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 arc length of the curve defined by the equation \( y = 2x^2 - 1 \) over the interval \( 0 \leq x \leq 3 \).
**Solution Explanation:**
To find the arc length of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \), we use the formula:
\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
### Steps:
1. **Differentiate \( y = 2x^2 - 1 \) with respect to \( x \):**
\[
\frac{dy}{dx} = 4x
\]
2. **Substitute into the arc length formula:**
\[
L = \int_0^3 \sqrt{1 + (4x)^2} \, dx = \int_0^3 \sqrt{1 + 16x^2} \, dx
\]
3. **Evaluate the integral:**
The integral \( \int_0^3 \sqrt{1 + 16x^2} \, dx \) can be solved using an appropriate substitution or numerical methods, leading to the final arc length.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2F76f3c26f-6e66-4f17-93b8-af075b63a594%2Fjykevj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the arc length of the curve defined by the equation \( y = 2x^2 - 1 \) over the interval \( 0 \leq x \leq 3 \).
**Solution Explanation:**
To find the arc length of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \), we use the formula:
\[
L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]
### Steps:
1. **Differentiate \( y = 2x^2 - 1 \) with respect to \( x \):**
\[
\frac{dy}{dx} = 4x
\]
2. **Substitute into the arc length formula:**
\[
L = \int_0^3 \sqrt{1 + (4x)^2} \, dx = \int_0^3 \sqrt{1 + 16x^2} \, dx
\]
3. **Evaluate the integral:**
The integral \( \int_0^3 \sqrt{1 + 16x^2} \, dx \) can be solved using an appropriate substitution or numerical methods, leading to the final arc length.
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