Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement**
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) or \( y \). Then find the area of the region.
Given Equations:
- \( 2y = 3\sqrt{x} \)
- \( y = 5 \)
- \( 2y + 2x = 5 \)
**Solution Attempt**
The answer provided: \(-7.47\) is incorrect.
*Explanation and Steps*
1. **Graphing the Curves**:
- **Equation 1**: Rearrange \(2y = 3\sqrt{x}\) to express \(y\) in terms of \(x\):
\[
y = \frac{3}{2}\sqrt{x}
\]
This is a square root function, which starts at the origin and increases to the right.
- **Equation 2**: The equation \(y = 5\) is a horizontal line across \(y = 5\).
- **Equation 3**: Rearrange \(2y + 2x = 5\) to express \(y\):
\[
y = -x + \frac{5}{2}
\]
This is a straight line with a negative slope.
2. **Decide Integration Method**:
- Identify where the curves intersect and the regions they enclose. This will guide whether to integrate with respect to \( x \) or \( y \). Set up the integrals to find the exact area of the enclosed region by calculating intersections and the enclosed area.
*Note: Make sure calculations yield a positive area value.*](https://content.bartleby.com/qna-images/question/29f9c049-e4a2-4005-a5d6-aa9835638972/d3858cc9-1711-44e5-86af-02b00633465a/v9kf9s_processed.jpeg)
Transcribed Image Text:**Problem Statement**
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) or \( y \). Then find the area of the region.
Given Equations:
- \( 2y = 3\sqrt{x} \)
- \( y = 5 \)
- \( 2y + 2x = 5 \)
**Solution Attempt**
The answer provided: \(-7.47\) is incorrect.
*Explanation and Steps*
1. **Graphing the Curves**:
- **Equation 1**: Rearrange \(2y = 3\sqrt{x}\) to express \(y\) in terms of \(x\):
\[
y = \frac{3}{2}\sqrt{x}
\]
This is a square root function, which starts at the origin and increases to the right.
- **Equation 2**: The equation \(y = 5\) is a horizontal line across \(y = 5\).
- **Equation 3**: Rearrange \(2y + 2x = 5\) to express \(y\):
\[
y = -x + \frac{5}{2}
\]
This is a straight line with a negative slope.
2. **Decide Integration Method**:
- Identify where the curves intersect and the regions they enclose. This will guide whether to integrate with respect to \( x \) or \( y \). Set up the integrals to find the exact area of the enclosed region by calculating intersections and the enclosed area.
*Note: Make sure calculations yield a positive area value.*
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