O e a double integral in polar coordinates to find the area of the region bounded on the inside by the circle of radius 3 and on the outside by the cardioid r = 3(1 + cos(0))
O e a double integral in polar coordinates to find the area of the region bounded on the inside by the circle of radius 3 and on the outside by the cardioid r = 3(1 + cos(0))
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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![**Title: Calculating Area Using Polar Coordinates**
**Graph Description:**
This diagram shows two curves on a polar coordinate grid.
- The first curve is a circle centered at the origin with a radius of 3.
- The second curve is a cardioid described by the polar equation \( r = 3(1 + \cos(\theta)) \).
The shaded region of interest is the area that is outside the cardioid but inside the circle, depicted in red.
**Problem Statement:**
Use a double integral in polar coordinates to find the area of the region bounded on the inside by the circle of radius 3 and on the outside by the cardioid \( r = 3(1 + \cos(\theta)) \).
**Solution Approach:**
To find the area of this region using polar coordinates, set up the double integral with appropriate limits for \( r \) and \( \theta \) that describe the region of interest. Calculate the area by integrating:
\[ A = \int_{\theta_1}^{\theta_2} \int_{r_{\text{inner}}}^{r_{\text{outer}}} r \, dr \, d\theta \]
Where:
- \( r_{\text{inner}} \) is the radius of the cardioid at a given \( \theta \).
- \( r_{\text{outer}} \) is the radius of the circle.
- \( \theta_1 \) and \( \theta_2 \) are the angular limits determined by the intersection points of the two curves.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0c8037-834b-4d36-b3c4-eda43340784b%2F6325887b-367a-41bc-a4a2-6d2920c05d1d%2Fk6mxj5t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Area Using Polar Coordinates**
**Graph Description:**
This diagram shows two curves on a polar coordinate grid.
- The first curve is a circle centered at the origin with a radius of 3.
- The second curve is a cardioid described by the polar equation \( r = 3(1 + \cos(\theta)) \).
The shaded region of interest is the area that is outside the cardioid but inside the circle, depicted in red.
**Problem Statement:**
Use a double integral in polar coordinates to find the area of the region bounded on the inside by the circle of radius 3 and on the outside by the cardioid \( r = 3(1 + \cos(\theta)) \).
**Solution Approach:**
To find the area of this region using polar coordinates, set up the double integral with appropriate limits for \( r \) and \( \theta \) that describe the region of interest. Calculate the area by integrating:
\[ A = \int_{\theta_1}^{\theta_2} \int_{r_{\text{inner}}}^{r_{\text{outer}}} r \, dr \, d\theta \]
Where:
- \( r_{\text{inner}} \) is the radius of the cardioid at a given \( \theta \).
- \( r_{\text{outer}} \) is the radius of the circle.
- \( \theta_1 \) and \( \theta_2 \) are the angular limits determined by the intersection points of the two curves.
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