**Problem Statement**Please give all the necessary details.Compute $\iint\limits_{D} (x^2 + y) \, dA$, where $D$ is the region between the circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 5$.**Explanation**The goal is to evaluate the double integral of the function $x^2 + y$ over the region $D$, which is the annular region between the inner circle with radius 1 and the outer circle with radius $\sqrt{5}$.**Methodology**1. **Identify the Region:** - The inner circle is defined by $x^2 + y^2 = 1$ with a radius of 1. - The outer circle is defined by $x^2 + y^2 = 5$ with a radius of $\sqrt{5}$.2. **Express in Polar Coordinates:** - Convert the Cartesian coordinates to polar coordinates: - $x = r \cos \theta$ - $y = r \sin \theta$ - $dA = r \, dr \, d\theta$3. **Set Up the Integral:** - The limits for $r$ are from 1 to $\sqrt{5}$. - The limits for $\theta$ are from 0 to $2\pi$.4. **Integrate the Function:** - Substitute $x = r \cos \theta$ and $y = r \sin \theta$ into the function: - $x^2 + y = (r \cos \theta)^2 + r \sin \theta$ - Evaluate the integral using polar limits.This provides a comprehensive breakdown of how to tackle the integral of given functions over circular regions using polar coordinates.

Consider the given Integral. I=∬Dx2+ydA And, x2+y2=1,x2+y2=5 Convert the equation in the polar…

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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**Problem Statement**

Please give all the necessary details.

Compute $\iint\limits_{D} (x^2 + y) \, dA$, where $D$ is the region between the circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 5$.

**Explanation**

The goal is to evaluate the double integral of the function $x^2 + y$ over the region $D$, which is the annular region between the inner circle with radius 1 and the outer circle with radius $\sqrt{5}$.

**Methodology**

1. **Identify the Region:**
- The inner circle is defined by $x^2 + y^2 = 1$ with a radius of 1.
- The outer circle is defined by $x^2 + y^2 = 5$ with a radius of $\sqrt{5}$.

2. **Express in Polar Coordinates:**
- Convert the Cartesian coordinates to polar coordinates:
- $x = r \cos \theta$
- $y = r \sin \theta$
- $dA = r \, dr \, d\theta$

3. **Set Up the Integral:**
- The limits for $r$ are from 1 to $\sqrt{5}$.
- The limits for $\theta$ are from 0 to $2\pi$.

4. **Integrate the Function:**
- Substitute $x = r \cos \theta$ and $y = r \sin \theta$ into the function:
- $x^2 + y = (r \cos \theta)^2 + r \sin \theta$
- Evaluate the integral using polar limits.

This provides a comprehensive breakdown of how to tackle the integral of given functions over circular regions using polar coordinates.

Expert Solution

Step 1

Consider the given Integral.

$I = \iint_{D} (x^{2} + y) d A$

And,

$x^{2} + y^{2} = 1, x^{2} + y^{2} = 5$

Convert the equation in the polar equation.

Step 2

Put $x = r \cos θ and y = r \sin θ$ in the given equation.

$\begin{array}{rcl} r^{2} \cos^{2} θ + r^{2} \sin^{2} θ & = & 1 \\ r^{2} (\cos^{2} θ + \sin^{2} θ) & = & 1 \\ r^{2} & = & 1 \\ r & = & 1 \end{array}$

And,

$\begin{array}{rcl} r^{2} \cos^{2} θ + r^{2} \sin^{2} θ & = & 5 \\ r^{2} (\cos^{2} θ + \sin^{2} θ) & = & 5 \\ r^{2} & = & 1 \\ r & = & \sqrt{5} \end{array}$

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