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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This question was asked before, but I'm being told its incorrect. Can I get someone else to answer this?
![**Problem Statement:**
Find the exact length of the curve for the given function.
**Function:**
\[ y = 1 + 8x^{3/2} \]
**Interval:**
\[ 0 \leq x \leq 1 \]
**Solution Approach:**
To find the exact length of the curve, we will use the arc length formula for a curve defined by \( y = f(x) \) on the interval \([a, b]\):
\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
1. **Differentiate the Function:**
Find the derivative \(\frac{dy}{dx}\) of the given function \( y = 1 + 8x^{3/2} \).
2. **Set Up the Integral:**
Substitute the derivative into the arc length formula and calculate the integral:
\[ L = \int_{0}^{1} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
This approach will yield the exact length of the curve from \( x = 0 \) to \( x = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8abda4d0-aef7-4bfa-8fb9-182b00f7a9e2%2F5eb020f1-5019-47af-a1f5-0a36f4b50f14%2Fckuc92j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the exact length of the curve for the given function.
**Function:**
\[ y = 1 + 8x^{3/2} \]
**Interval:**
\[ 0 \leq x \leq 1 \]
**Solution Approach:**
To find the exact length of the curve, we will use the arc length formula for a curve defined by \( y = f(x) \) on the interval \([a, b]\):
\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
1. **Differentiate the Function:**
Find the derivative \(\frac{dy}{dx}\) of the given function \( y = 1 + 8x^{3/2} \).
2. **Set Up the Integral:**
Substitute the derivative into the arc length formula and calculate the integral:
\[ L = \int_{0}^{1} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \]
This approach will yield the exact length of the curve from \( x = 0 \) to \( x = 1 \).
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