Sketch the region of integration and change the order of integration. So SN f(x, y)dyd.r f (x, y) dxdy
Sketch the region of integration and change the order of integration. So SN f(x, y)dyd.r f (x, y) dxdy
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
![**Sketch the Region of Integration and Change the Order of Integration**
The given double integral is:
\[
\int_{0}^{4} \int_{0}^{\sqrt{x}} f(x, y) \, dy \, dx
\]
**Task:** Sketch the region of integration and change the order of integration.
### Explanation
The limits of integration for \( y \) are from \( 0 \) to \( \sqrt{x} \), and for \( x \) are from \( 0 \) to \( 4 \).
### Region of Integration
To visualize the region:
1. **x limits:** From 0 to 4.
2. **y limits:** From 0 to \(\sqrt{x}\).
This represents the area under the curve \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \).
### New Order of Integration
We need to express \( x \) as a function of \( y \) by solving \( y = \sqrt{x} \) for \( x \):
\[ x = y^2 \]
**New limits:**
- For \( x \): from \( y^2 \) to 4.
- For \( y \): from 0 to 2 (since \( \sqrt{4} = 2 \)).
This changes the order of the integral to:
\[
\int_{0}^{2} \int_{y^2}^{4} f(x, y) \, dx \, dy
\]
### Summary
The integration order is switched from \(\int \int dydx\) to \(\int \int dxdy\), with the newly determined limits reflecting the same region of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F763773a4-b59e-428b-8a3f-bc3dd5fb97d6%2Fa902dd5c-523b-4933-8a8c-e0ecacb4fdfc%2Fqxdn0pv_processed.png&w=3840&q=75)
Transcribed Image Text:**Sketch the Region of Integration and Change the Order of Integration**
The given double integral is:
\[
\int_{0}^{4} \int_{0}^{\sqrt{x}} f(x, y) \, dy \, dx
\]
**Task:** Sketch the region of integration and change the order of integration.
### Explanation
The limits of integration for \( y \) are from \( 0 \) to \( \sqrt{x} \), and for \( x \) are from \( 0 \) to \( 4 \).
### Region of Integration
To visualize the region:
1. **x limits:** From 0 to 4.
2. **y limits:** From 0 to \(\sqrt{x}\).
This represents the area under the curve \( y = \sqrt{x} \) from \( x = 0 \) to \( x = 4 \).
### New Order of Integration
We need to express \( x \) as a function of \( y \) by solving \( y = \sqrt{x} \) for \( x \):
\[ x = y^2 \]
**New limits:**
- For \( x \): from \( y^2 \) to 4.
- For \( y \): from 0 to 2 (since \( \sqrt{4} = 2 \)).
This changes the order of the integral to:
\[
\int_{0}^{2} \int_{y^2}^{4} f(x, y) \, dx \, dy
\]
### Summary
The integration order is switched from \(\int \int dydx\) to \(\int \int dxdy\), with the newly determined limits reflecting the same region of integration.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
