b) use Polar Coordinates to compute SS. (x+y) JA the first quadrant where R is the region in between the circle of radius I and radius 3 Centered at the origin
b) use Polar Coordinates to compute SS. (x+y) JA the first quadrant where R is the region in between the circle of radius I and radius 3 Centered at the origin
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Using Polar Coordinates to Compute the Double Integral**
In this exercise, we are asked to use polar coordinates to compute the double integral \(\iint_R (x + y) \, dA\), where \(R\) is the region in the first quadrant between the circle of radius 1 and radius 3 centered at the origin.
Here's the detailed transcription and the explanation:
### Problem Statement:
Use polar coordinates to compute \(\iint_R (x + y) \, dA\) where \(R\) is the region in the first quadrant between the circle of radius 1 and radius 3 centered at the origin.
### Detailed Solution:
To solve this problem, we need to:
1. **Convert the Cartesian coordinates to polar coordinates:**
- In polar coordinates, the variables \(x\) and \(y\) can be expressed as:
\[
x = r \cos(\theta)
\]
\[
y = r \sin(\theta)
\]
- The area element \(dA\) in polar coordinates is given by \(r \, dr \, d\theta\).
2. **Set up the integral limits:**
- Since \(R\) is the region in the first quadrant between the circles of radius 1 and 3:
- The limits for \(r\) will be from 1 to 3.
- The limits for \(\theta\) will be from 0 to \(\frac{\pi}{2}\) (since we are in the first quadrant).
3. **Convert the integrand \(x + y\) to polar coordinates:**
- Using the expressions for \(x\) and \(y\), we have:
\[
x + y = r \cos(\theta) + r \sin(\theta) = r (\cos(\theta) + \sin(\theta))
\]
4. **Set up the double integral in polar coordinates:**
\[
\iint_R (x + y) \, dA = \int_0^{\frac{\pi}{2}} \int_1^3 r (\cos(\theta) + \sin(\theta)) \, r \, dr \, d\theta
\]
5. **Evaluate the integral:**
- First, simplify the integrand:
\[
\int_0^{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53b47ed7-4497-4fb9-9d0f-c41504626a91%2Fa44d0c2d-e425-4383-820b-8488182e05ec%2F6f0dm6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Using Polar Coordinates to Compute the Double Integral**
In this exercise, we are asked to use polar coordinates to compute the double integral \(\iint_R (x + y) \, dA\), where \(R\) is the region in the first quadrant between the circle of radius 1 and radius 3 centered at the origin.
Here's the detailed transcription and the explanation:
### Problem Statement:
Use polar coordinates to compute \(\iint_R (x + y) \, dA\) where \(R\) is the region in the first quadrant between the circle of radius 1 and radius 3 centered at the origin.
### Detailed Solution:
To solve this problem, we need to:
1. **Convert the Cartesian coordinates to polar coordinates:**
- In polar coordinates, the variables \(x\) and \(y\) can be expressed as:
\[
x = r \cos(\theta)
\]
\[
y = r \sin(\theta)
\]
- The area element \(dA\) in polar coordinates is given by \(r \, dr \, d\theta\).
2. **Set up the integral limits:**
- Since \(R\) is the region in the first quadrant between the circles of radius 1 and 3:
- The limits for \(r\) will be from 1 to 3.
- The limits for \(\theta\) will be from 0 to \(\frac{\pi}{2}\) (since we are in the first quadrant).
3. **Convert the integrand \(x + y\) to polar coordinates:**
- Using the expressions for \(x\) and \(y\), we have:
\[
x + y = r \cos(\theta) + r \sin(\theta) = r (\cos(\theta) + \sin(\theta))
\]
4. **Set up the double integral in polar coordinates:**
\[
\iint_R (x + y) \, dA = \int_0^{\frac{\pi}{2}} \int_1^3 r (\cos(\theta) + \sin(\theta)) \, r \, dr \, d\theta
\]
5. **Evaluate the integral:**
- First, simplify the integrand:
\[
\int_0^{\
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