Find the arc length of the curve y =(2²-8 In(z)) from z = 2 to z = 5. Length = 0
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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![**Calculating the Arc Length of a Curve**
**Problem Statement:**
Find the arc length of the curve \( y = \frac{1}{8} \left( x^2 - 8 \ln(x) \right) \) from \( x = 2 \) to \( x = 5 \).
**Solution:**
1. **Identify the Curve Equation:**
The given curve is \( y = \frac{1}{8} (x^2 - 8 \ln(x)) \).
2. **Apply the Arc Length Formula:**
For a curve \( y = f(x) \) from \( x = a \) to \( x = b \), the arc length \( L \) is given by:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
3. **Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \):**
\( y = \frac{1}{8} (x^2 - 8 \ln(x)) \)
First, simplify the function:
\[
y = \frac{1}{8} x^2 - \ln(x)
\]
Differentiate \( y \):
\[
\frac{dy}{dx} = \frac{d}{dx} \left( \frac{1}{8} x^2 - \ln(x) \right) = \frac{1}{4} x - \frac{1}{x}
\]
4. **Substitute \( \frac{dy}{dx} \) into the Arc Length Formula:**
\[
\left( \frac{dy}{dx} \right)^2 = \left( \frac{1}{4} x - \frac{1}{x} \right)^2
\]
Hence,
\[
L = \int_{2}^{5} \sqrt{1 + \left( \frac{1}{4} x - \frac{1}{x} \right)^2} \, dx
\]
5. **Evaluate the Integral:**
The evaluation of this integral requires calculus techniques, which may involve numerical methods or software tools for accurate and precise](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa59c8a62-6d27-4c14-9dc5-c4241c4a7fe0%2F575912ac-18f6-4698-be5c-632ab9514afa%2F35rx3jp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculating the Arc Length of a Curve**
**Problem Statement:**
Find the arc length of the curve \( y = \frac{1}{8} \left( x^2 - 8 \ln(x) \right) \) from \( x = 2 \) to \( x = 5 \).
**Solution:**
1. **Identify the Curve Equation:**
The given curve is \( y = \frac{1}{8} (x^2 - 8 \ln(x)) \).
2. **Apply the Arc Length Formula:**
For a curve \( y = f(x) \) from \( x = a \) to \( x = b \), the arc length \( L \) is given by:
\[
L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
3. **Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \):**
\( y = \frac{1}{8} (x^2 - 8 \ln(x)) \)
First, simplify the function:
\[
y = \frac{1}{8} x^2 - \ln(x)
\]
Differentiate \( y \):
\[
\frac{dy}{dx} = \frac{d}{dx} \left( \frac{1}{8} x^2 - \ln(x) \right) = \frac{1}{4} x - \frac{1}{x}
\]
4. **Substitute \( \frac{dy}{dx} \) into the Arc Length Formula:**
\[
\left( \frac{dy}{dx} \right)^2 = \left( \frac{1}{4} x - \frac{1}{x} \right)^2
\]
Hence,
\[
L = \int_{2}^{5} \sqrt{1 + \left( \frac{1}{4} x - \frac{1}{x} \right)^2} \, dx
\]
5. **Evaluate the Integral:**
The evaluation of this integral requires calculus techniques, which may involve numerical methods or software tools for accurate and precise
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