Use a change to polar coordinates to compute the double integral S S dA, where R is R the region between two circles centered at the origin with radii a and b, where 0 < a < b.
Use a change to polar coordinates to compute the double integral S S dA, where R is R the region between two circles centered at the origin with radii a and b, where 0 < a < b.
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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![### Problem Statement
**Topic: Polar Coordinates and Double Integration**
Use a change to polar coordinates to compute the double integral
\[ \iint_{R} \frac{y^2}{x^2 + y^2} \, dA, \]
where \( R \) is the region between two circles centered at the origin with radii \( a \) and \( b \), where \( 0 < a < b \).
### Detailed Explanation
**Transforming to Polar Coordinates:**
In polar coordinates, any point \((x, y)\) in the Cartesian plane can be represented as \((r \cos \theta, r \sin \theta)\), where \( r \) is the radius, or the distance from the origin, and \( \theta \) is the angle formed with the positive \(x\)-axis.
The region \( R \) described is the annular region between two circles with radii \( a \) and \( b \). This can be mathematically expressed as:
\[ a \leq r \leq b,\]
\[ 0 \leq \theta < 2\pi. \]
**Integrand in Polar Coordinates:**
Given the integrand \(\frac{y^2}{x^2 + y^2}\):
\[ y = r \sin \theta, \]
\[ x^2 + y^2 = r^2. \]
Thus, the given integrand in polar coordinates becomes:
\[ \frac{y^2}{x^2 + y^2} = \frac{(r \sin \theta)^2}{r^2} = \sin^2 \theta. \]
The differential element \( dA \) in polar coordinates is \( r \, dr \, d\theta \).
**Integral Transformation:**
The double integral can be transformed as:
\[ \iint_{R} \frac{y^2}{x^2 + y^2} \, dA = \int_{0}^{2\pi} \int_{a}^{b} \sin^2 \theta \, r \, dr \, d\theta. \]
Here, the limits of integration for \( r \) are from \( a \) to \( b \), and for \( \theta \) are from \( 0 \) to \( 2\pi \).
### Detailed Steps to Solve the Integral:
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1aef51c-637d-44a7-9ef1-4778df07fb22%2F8bf301ac-7e5f-42e3-9b58-15cb3f8f6ad8%2Fbsj5k5_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Topic: Polar Coordinates and Double Integration**
Use a change to polar coordinates to compute the double integral
\[ \iint_{R} \frac{y^2}{x^2 + y^2} \, dA, \]
where \( R \) is the region between two circles centered at the origin with radii \( a \) and \( b \), where \( 0 < a < b \).
### Detailed Explanation
**Transforming to Polar Coordinates:**
In polar coordinates, any point \((x, y)\) in the Cartesian plane can be represented as \((r \cos \theta, r \sin \theta)\), where \( r \) is the radius, or the distance from the origin, and \( \theta \) is the angle formed with the positive \(x\)-axis.
The region \( R \) described is the annular region between two circles with radii \( a \) and \( b \). This can be mathematically expressed as:
\[ a \leq r \leq b,\]
\[ 0 \leq \theta < 2\pi. \]
**Integrand in Polar Coordinates:**
Given the integrand \(\frac{y^2}{x^2 + y^2}\):
\[ y = r \sin \theta, \]
\[ x^2 + y^2 = r^2. \]
Thus, the given integrand in polar coordinates becomes:
\[ \frac{y^2}{x^2 + y^2} = \frac{(r \sin \theta)^2}{r^2} = \sin^2 \theta. \]
The differential element \( dA \) in polar coordinates is \( r \, dr \, d\theta \).
**Integral Transformation:**
The double integral can be transformed as:
\[ \iint_{R} \frac{y^2}{x^2 + y^2} \, dA = \int_{0}^{2\pi} \int_{a}^{b} \sin^2 \theta \, r \, dr \, d\theta. \]
Here, the limits of integration for \( r \) are from \( a \) to \( b \), and for \( \theta \) are from \( 0 \) to \( 2\pi \).
### Detailed Steps to Solve the Integral:
1
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