Arc length of a parabola Evaluate GV1 + 4x? dx, the arc length ofthe segment of the parabola y = x² on [0, 2].

Answered: Arc length of a parabola Evaluate GV1 +…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Arc length of a parabola Evaluate GV1 + 4x? dx, the arc length of
the segment of the parabola y = x² on [0, 2].

Expert Solution

Step 1

To Determine: evaluate the arc length of the segment of the parabola $y = x^{2} o n [0, 2]$ .

Given: we have $\int_{0}^{2} \sqrt{1 + 4 x^{2} d x}$

Explanation: we have an equation of parabola $y = x^{2} o n [0, 2]$

on differentiating we have $y' = 2 x$

on squaring we have ${(y')}^{2} = {(2 x)}^{2} = 4 x^{2}$

so we have the integral

L= $\int_{0}^{2} \sqrt{1 + 4 x^{2} d x}$

let $2 x = \tan θ$ then $2 d x = s e c^{2} θ d θ$

also we can write

$\begin{array}{rcl} 4 x^{2} & = & \tan^{2} θ t h e n 1 + 4 x^{2} = 1 + \tan^{2} θ = s e c^{2} θ \end{array}$

so the integral becomes

$\int \frac{1}{2} s e c^{2} θ \sqrt{s e c^{2} θ} d θ = \frac{1}{2} \int s e c^{3} θ d θ w e k n o w t h a t \int \sec^{n} (x) d x = \frac{\sec^{n - 1} (x) \sin (x)}{n - 1} + + \frac{n - 2}{n - 1} \int \sec^{n - 2} (x) d x$

now

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Arc length of a parabola Evaluate GV1 + 4x? dx, the arc length of the segment of the parabola y = x² on [0, 2].