10 12 Find ¹0¹²(x + In y) dydx
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Problem Statement
Find the value of the double integral:
\[
\int_{4}^{10} \int_{6}^{12} (x + \ln y) \, dy \, dx
\]
### Explanation
This problem involves calculating a double integral over a specified region. The integrand is \(x + \ln y\), and the limits of integration are from 6 to 12 for \(y\), and from 4 to 10 for \(x\).
### Steps for Solution
1. **Integration with Respect to \(y\):**
- Integrate the function \(x + \ln y\) with respect to \(y\) while treating \(x\) as a constant.
2. **Apply the Limits for \(y\):**
- Substituting the limits 6 and 12 into the resulting expression from step 1.
3. **Integration with Respect to \(x\):**
- Integrate the simplified expression obtained after substituting the limits for \(y\) with respect to \(x\).
4. **Apply the Limits for \(x\):**
- Substituting the limits 4 and 10 into the expression obtained from the integration with respect to \(x\).
The final result will provide the value of the double integral over the specified region.
### Visualization
No specific graphs or diagrams are included in the image, but if needed, a diagram of the integration region defined by the limits can be drawn to aid visualization. The region would be a rectangle in the \(xy\)-plane, bounded by \(x\) from 4 to 10 and \(y\) from 6 to 12.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fe6d10-65c6-4dea-a463-83c487832ab0%2Fc615faad-0d32-430e-a9e6-7a3056485545%2Fpgizfr_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the value of the double integral:
\[
\int_{4}^{10} \int_{6}^{12} (x + \ln y) \, dy \, dx
\]
### Explanation
This problem involves calculating a double integral over a specified region. The integrand is \(x + \ln y\), and the limits of integration are from 6 to 12 for \(y\), and from 4 to 10 for \(x\).
### Steps for Solution
1. **Integration with Respect to \(y\):**
- Integrate the function \(x + \ln y\) with respect to \(y\) while treating \(x\) as a constant.
2. **Apply the Limits for \(y\):**
- Substituting the limits 6 and 12 into the resulting expression from step 1.
3. **Integration with Respect to \(x\):**
- Integrate the simplified expression obtained after substituting the limits for \(y\) with respect to \(x\).
4. **Apply the Limits for \(x\):**
- Substituting the limits 4 and 10 into the expression obtained from the integration with respect to \(x\).
The final result will provide the value of the double integral over the specified region.
### Visualization
No specific graphs or diagrams are included in the image, but if needed, a diagram of the integration region defined by the limits can be drawn to aid visualization. The region would be a rectangle in the \(xy\)-plane, bounded by \(x\) from 4 to 10 and \(y\) from 6 to 12.
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