Find the exact length of the curve y = In(1 – x²), 0 Sx S
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Find the exact length of the curve defined by the equation:
\[ y = \ln(1 - x^2) \]
for the interval:
\[ 0 \leq x \leq \frac{1}{8} \]
**Instructions:**
To find the length of the curve, you will need to use the arc length formula for a function \( y = f(x) \) over a given interval \([a, b]\):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Start by computing the derivative \(\frac{dy}{dx}\) of the given function, then substitute \(\frac{dy}{dx}\) into the arc length formula. Evaluate the integral to determine the exact length of the curve from \(x = 0\) to \(x = \frac{1}{8}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa836ac4e-4db0-475d-87c6-ece7bcb44ff3%2F816e1881-7988-4b35-a16b-81cff962472e%2Fas5ixtm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the exact length of the curve defined by the equation:
\[ y = \ln(1 - x^2) \]
for the interval:
\[ 0 \leq x \leq \frac{1}{8} \]
**Instructions:**
To find the length of the curve, you will need to use the arc length formula for a function \( y = f(x) \) over a given interval \([a, b]\):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Start by computing the derivative \(\frac{dy}{dx}\) of the given function, then substitute \(\frac{dy}{dx}\) into the arc length formula. Evaluate the integral to determine the exact length of the curve from \(x = 0\) to \(x = \frac{1}{8}\).
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