Find the volume of the solid generated by revolving the region bounded by y = 2x, y=0, x=0, and x = 4 about the x-axis. The volume of the solid generated is cubic units. (Type an exact answer, using as needed.)
Find the volume of the solid generated by revolving the region bounded by y = 2x, y=0, x=0, and x = 4 about the x-axis. The volume of the solid generated is cubic units. (Type an exact answer, using as needed.)
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...
Related questions
Question
![**Problem Statement**
Find the volume of the solid generated by revolving the region bounded by \( y = 2x \), \( y = 0 \), \( x = 0 \), and \( x = 4 \) about the x-axis.
---
**Answer Section**
The volume of the solid generated is \( \boxed{\quad } \) cubic units.
(Type an exact answer, using \(\pi\) as needed.)
---
**Detailed Explanation**
To elaborate on this problem, consider the process of integrating to find the volume of a solid of revolution. The region defined by the given boundaries is rotated around the x-axis.
1. **Function and Boundaries**:
- The function \( y = 2x \) specifies the upper boundary of the region.
- The x-axis \( y = 0 \) serves as the lower boundary.
- The vertical lines \( x = 0 \) (left boundary) and \( x = 4 \) (right boundary) provide the limits for the x-values.
2. **Integration Setup**:
- The volume of a solid of revolution can be calculated using the disk method, where the volume \( V \) is given by the integral:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
- In this case, \( f(x) = 2x \), and the limits of integration are from \( x = 0 \) to \( x = 4 \). Therefore, the volume integral becomes:
\[
V = \pi \int_{0}^{4} (2x)^2 \, dx
\]
3. **Simplify the Integral**:
- Simplify the integrand:
\[
V = \pi \int_{0}^{4} 4x^2 \, dx
\]
4. **Calculate the Definite Integral**:
- Integrate \( 4x^2 \):
\[
V = 4\pi \int_{0}^{4} x^2 \, dx = 4\pi \left[ \frac{x^3}{3} \right]_{0}^{4}
\]
- Evaluate the definite integral:
\[
V = 4\pi \left( \frac{4^3}{3} - \frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8558f12-eb43-4f23-8df4-76c708dcf941%2F85ba0bec-b384-498d-97c8-06a59a41d633%2Fzw1g54_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the volume of the solid generated by revolving the region bounded by \( y = 2x \), \( y = 0 \), \( x = 0 \), and \( x = 4 \) about the x-axis.
---
**Answer Section**
The volume of the solid generated is \( \boxed{\quad } \) cubic units.
(Type an exact answer, using \(\pi\) as needed.)
---
**Detailed Explanation**
To elaborate on this problem, consider the process of integrating to find the volume of a solid of revolution. The region defined by the given boundaries is rotated around the x-axis.
1. **Function and Boundaries**:
- The function \( y = 2x \) specifies the upper boundary of the region.
- The x-axis \( y = 0 \) serves as the lower boundary.
- The vertical lines \( x = 0 \) (left boundary) and \( x = 4 \) (right boundary) provide the limits for the x-values.
2. **Integration Setup**:
- The volume of a solid of revolution can be calculated using the disk method, where the volume \( V \) is given by the integral:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
- In this case, \( f(x) = 2x \), and the limits of integration are from \( x = 0 \) to \( x = 4 \). Therefore, the volume integral becomes:
\[
V = \pi \int_{0}^{4} (2x)^2 \, dx
\]
3. **Simplify the Integral**:
- Simplify the integrand:
\[
V = \pi \int_{0}^{4} 4x^2 \, dx
\]
4. **Calculate the Definite Integral**:
- Integrate \( 4x^2 \):
\[
V = 4\pi \int_{0}^{4} x^2 \, dx = 4\pi \left[ \frac{x^3}{3} \right]_{0}^{4}
\]
- Evaluate the definite integral:
\[
V = 4\pi \left( \frac{4^3}{3} - \frac{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning