a curve is 56.67 units.

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To find:The length of the arc of the curve f(x)=x32−13x12 on [4,16] .

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Chapter 6 Solutions