2) Example: Region R is bound by the function r(x) = 13x+11, the x-axis, and the vertical lines x=5 and x = 2. Determine the volume of the solid formed as region R is revolved by 2 about the x-axis
2) Example: Region R is bound by the function r(x) = 13x+11, the x-axis, and the vertical lines x=5 and x = 2. Determine the volume of the solid formed as region R is revolved by 2 about the x-axis
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
#### Graph Description:
The provided graph illustrates the function \( r(x) = |3x + 1| \) creating a conical shape when revolved around the \( x \)-axis. The region between \( x = -5 \) and \( x = 2 \) is shown, bounded by the \( x \)-axis, with vertical lines at these points. The rotations create a series of cylindrical disks, demonstrating the solid of revolution.
#### Formulae:
**Volume of a solid of revolution formed by rotating a region \( R \) about the \( x \)-axis:**
When region \( R \) is bound by the function \( f(x) \) about the \( x \)-axis and the vertical lines \( x = A \) and \( x = B \) where \( A < B \):
\[
V_{olume_{disc}} = \pi \int^B_A [f(x)]^2 \, dx
\]
**Volume of a solid of revolution formed by rotating a region \( R \) about the \( x \)-axis:**
When region \( R \) is bound by the upper function \( f(x) \), lower function \( g(x) \), and the vertical lines \( x = A \) and \( x = B \) where \( A < B \):
\[ V_{olume_{washer}} = \pi \int^B_A \left([f(x)]^2 - [g(x)]^2\right) \, dx \]
These formulae can be used to determine the volume of the solid formed by revolving the given region around the \( x \)-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec4afb63-02f6-48e1-bd91-d1418f4294a6%2F6ffe4bb9-c300-4b10-959a-de5a2b24ffec%2F6sr5bk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Guided Example
**Example:**
Region \( R \) is bound by the function \( r(x) = |3x + 1| \), the \( x \)-axis, and the vertical lines \( x = -5 \) and \( x = 2 \). Determine the volume of the solid formed as region \( R \) is revolved by \( 2\pi \) about the \( x \)-axis.
#### Graph Description:
The provided graph illustrates the function \( r(x) = |3x + 1| \) creating a conical shape when revolved around the \( x \)-axis. The region between \( x = -5 \) and \( x = 2 \) is shown, bounded by the \( x \)-axis, with vertical lines at these points. The rotations create a series of cylindrical disks, demonstrating the solid of revolution.
#### Formulae:
**Volume of a solid of revolution formed by rotating a region \( R \) about the \( x \)-axis:**
When region \( R \) is bound by the function \( f(x) \) about the \( x \)-axis and the vertical lines \( x = A \) and \( x = B \) where \( A < B \):
\[
V_{olume_{disc}} = \pi \int^B_A [f(x)]^2 \, dx
\]
**Volume of a solid of revolution formed by rotating a region \( R \) about the \( x \)-axis:**
When region \( R \) is bound by the upper function \( f(x) \), lower function \( g(x) \), and the vertical lines \( x = A \) and \( x = B \) where \( A < B \):
\[ V_{olume_{washer}} = \pi \int^B_A \left([f(x)]^2 - [g(x)]^2\right) \, dx \]
These formulae can be used to determine the volume of the solid formed by revolving the given region around the \( x \)-axis.
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