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. 2y = 3√x, y = 5, and 2y + 2x = 5 -7.47 X
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. 2y = 3√x, y = 5, and 2y + 2x = 5 -7.47 X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**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.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29f9c049-e4a2-4005-a5d6-aa9835638972%2Fd3858cc9-1711-44e5-86af-02b00633465a%2Fv9kf9s_processed.jpeg&w=3840&q=75)
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.*
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 now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
