dz²dA where R = {(x, y) | 4z² + 25y² ≤ 100} Find Round your answer to four decimal places.
dz²dA where R = {(x, y) | 4z² + 25y² ≤ 100} Find Round your answer to four decimal places.
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 double integral
\[
\iint_R x^2 \, dA
\]
where \( R = \{(x, y) \mid 4x^2 + 25y^2 \leq 100\} \).
**Instructions:**
Round your answer to four decimal places.
**Explanation:**
The region \( R \) is defined by the inequality \( 4x^2 + 25y^2 \leq 100 \), which represents an ellipse. The ellipse is centered at the origin \((0,0)\) with \( a^2 = 25 \) and \( b^2 = 4 \), giving semi-major axis \( a = 5 \) and semi-minor axis \( b = 2 \).
To solve the integral, consider using a coordinate transformation suitable for ellipses, such as changing to polar coordinates. This helps simplify the integration process over regions with elliptical boundaries. After setup, integrate \( x^2 \) over the defined region and round the result to four decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd20dfe5a-a4c1-4793-9f05-a80ad59a67d4%2F47aacb89-dac5-4d8a-a76e-59dbf0d272f0%2F8iajgk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the double integral
\[
\iint_R x^2 \, dA
\]
where \( R = \{(x, y) \mid 4x^2 + 25y^2 \leq 100\} \).
**Instructions:**
Round your answer to four decimal places.
**Explanation:**
The region \( R \) is defined by the inequality \( 4x^2 + 25y^2 \leq 100 \), which represents an ellipse. The ellipse is centered at the origin \((0,0)\) with \( a^2 = 25 \) and \( b^2 = 4 \), giving semi-major axis \( a = 5 \) and semi-minor axis \( b = 2 \).
To solve the integral, consider using a coordinate transformation suitable for ellipses, such as changing to polar coordinates. This helps simplify the integration process over regions with elliptical boundaries. After setup, integrate \( x^2 \) over the defined region and round the result to four decimal places.
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