   Chapter 8, Problem 6PS

Chapter
Section
Textbook Problem

Arc Length Find the arc length of the graph of the function y = ln ( 1 − x 2 ) on the interval 0 ≤ x ≤ 1 2 (see figure). To determine

To calculate: The arc length of the provided graph of function y=ln(1x2) on [0,12]. Explanation

Given:

The function y=ln(1x2) on interval [0,12] with graph

Formula Used:

The arc length is determined using the formula

Arc=ab1+(y)2dx

The Derivative formula is

ddxxn=nxn1

ddxlogx=1x

Calculation:

The function is y=ln(1x2).

Differentiating with respect to x, we get

y=ddx(ln(1x2))=1(1x2)(2x)=2x(1x2)

Squaring and adding 1 on both sides, we get

1+(y)2=1+(2x1x2)2=1+4x2(1x2)2=12x2+x4+4x2(1x2)2=1+2x2+x4(1x2)2

Simplify further to get

1+(y)2=1+2x2+(x2)2(1x2)2=(1+x2)2(1x2)2=(1+x21x2)2

Now, find the arc length as shown below

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