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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Question
![**Problem Statement:**
Find the length of the curve given by the equation
\[ y = \frac{x^{3/2}}{18} - 6\sqrt{x} \]
within the interval
\[ 16 \leq x \leq 81. \]
**Solution:**
This requires the application of the arc length formula for a function \( y = f(x) \), given by
\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx, \]
where \( \frac{dy}{dx} \) is the derivative of the function and \([a, b]\) is the interval over which you are measuring the arc length.
1. **Differentiate \( y \) with respect to \( x \).**
2. **Substitute \( \frac{dy}{dx} \) into the arc length formula.**
3. **Evaluate the integral from \( x = 16 \) to \( x = 81 \).**
**Result:**
Provide the computed length of the curve in the box labeled "Length is."](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26fc455c-5ac8-41d9-a0ac-3ac28f9bfd1d%2Ff3c6e0a2-8aa6-49f0-8f09-618ce31405f3%2Fc6zd5mt_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the length of the curve given by the equation
\[ y = \frac{x^{3/2}}{18} - 6\sqrt{x} \]
within the interval
\[ 16 \leq x \leq 81. \]
**Solution:**
This requires the application of the arc length formula for a function \( y = f(x) \), given by
\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx, \]
where \( \frac{dy}{dx} \) is the derivative of the function and \([a, b]\) is the interval over which you are measuring the arc length.
1. **Differentiate \( y \) with respect to \( x \).**
2. **Substitute \( \frac{dy}{dx} \) into the arc length formula.**
3. **Evaluate the integral from \( x = 16 \) to \( x = 81 \).**
**Result:**
Provide the computed length of the curve in the box labeled "Length is."
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