A region bounded by f(x) = 3, y = 0, and x = 0 is shown below. Find the volume of the solid formed by revolving the region about the x-axis. Y 5 4 3 2+ 1 X 4-3 -2 -11 1 2 3 4 5 6 -2- AWN -3 о 18л 80 -π 3 о 27п 85 0 T
A region bounded by f(x) = 3, y = 0, and x = 0 is shown below. Find the volume of the solid formed by revolving the region about the x-axis. Y 5 4 3 2+ 1 X 4-3 -2 -11 1 2 3 4 5 6 -2- AWN -3 о 18л 80 -π 3 о 27п 85 0 T
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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![## Volume of Solid of Revolution
### Problem Statement:
A region bounded by \( f(x) = 3 \), \( y = 0 \), and \( x = 0 \) is shown below.
**Question:** Find the volume of the solid formed by revolving the region about the x-axis.
### Graph Explanation:
- The graph features a rectangular region bounded horizontally from \( x = 0 \) to \( x = 3 \) and vertically from \( y = 0 \) to \( y = 3 \).
- The shaded region under \( f(x) = 3 \) (a horizontal line at \( y = 3 \)) is between \( x = 0 \) and \( x = 3 \).
- This region is rotated about the x-axis to form a solid.
### Diagram Interpretation:
- Vertical axis (y-axis) ranges from -4 to 6.
- Horizontal axis (x-axis) ranges from -4 to 6.
- Notice the green horizontal line at \( y = 3 \) and the green vertical line at \( x = 3 \).
### Choices:
1. \( 18\pi \)
2. \( \dfrac{80}{3}\pi \)
3. \( 27\pi \)
4. \( \dfrac{85}{3}\pi \)
### Solution Approach:
1. The volume \( V \) of the solid obtained by rotating the region bounded by \( y = f(x) \), \( y = 0 \), \( x = a \), and \( x = b \) about the x-axis is given by the disk method formula:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
2. For this problem \( f(x)=3 \), \( a=0 \), and \( b=3 \):
\[
V = \pi \int_{0}^{3} [3]^2 \, dx
\]
3. Calculating the integral:
\[
V = \pi \int_{0}^{3} 9 \, dx
\]
\[
V = \pi \left[ 9x \right]_{0}^{3}
\]
\[
V = \pi \left[ 9(3) - 9(0) \right]
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf75a45c-687d-4994-8001-f519eebb3c9c%2F32aed57a-b9e3-4c8f-a3e2-88bb5cdb21ea%2Fwggh2r_processed.png&w=3840&q=75)
Transcribed Image Text:## Volume of Solid of Revolution
### Problem Statement:
A region bounded by \( f(x) = 3 \), \( y = 0 \), and \( x = 0 \) is shown below.
**Question:** Find the volume of the solid formed by revolving the region about the x-axis.
### Graph Explanation:
- The graph features a rectangular region bounded horizontally from \( x = 0 \) to \( x = 3 \) and vertically from \( y = 0 \) to \( y = 3 \).
- The shaded region under \( f(x) = 3 \) (a horizontal line at \( y = 3 \)) is between \( x = 0 \) and \( x = 3 \).
- This region is rotated about the x-axis to form a solid.
### Diagram Interpretation:
- Vertical axis (y-axis) ranges from -4 to 6.
- Horizontal axis (x-axis) ranges from -4 to 6.
- Notice the green horizontal line at \( y = 3 \) and the green vertical line at \( x = 3 \).
### Choices:
1. \( 18\pi \)
2. \( \dfrac{80}{3}\pi \)
3. \( 27\pi \)
4. \( \dfrac{85}{3}\pi \)
### Solution Approach:
1. The volume \( V \) of the solid obtained by rotating the region bounded by \( y = f(x) \), \( y = 0 \), \( x = a \), and \( x = b \) about the x-axis is given by the disk method formula:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]
2. For this problem \( f(x)=3 \), \( a=0 \), and \( b=3 \):
\[
V = \pi \int_{0}^{3} [3]^2 \, dx
\]
3. Calculating the integral:
\[
V = \pi \int_{0}^{3} 9 \, dx
\]
\[
V = \pi \left[ 9x \right]_{0}^{3}
\]
\[
V = \pi \left[ 9(3) - 9(0) \right]
\
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