Evaluate the given integral by changing to polar coordinates. Sk (4x - y) dA, where R is the region in the first quadrant enclosed by the circle x² + y² = 4 and the lines x = 0 and y = x
Evaluate the given integral by changing to polar coordinates. Sk (4x - y) dA, where R is the region in the first quadrant enclosed by the circle x² + y² = 4 and the lines x = 0 and y = x
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...
Related questions
Question
![### Evaluating Integrals Using Polar Coordinates
In this exercise, we are asked to evaluate the given integral by changing to polar coordinates.
#### Given Integral
\[ \iint_{R} (4x - y) \, dA \]
where \( R \) is the region in the first quadrant enclosed by:
- The circle \( x^2 + y^2 = 4 \)
- The lines \( x = 0 \) and \( y = x \)
### Steps to Convert to Polar Coordinates
1. **Identify the region \( R \)**:
- The region \( R \) is bounded by the circle of radius 2, i.e., \( x^2 + y^2 = 4 \).
- The line \( y = x \), which forms a 45-degree angle (\( \frac{\pi}{4} \) radians) with the x-axis.
- The line \( x = 0 \).
2. **Set up the polar coordinates**:
- In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \).
- The differential area element \( dA \) in polar coordinates is \( r \, dr \, d\theta \).
3. **Determine the limits of integration**:
- \( r \) ranges from 0 to the radius of the circle 2.
- \( \theta \) ranges from 0 to \( \frac{\pi}{4} \).
4. **Convert the integrand**:
- The integrand \( 4x - y \) in polar coordinates becomes \( 4(r \cos \theta) - (r \sin \theta) \).
- This simplifies to \( r(4 \cos \theta - \sin \theta) \).
5. **Set up the integral in polar coordinates**:
\[ \iint_{R} (4x - y) \, dA = \int_{0}^{\frac{\pi}{4}} \int_{0}^{2} r(4 \cos \theta - \sin \theta) \, r \, dr \, d\theta \]
6. **Evaluate the integral**:
First, evaluate the inner integral with respect to \( r \):
\[ \int_{0}^{2} r^2 (4 \cos \theta - \sin](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab6b79a9-c663-4f87-993a-883a678be91b%2Fb74343a3-d0a9-4696-b6d2-45b3ba8f11b9%2Fey80jw9_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluating Integrals Using Polar Coordinates
In this exercise, we are asked to evaluate the given integral by changing to polar coordinates.
#### Given Integral
\[ \iint_{R} (4x - y) \, dA \]
where \( R \) is the region in the first quadrant enclosed by:
- The circle \( x^2 + y^2 = 4 \)
- The lines \( x = 0 \) and \( y = x \)
### Steps to Convert to Polar Coordinates
1. **Identify the region \( R \)**:
- The region \( R \) is bounded by the circle of radius 2, i.e., \( x^2 + y^2 = 4 \).
- The line \( y = x \), which forms a 45-degree angle (\( \frac{\pi}{4} \) radians) with the x-axis.
- The line \( x = 0 \).
2. **Set up the polar coordinates**:
- In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \).
- The differential area element \( dA \) in polar coordinates is \( r \, dr \, d\theta \).
3. **Determine the limits of integration**:
- \( r \) ranges from 0 to the radius of the circle 2.
- \( \theta \) ranges from 0 to \( \frac{\pi}{4} \).
4. **Convert the integrand**:
- The integrand \( 4x - y \) in polar coordinates becomes \( 4(r \cos \theta) - (r \sin \theta) \).
- This simplifies to \( r(4 \cos \theta - \sin \theta) \).
5. **Set up the integral in polar coordinates**:
\[ \iint_{R} (4x - y) \, dA = \int_{0}^{\frac{\pi}{4}} \int_{0}^{2} r(4 \cos \theta - \sin \theta) \, r \, dr \, d\theta \]
6. **Evaluate the integral**:
First, evaluate the inner integral with respect to \( r \):
\[ \int_{0}^{2} r^2 (4 \cos \theta - \sin
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning