Evaluate the following double integral over the region R. 2 ſſ⁹(x³ – y5) ²dA; R = {(x,y): 0≤x≤ 1, − 1 ≤y≤1} R
Evaluate the following double integral over the region R. 2 ſſ⁹(x³ – y5) ²dA; R = {(x,y): 0≤x≤ 1, − 1 ≤y≤1} R
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
![**Title: Evaluating a Double Integral Over a Defined Region**
**Problem Statement:**
Evaluate the following double integral over the region \( R \):
\[ \iint\limits_R 9 \left( x^5 - y^5 \right)^2 \, dA \]
where \( R = \{(x, y) : 0 \leq x \leq 1, -1 \leq y \leq 1\} \).
**Solution Outline:**
To solve this double integral, we will follow these steps:
1. **Identify the Region \( R \)**:
The region \( R \) is defined as:
\[
R = \{(x, y) : 0 \leq x \leq 1, -1 \leq y \leq 1\}
\]
It signifies a rectangular region in the \( xy \)-plane from \( x = 0 \) to \( x = 1 \) and from \( y = -1 \) to \( y = 1 \).
2. **Set Up the Integral**:
Convert the given double integral into two iterated integrals.
3. **Evaluate the Integral**:
Compute the integral in terms of \( x \) and \( y \).
### Step-by-Step Solution:
**Integral Setup:**
Given the region \( R \), we can set up the integral as follows:
\[
\iint\limits_R 9 \left( x^5 - y^5 \right)^2 \, dA = \int_{0}^{1} \int_{-1}^{1} 9 \left( x^5 - y^5 \right)^2 \, dy \, dx
\]
**Evaluating the Inner Integral:**
Evaluating the inner integral with respect to \( y \):
\[
\int_{-1}^{1} 9 \left( x^5 - y^5 \right)^2 \, dy
\]
Expanding \( \left( x^5 - y^5 \right)^2 \):
\[
\left( x^5 - y^5 \right)^2 = x^{10} - 2x^5y^5 + y^{10}
\]
Thus, the inner integral becomes:
\[
\int_{-1}^{1} 9 \left( x^{10](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ff120f1-bc43-4dfa-8402-4dae27bc4a74%2F86ba68fc-9ae6-4b19-9049-217f2d5329ae%2F6j2d6_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating a Double Integral Over a Defined Region**
**Problem Statement:**
Evaluate the following double integral over the region \( R \):
\[ \iint\limits_R 9 \left( x^5 - y^5 \right)^2 \, dA \]
where \( R = \{(x, y) : 0 \leq x \leq 1, -1 \leq y \leq 1\} \).
**Solution Outline:**
To solve this double integral, we will follow these steps:
1. **Identify the Region \( R \)**:
The region \( R \) is defined as:
\[
R = \{(x, y) : 0 \leq x \leq 1, -1 \leq y \leq 1\}
\]
It signifies a rectangular region in the \( xy \)-plane from \( x = 0 \) to \( x = 1 \) and from \( y = -1 \) to \( y = 1 \).
2. **Set Up the Integral**:
Convert the given double integral into two iterated integrals.
3. **Evaluate the Integral**:
Compute the integral in terms of \( x \) and \( y \).
### Step-by-Step Solution:
**Integral Setup:**
Given the region \( R \), we can set up the integral as follows:
\[
\iint\limits_R 9 \left( x^5 - y^5 \right)^2 \, dA = \int_{0}^{1} \int_{-1}^{1} 9 \left( x^5 - y^5 \right)^2 \, dy \, dx
\]
**Evaluating the Inner Integral:**
Evaluating the inner integral with respect to \( y \):
\[
\int_{-1}^{1} 9 \left( x^5 - y^5 \right)^2 \, dy
\]
Expanding \( \left( x^5 - y^5 \right)^2 \):
\[
\left( x^5 - y^5 \right)^2 = x^{10} - 2x^5y^5 + y^{10}
\]
Thus, the inner integral becomes:
\[
\int_{-1}^{1} 9 \left( x^{10
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