Suppose R is the shaded region in the figure. As an iterated integral in polar coordinates, B D f f(x, y) dA= [₁ fº R with limits of integration A = ? B = ? C = 0 D = 3sqrt(2) ← - ▶ 2 -3 [ f( cos(9), 7 sin(0)) r dr de -2 2 3
Suppose R is the shaded region in the figure. As an iterated integral in polar coordinates, B D f f(x, y) dA= [₁ fº R with limits of integration A = ? B = ? C = 0 D = 3sqrt(2) ← - ▶ 2 -3 [ f( cos(9), 7 sin(0)) r dr de -2 2 3
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
![Suppose \( R \) is the shaded region in the figure. As an iterated integral in polar coordinates,
\[
\iint_R f(x, y) \, dA = \int_A^B \int_C^D f(r \cos(\theta), r \sin(\theta)) \, r \, dr \, d\theta
\]
with limits of integration
- \( A = ? \)
- \( B = ? \)
- \( C = 0 \)
- \( D = 3\sqrt{2} \)
**Diagram Explanation:**
The graph is a polar coordinate plot showing a sector of a circle. The sector is bounded by two lines forming an angle at the origin and a curved arc at the top. The bounding lines create an angle with the x-axis, forming the sector shown in the shaded area.
The shaded area forms a sector that is symmetric about the y-axis, extending from \(-\pi/4\) to \(\pi/4\) in terms of angle \(\theta\). The arc at the top of the shaded region is part of a circle with radius \(3\sqrt{2}\).
To complete the integration limits:
- \( A = -\frac{\pi}{4} \)
- \( B = \frac{\pi}{4} \)
This corresponds to complete coverage of the shaded sector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3444a3f-7d5c-4474-ab08-7930ed9e066d%2F3d0445d8-4a2d-4cd8-8b7d-14bab5e4ba1f%2F8z4pr4j_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose \( R \) is the shaded region in the figure. As an iterated integral in polar coordinates,
\[
\iint_R f(x, y) \, dA = \int_A^B \int_C^D f(r \cos(\theta), r \sin(\theta)) \, r \, dr \, d\theta
\]
with limits of integration
- \( A = ? \)
- \( B = ? \)
- \( C = 0 \)
- \( D = 3\sqrt{2} \)
**Diagram Explanation:**
The graph is a polar coordinate plot showing a sector of a circle. The sector is bounded by two lines forming an angle at the origin and a curved arc at the top. The bounding lines create an angle with the x-axis, forming the sector shown in the shaded area.
The shaded area forms a sector that is symmetric about the y-axis, extending from \(-\pi/4\) to \(\pi/4\) in terms of angle \(\theta\). The arc at the top of the shaded region is part of a circle with radius \(3\sqrt{2}\).
To complete the integration limits:
- \( A = -\frac{\pi}{4} \)
- \( B = \frac{\pi}{4} \)
This corresponds to complete coverage of the shaded sector.
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