Chapter 8, Problem 6PS

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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(1âˆ’x2) on interval [0,12] with graph

Formula Used:

The arc length is determined using the formula

Arc=âˆ«ab1+(yâ€²)2dx

The Derivative formula is

ddxxn=nxnâˆ’1

ddxlogx=1x

Calculation:

The function is y=ln(1âˆ’x2).

Differentiating with respect to x, we get

yâ€²=ddx(ln(1âˆ’x2))=1(1âˆ’x2)(âˆ’2x)=âˆ’2x(1âˆ’x2)

Squaring and adding 1 on both sides, we get

1+(yâ€²)2=1+(âˆ’2x1âˆ’x2)2=1+4x2(1âˆ’x2)2=1âˆ’2x2+x4+4x2(1âˆ’x2)2=1+2x2+x4(1âˆ’x2)2

Simplify further to get

1+(yâ€²)2=1+2x2+(x2)2(1âˆ’x2)2=(1+x2)2(1âˆ’x2)2=(1+x21âˆ’x2)2

Now, find the arc length as shown below

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