Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Volume of Generated Solid by Rotation
#### Problem Statement:
Refer to the figure and find the volume \( V \) generated by rotating the given region about the specified line.
#### Rotation Region:
Rotate \( R_3 \) about the line \( AB \).
#### Volume Formula:
\[ V = \]
#### Diagram Explanation:
- The diagram is set on a coordinate system with x-axis (\( x \)) and y-axis (\( y \)).
- Point \( O \) is at the origin \((0, 0)\).
- The points marked on the diagram are \( A(1, 0) \), \( B(1, 4) \), and \( C(0, 4) \).
- There are three marked regions:
- Region \( R_1 \) is the bottom right triangular area, marked in blue.
- Region \( R_2 \) is the upper left area, marked in green, formed under the curve \( y = 4 \sqrt[4]{x} \) and above the x-axis.
- Region \( R_3 \) is the area in yellow, formed between region \( R_1 \) and region \( R_2 \). This region is bounded by the line \( OB \) and the curve \( y = 4 \sqrt[4]{x} \).
#### Key Elements and Boundaries:
- The curve \( y = 4 \sqrt[4]{x} \) defines part of the boundary for \( R_2 \) and \( R_3 \).
- The rectangle \( OACB \) encompasses all regions \( R_1 \), \( R_2 \), and \( R_3 \).
- The line segment \( AB \) acts as the axis of rotation.
#### Integration and Calculation:
To solve for the volume \( V \) generated by the rotation of region \( R_3 \) around line \( AB \), set up and evaluate the integral using the Disk/Washer or Cylindrical Shell method as appropriate for the given boundaries and axis of rotation. This involves determining the radii and heights pertinent to the areas being rotated.](https://content.bartleby.com/qna-images/question/64a32957-950f-49d5-802d-a933129b388e/31637827-d1a9-4863-9fa4-63816e7e0142/3lsh5p7_processed.png)
Transcribed Image Text:### Volume of Generated Solid by Rotation
#### Problem Statement:
Refer to the figure and find the volume \( V \) generated by rotating the given region about the specified line.
#### Rotation Region:
Rotate \( R_3 \) about the line \( AB \).
#### Volume Formula:
\[ V = \]
#### Diagram Explanation:
- The diagram is set on a coordinate system with x-axis (\( x \)) and y-axis (\( y \)).
- Point \( O \) is at the origin \((0, 0)\).
- The points marked on the diagram are \( A(1, 0) \), \( B(1, 4) \), and \( C(0, 4) \).
- There are three marked regions:
- Region \( R_1 \) is the bottom right triangular area, marked in blue.
- Region \( R_2 \) is the upper left area, marked in green, formed under the curve \( y = 4 \sqrt[4]{x} \) and above the x-axis.
- Region \( R_3 \) is the area in yellow, formed between region \( R_1 \) and region \( R_2 \). This region is bounded by the line \( OB \) and the curve \( y = 4 \sqrt[4]{x} \).
#### Key Elements and Boundaries:
- The curve \( y = 4 \sqrt[4]{x} \) defines part of the boundary for \( R_2 \) and \( R_3 \).
- The rectangle \( OACB \) encompasses all regions \( R_1 \), \( R_2 \), and \( R_3 \).
- The line segment \( AB \) acts as the axis of rotation.
#### Integration and Calculation:
To solve for the volume \( V \) generated by the rotation of region \( R_3 \) around line \( AB \), set up and evaluate the integral using the Disk/Washer or Cylindrical Shell method as appropriate for the given boundaries and axis of rotation. This involves determining the radii and heights pertinent to the areas being rotated.
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