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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![### Double Integral Calculation
#### Problem Statement:
Evaluate the double integral:
\[
\int \int_D (3x + y) \, dA
\]
where \( D \) is defined by the region:
\[
D = \{ (x, y) \mid x^2 + y^2 \leq 4, \, x \geq 0 \}
\]
#### Solution Attempt:
The provided answer is 80, which is indicated as incorrect by a red "X" symbol.
---
#### Detailed Explanation:
1. **Region \( D \)**:
- The region \( D \) is the right half of the disk \( x^2 + y^2 \leq 4 \).
- This is because \( x \geq 0 \) restricts the region to the right of the y-axis.
2. **Strategy for Solution**:
- Convert the integral to polar coordinates since the region \( D \) is circular.
- Use the transformations \( x = r \cos \theta \) and \( y = r \sin \theta \).
- The area element \( dA \) in polar coordinates is \( r \, dr \, d\theta \).
3. **Limits of Integration**:
- For \( r \): from 0 to 2 (as \( x^2 + y^2 \leq 4 \) describes a circle of radius 2).
- For \( \theta \): from 0 to \( \pi \) (as \( x \geq 0 \) limits to the right half).
4. **Rewrite the Integral**:
\[
\int_0^\pi \int_0^2 (3r \cos \theta + r \sin \theta) r \, dr \, d\theta
\]
5. **Evaluate the Integral**:
- Perform the integration with respect to \( r \) first, and then integrate with respect to \( \theta \).
- Ensure transformation of the integrand correctly accounts for \( r \).
---
This problem is an excellent example of using polar coordinates to simplify integration over circular regions. The key steps include setting up the correct transformation and carefully defining the limits for \( r \) and \( \theta \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc05bac74-9392-4bb0-9ff6-f7e027936b2b%2Fd2c90139-de6c-4d77-aa47-9ee78aa0fd11%2Frrzo8i_processed.png&w=3840&q=75)
Transcribed Image Text:### Double Integral Calculation
#### Problem Statement:
Evaluate the double integral:
\[
\int \int_D (3x + y) \, dA
\]
where \( D \) is defined by the region:
\[
D = \{ (x, y) \mid x^2 + y^2 \leq 4, \, x \geq 0 \}
\]
#### Solution Attempt:
The provided answer is 80, which is indicated as incorrect by a red "X" symbol.
---
#### Detailed Explanation:
1. **Region \( D \)**:
- The region \( D \) is the right half of the disk \( x^2 + y^2 \leq 4 \).
- This is because \( x \geq 0 \) restricts the region to the right of the y-axis.
2. **Strategy for Solution**:
- Convert the integral to polar coordinates since the region \( D \) is circular.
- Use the transformations \( x = r \cos \theta \) and \( y = r \sin \theta \).
- The area element \( dA \) in polar coordinates is \( r \, dr \, d\theta \).
3. **Limits of Integration**:
- For \( r \): from 0 to 2 (as \( x^2 + y^2 \leq 4 \) describes a circle of radius 2).
- For \( \theta \): from 0 to \( \pi \) (as \( x \geq 0 \) limits to the right half).
4. **Rewrite the Integral**:
\[
\int_0^\pi \int_0^2 (3r \cos \theta + r \sin \theta) r \, dr \, d\theta
\]
5. **Evaluate the Integral**:
- Perform the integration with respect to \( r \) first, and then integrate with respect to \( \theta \).
- Ensure transformation of the integrand correctly accounts for \( r \).
---
This problem is an excellent example of using polar coordinates to simplify integration over circular regions. The key steps include setting up the correct transformation and carefully defining the limits for \( r \) and \( \theta \).
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