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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![**Problem Statement:**
Find the arc length of the function
\[ y = \frac{2}{3}(x^2 + 1)^{3/2} \]
from \( x = 1 \) to \( x = 4 \).
**Solution Steps:**
To find the arc length of a curve \( y = f(x) \) from \( x = a \) to \( x = b \), use the arc length formula:
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
1. **Find the Derivative:** Calculate \(\frac{dy}{dx}\) for the given function.
2. **Substitute and Simplify:** Substitute \(\frac{dy}{dx}\) into the arc length formula and simplify.
3. **Evaluate the Integral:** Compute the integral over the interval from \( x = 1 \) to \( x = 4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ea55b65-6a30-46ce-96b8-2b732a599bb3%2Fb16afc61-b661-4af9-a5da-b3a7b73162c5%2Fxohr2o_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the arc length of the function
\[ y = \frac{2}{3}(x^2 + 1)^{3/2} \]
from \( x = 1 \) to \( x = 4 \).
**Solution Steps:**
To find the arc length of a curve \( y = f(x) \) from \( x = a \) to \( x = b \), use the arc length formula:
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
1. **Find the Derivative:** Calculate \(\frac{dy}{dx}\) for the given function.
2. **Substitute and Simplify:** Substitute \(\frac{dy}{dx}\) into the arc length formula and simplify.
3. **Evaluate the Integral:** Compute the integral over the interval from \( x = 1 \) to \( x = 4 \).
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