Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Related questions
Question
![**Volume of a Solid of Revolution Using the Disk Method**
**Problem Statement:**
A solid of revolution is generated by rotating the region between the x-axis and the graphs of \( f(x) = \sqrt{x} + 1 \), \( x = 3 \), and \( x = 8 \) about the x-axis. Using the disk method, what is the volume of the solid?
Enter only the value of the volume and do not include units in your answer. Submit your answer in terms of \(\pi\), if necessary.
**Provide your answer below:**
[Input Field]
**Feedback Button:**
[FEEDBACK Button]
**Explanation:**
The above problem involves finding the volume of a solid created by rotating a given region around the x-axis. The function provided is \( f(x) = \sqrt{x} + 1 \). We are given the limits \( x = 3 \) and \( x = 8 \).
To solve this problem using the disk method, we integrate the area of the circular cross-sections of the solid along the axis of rotation. In the disk method, these cross-sections are circles with radius \( f(x) \).
The volume \( V \) of the solid of revolution is calculated using the integral:
\[ V = \pi \int_{3}^{8} [f(x)]^2 \, dx \]
Substituting \( f(x) = \sqrt{x} + 1 \):
\[ V = \pi \int_{3}^{8} (\sqrt{x} + 1)^2 \, dx \]
We then need to expand the integrand and evaluate the integral:
\[ (\sqrt{x} + 1)^2 = x + 2\sqrt{x} + 1 \]
So the integral becomes:
\[ V = \pi \int_{3}^{8} (x + 2\sqrt{x} + 1) \, dx \]
Evaluating this integral will give the volume of the solid.
**Further Steps:**
1. Integrate \( \pi \int_{3}^{8} x \, dx \)
2. Integrate \( \pi \int_{3}^{8} 2\sqrt{x} \, dx \)
3. Integrate \( \pi \int_{3}^{8} 1 \, dx \)
4. Sum the results and simplify to find the volume.
Finally, make sure to submit your](https://content.bartleby.com/qna-images/question/a79c18d0-6a0e-4a38-9496-04f82bd62d69/d6c114e1-be71-4b1d-8635-3d56fd63e925/0peq2kj_processed.jpeg)
Transcribed Image Text:**Volume of a Solid of Revolution Using the Disk Method**
**Problem Statement:**
A solid of revolution is generated by rotating the region between the x-axis and the graphs of \( f(x) = \sqrt{x} + 1 \), \( x = 3 \), and \( x = 8 \) about the x-axis. Using the disk method, what is the volume of the solid?
Enter only the value of the volume and do not include units in your answer. Submit your answer in terms of \(\pi\), if necessary.
**Provide your answer below:**
[Input Field]
**Feedback Button:**
[FEEDBACK Button]
**Explanation:**
The above problem involves finding the volume of a solid created by rotating a given region around the x-axis. The function provided is \( f(x) = \sqrt{x} + 1 \). We are given the limits \( x = 3 \) and \( x = 8 \).
To solve this problem using the disk method, we integrate the area of the circular cross-sections of the solid along the axis of rotation. In the disk method, these cross-sections are circles with radius \( f(x) \).
The volume \( V \) of the solid of revolution is calculated using the integral:
\[ V = \pi \int_{3}^{8} [f(x)]^2 \, dx \]
Substituting \( f(x) = \sqrt{x} + 1 \):
\[ V = \pi \int_{3}^{8} (\sqrt{x} + 1)^2 \, dx \]
We then need to expand the integrand and evaluate the integral:
\[ (\sqrt{x} + 1)^2 = x + 2\sqrt{x} + 1 \]
So the integral becomes:
\[ V = \pi \int_{3}^{8} (x + 2\sqrt{x} + 1) \, dx \]
Evaluating this integral will give the volume of the solid.
**Further Steps:**
1. Integrate \( \pi \int_{3}^{8} x \, dx \)
2. Integrate \( \pi \int_{3}^{8} 2\sqrt{x} \, dx \)
3. Integrate \( \pi \int_{3}^{8} 1 \, dx \)
4. Sum the results and simplify to find the volume.
Finally, make sure to submit your
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 3 steps with 4 images
Knowledge Booster
Similar questions
- Find the volume of the solid obtained by rotating the region enclosed by the curves y = x² + 2 and y axis. (Use symbolic notation and fractions where needed.) volume: 10x² about the x-arrow_forwardA volume is described as follows: 1. the base is the region bounded by y = 9 - 9x² and y = 0 2. every cross section parallel to the x-axis is a triangle whose height and base are equal. Find the volume of this object. volume =arrow_forward
Recommended textbooks for you
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage Learning
Thomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSON
Calculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON 
Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. Freeman
Calculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning
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