• Find In (6) In (4) 2 S 0 e - X- 3y 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
5.1.4
![**Problem Statement:**
Evaluate the double integral:
\[
\int_{\ln(4)}^{\ln(6)} \int_{0}^{2} e^{-x - 3y} \, dy \, dx
\]
**Detailed Explanation:**
This problem requires calculating the value of a double integral over the region defined by the given limits of integration. The integrand is the exponential function \(e^{-x - 3y}\).
1. **Inner Integral**: The inner integral \(\int_{0}^{2} e^{-x - 3y} \, dy\) is evaluated first, with respect to \(y\), keeping \(x\) constant.
2. **Outer Integral**: Once the inner integral is solved, the resulting expression is then integrated with respect to \(x\) over the limits \(\ln(4)\) to \(\ln(6)\).
**Note:**
- The limits of integration for the inner integral are from \(0\) to \(2\).
- The limits for the outer integral are from \(\ln(4)\) to \(\ln(6)\).
**Application:**
Double integrals like this are used in various applications, including calculating volume under a surface over a specified region, and in probability, to find joint probabilities for continuous random variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F876c9bbd-0074-4123-babe-119feafc9f73%2Fd94ebc99-c31d-4d45-9149-a79fcf9d208c%2Ftw73hrj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Evaluate the double integral:
\[
\int_{\ln(4)}^{\ln(6)} \int_{0}^{2} e^{-x - 3y} \, dy \, dx
\]
**Detailed Explanation:**
This problem requires calculating the value of a double integral over the region defined by the given limits of integration. The integrand is the exponential function \(e^{-x - 3y}\).
1. **Inner Integral**: The inner integral \(\int_{0}^{2} e^{-x - 3y} \, dy\) is evaluated first, with respect to \(y\), keeping \(x\) constant.
2. **Outer Integral**: Once the inner integral is solved, the resulting expression is then integrated with respect to \(x\) over the limits \(\ln(4)\) to \(\ln(6)\).
**Note:**
- The limits of integration for the inner integral are from \(0\) to \(2\).
- The limits for the outer integral are from \(\ln(4)\) to \(\ln(6)\).
**Application:**
Double integrals like this are used in various applications, including calculating volume under a surface over a specified region, and in probability, to find joint probabilities for continuous random variables.
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