Sex²+y² dxdy R= {(r,0): 0≤ r ≤ 2, 0≤ 0 ≤ 2}
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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![The image contains a mathematical expression, which is a double integral:
\[
\iint e^{x^2 + y^2} \, dx \, dy
\]
It is stated over the region \( R \), defined in polar coordinates as:
\[
R = \{(r, \theta) : 0 \leq r \leq 2, \, 0 \leq \theta \leq 2\pi\}
\]
### Explanation:
- **Double Integral:** The integral \(\iint e^{x^2 + y^2} \, dx \, dy\) calculates the volume under the surface defined by \(e^{x^2 + y^2}\) over the specified region \(R\).
- **Region \(R\):** The region is defined in polar coordinates.
- \(r\) is the radial distance from the origin, ranging from 0 to 2.
- \(\theta\) is the angle in radians, ranging from 0 to \(2\pi\), representing a full circle.
This setup describes a circular region with radius 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ad28635-52ce-4e5e-984f-c041c5ae53dd%2F1e52e837-111c-473b-8c9a-9d0eef81fc80%2Fs4fxzqm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression, which is a double integral:
\[
\iint e^{x^2 + y^2} \, dx \, dy
\]
It is stated over the region \( R \), defined in polar coordinates as:
\[
R = \{(r, \theta) : 0 \leq r \leq 2, \, 0 \leq \theta \leq 2\pi\}
\]
### Explanation:
- **Double Integral:** The integral \(\iint e^{x^2 + y^2} \, dx \, dy\) calculates the volume under the surface defined by \(e^{x^2 + y^2}\) over the specified region \(R\).
- **Region \(R\):** The region is defined in polar coordinates.
- \(r\) is the radial distance from the origin, ranging from 0 to 2.
- \(\theta\) is the angle in radians, ranging from 0 to \(2\pi\), representing a full circle.
This setup describes a circular region with radius 2.
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