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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![### Calculating Arc Length
**Problem Statement:**
Calculate the arc length of \( y = 3x + 1 \) as \( x \) varies from 0 to 4.
To find the arc length \( s \):
\[ s = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Given:
- \( y = 3x + 1 \)
- \( a = 0 \)
- \( b = 4 \)
First, calculate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 3 \]
Next, plug \( \frac{dy}{dx} \) into the formula:
\[ s = \int_{0}^{4} \sqrt{1 + (3)^2} \, dx \]
\[ s = \int_{0}^{4} \sqrt{1 + 9} \, dx \]
\[ s = \int_{0}^{4} \sqrt{10} \, dx \]
Since \( \sqrt{10} \) is a constant:
\[ s = \sqrt{10} \int_{0}^{4} dx \]
\[ s = \sqrt{10} [x]_{0}^{4} \]
\[ s = \sqrt{10} (4 - 0) \]
\[ s = 4 \sqrt{10} \]
Therefore, the arc length \( s \) is:
\[ s = 4 \sqrt{10} \]
This value represents the length of the curve \( y = 3x + 1 \) from \( x = 0 \) to \( x = 4 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85a13651-0641-4e1f-9366-64e873b85445%2F02a3df9e-c1b4-4605-a5b6-0b3bc787f443%2Fel8c1u_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Arc Length
**Problem Statement:**
Calculate the arc length of \( y = 3x + 1 \) as \( x \) varies from 0 to 4.
To find the arc length \( s \):
\[ s = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Given:
- \( y = 3x + 1 \)
- \( a = 0 \)
- \( b = 4 \)
First, calculate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 3 \]
Next, plug \( \frac{dy}{dx} \) into the formula:
\[ s = \int_{0}^{4} \sqrt{1 + (3)^2} \, dx \]
\[ s = \int_{0}^{4} \sqrt{1 + 9} \, dx \]
\[ s = \int_{0}^{4} \sqrt{10} \, dx \]
Since \( \sqrt{10} \) is a constant:
\[ s = \sqrt{10} \int_{0}^{4} dx \]
\[ s = \sqrt{10} [x]_{0}^{4} \]
\[ s = \sqrt{10} (4 - 0) \]
\[ s = 4 \sqrt{10} \]
Therefore, the arc length \( s \) is:
\[ s = 4 \sqrt{10} \]
This value represents the length of the curve \( y = 3x + 1 \) from \( x = 0 \) to \( x = 4 \).
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