Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Transcription for Educational Website:**
---
**Problem Statement:**
Evaluate the double integral:
\[
\int_{1}^{5} \int_{1}^{8} \frac{\ln y}{xy} \, dy \, dx.
\]
**Answer:** [ ]
---
**Explanation:**
This problem involves evaluating a double integral over a specific region with the given limits for \(x\) and \(y\). The bounds for \(x\) range from 1 to 5, while the bounds for \(y\) range from 1 to 8. The integrand is \(\frac{\ln y}{xy}\).
To solve this problem, you would first integrate with respect to \(y\) and then with respect to \(x\).
The process involves the following steps:
1. **Integrate \(\frac{\ln y}{xy}\) with respect to \(y\)**:
- Treat \(x\) as a constant since the integration is with respect to \(y\).
- Determine the antiderivative of the integrand.
2. **Substitute the limits of integration for \(y\) (from 1 to 8)**:
- Evaluate the result of the integration at the upper and lower limits.
3. **Integrate the result with respect to \(x\)**:
- Use the limits from 1 to 5.
- Evaluate the integral with respect to \(x\).
4. **Compute the final value**:
- Perform any necessary arithmetic to find the numerical value of the double integral.
This approach will yield the value of the given integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff35f8f89-65fc-4b90-98e9-68f8eaa1b3da%2Febeaab09-2ca3-43b6-9667-49c06d8b23a9%2Fgnig88o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Problem Statement:**
Evaluate the double integral:
\[
\int_{1}^{5} \int_{1}^{8} \frac{\ln y}{xy} \, dy \, dx.
\]
**Answer:** [ ]
---
**Explanation:**
This problem involves evaluating a double integral over a specific region with the given limits for \(x\) and \(y\). The bounds for \(x\) range from 1 to 5, while the bounds for \(y\) range from 1 to 8. The integrand is \(\frac{\ln y}{xy}\).
To solve this problem, you would first integrate with respect to \(y\) and then with respect to \(x\).
The process involves the following steps:
1. **Integrate \(\frac{\ln y}{xy}\) with respect to \(y\)**:
- Treat \(x\) as a constant since the integration is with respect to \(y\).
- Determine the antiderivative of the integrand.
2. **Substitute the limits of integration for \(y\) (from 1 to 8)**:
- Evaluate the result of the integration at the upper and lower limits.
3. **Integrate the result with respect to \(x\)**:
- Use the limits from 1 to 5.
- Evaluate the integral with respect to \(x\).
4. **Compute the final value**:
- Perform any necessary arithmetic to find the numerical value of the double integral.
This approach will yield the value of the given integral.
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