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 the Surface Area of a Revolved Function**
To determine the surface area of a solid formed by revolving the function \( y = \frac{2x^3}{3} \) around the x-axis from \( x = 1 \) to \( x = 2 \):
### Problem Statement:
Find the area of the surface that results when \( y = \frac{2x^3}{3} \) is revolved around the x-axis from \( x = 1 \) to \( x = 2 \).
### Solution Approach:
To solve this problem, we use the formula for the surface area of a solid of revolution around the x-axis:
\[
S = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
### Steps:
1. **Determine \( y \) and \( \frac{dy}{dx} \)**:
- \( y = \frac{2x^3}{3} \)
- Differentiate \( y \) with respect to \( x \):
\[
\frac{dy}{dx} = 2 x^2
\]
2. **Formula substitution**:
Plug into the surface area formula:
\[
S = 2\pi \int_{1}^{2} \left(\frac{2x^3}{3}\right) \sqrt{1 + (2x^2)^2} \, dx
\]
3. **Simplify the integrand**:
\[
S = 2\pi \int_{1}^{2} \left(\frac{2x^3}{3}\right) \sqrt{1 + 4x^4} \, dx
\]
4. **Integrate**:
This step involves performing the integration, which requires further simplification or the use of computational tools to achieve the final surface area.
By following these steps, the area of the surface of the solid of revolution can be calculated, helping to understand integrals in the context of real-world geometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f0128e8-6553-498c-afa0-dd14e88e258d%2Fb43df029-b8ff-459d-9ab7-df4d25282b76%2Fe1qr7am_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculating the Surface Area of a Revolved Function**
To determine the surface area of a solid formed by revolving the function \( y = \frac{2x^3}{3} \) around the x-axis from \( x = 1 \) to \( x = 2 \):
### Problem Statement:
Find the area of the surface that results when \( y = \frac{2x^3}{3} \) is revolved around the x-axis from \( x = 1 \) to \( x = 2 \).
### Solution Approach:
To solve this problem, we use the formula for the surface area of a solid of revolution around the x-axis:
\[
S = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]
### Steps:
1. **Determine \( y \) and \( \frac{dy}{dx} \)**:
- \( y = \frac{2x^3}{3} \)
- Differentiate \( y \) with respect to \( x \):
\[
\frac{dy}{dx} = 2 x^2
\]
2. **Formula substitution**:
Plug into the surface area formula:
\[
S = 2\pi \int_{1}^{2} \left(\frac{2x^3}{3}\right) \sqrt{1 + (2x^2)^2} \, dx
\]
3. **Simplify the integrand**:
\[
S = 2\pi \int_{1}^{2} \left(\frac{2x^3}{3}\right) \sqrt{1 + 4x^4} \, dx
\]
4. **Integrate**:
This step involves performing the integration, which requires further simplification or the use of computational tools to achieve the final surface area.
By following these steps, the area of the surface of the solid of revolution can be calculated, helping to understand integrals in the context of real-world geometry.
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