The region is D: x 2 0, y > 0, x² + y² < 4. Choose the sketch of the region. 4 D -4 D 4 O y 2 D 2 D 2 Integrate f(x, y) = 6xy over D using polar coordinates.
The region is D: x 2 0, y > 0, x² + y² < 4. Choose the sketch of the region. 4 D -4 D 4 O y 2 D 2 D 2 Integrate f(x, y) = 6xy over D using polar coordinates.
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 problem defines the region \( D \) as follows: \( x \geq 0 \), \( y \geq 0 \), \( x^2 + y^2 \leq 4 \).
### Diagram Explanation
The task is to choose the correct sketch of the region \( D \) from four options.
1. **Option 1** (top left) shows a semicircular region in the first and fourth quadrants, with the semicircle centered at the origin, extending along the x-axis from -4 to 4.
2. **Option 2** (top right) displays a quarter-circle in the first quadrant with radius 4, spanning along the x-axis and y-axis up to 4.
3. **Option 3** (bottom left) displays a quarter-circle in the first quadrant with radius 2, spanning along the x-axis and y-axis up to 2. This option is marked as the correct choice.
4. **Option 4** (bottom right) shows a semicircular region centered on the x-axis, extending from -2 to 2.
### Integration Task
The next step is to integrate the function \( f(x, y) = 6xy \) over the region \( D \) using polar coordinates. Use symbolic notation and fractions where necessary.
The integral is represented as:
\[
\iint_D 6xy \, dA = \underline{\hspace{2cm}}
\]
In polar coordinates, transformation equations \( x = r \cos \theta \) and \( y = r \sin \theta \) should be used, where \( r \leq 2 \) due to the circle equation \( x^2 + y^2 = 4 \).
This setup is useful for understanding regions and calculating double integrals in mathematical analysis, specifically using polar coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23ed5c9f-03d0-41ef-878f-b64ab507220e%2F60ec9f21-ee3a-4f55-b0bf-418e065272b3%2F0wpgjwl_processed.png&w=3840&q=75)
Transcribed Image Text:The problem defines the region \( D \) as follows: \( x \geq 0 \), \( y \geq 0 \), \( x^2 + y^2 \leq 4 \).
### Diagram Explanation
The task is to choose the correct sketch of the region \( D \) from four options.
1. **Option 1** (top left) shows a semicircular region in the first and fourth quadrants, with the semicircle centered at the origin, extending along the x-axis from -4 to 4.
2. **Option 2** (top right) displays a quarter-circle in the first quadrant with radius 4, spanning along the x-axis and y-axis up to 4.
3. **Option 3** (bottom left) displays a quarter-circle in the first quadrant with radius 2, spanning along the x-axis and y-axis up to 2. This option is marked as the correct choice.
4. **Option 4** (bottom right) shows a semicircular region centered on the x-axis, extending from -2 to 2.
### Integration Task
The next step is to integrate the function \( f(x, y) = 6xy \) over the region \( D \) using polar coordinates. Use symbolic notation and fractions where necessary.
The integral is represented as:
\[
\iint_D 6xy \, dA = \underline{\hspace{2cm}}
\]
In polar coordinates, transformation equations \( x = r \cos \theta \) and \( y = r \sin \theta \) should be used, where \( r \leq 2 \) due to the circle equation \( x^2 + y^2 = 4 \).
This setup is useful for understanding regions and calculating double integrals in mathematical analysis, specifically using polar coordinates.
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