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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Question
![**Title: Calculating the Area of a Region Defined by a Polar Equation**
**Objective:**
Learn how to compute the area enclosed by a polar coordinate equation.
**Problem Statement:**
Compute the area of the region enclosed by the polar coordinate equation \( r = 2 - \sin(\theta) \).
**Detailed Explanation:**
When dealing with polar coordinates, the area \( A \) enclosed by a curve from \( \theta = a \) to \( \theta = b \) is given by the integral:
\[
A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta
\]
For the equation \( r = 2 - \sin(\theta) \), the process to find the area involves the following steps:
1. **Determine the limits of integration**: Analyze the behavior of the equation \( r(\theta) = 2 - \sin(\theta) \) over the interval that completes one full shape (often \( 0 \) to \( 2\pi \) for common curves like limacons).
2. **Evaluate the Integral**: Substitute \( r \) into the area formula and evaluate the integral over the specified interval.
3. **Simplify and Solve**: Simplify the expression if possible and find the definite integral to compute the total area.
**Graphical Representation:**
For deeper understanding, it’s beneficial to graph the equation \( r = 2 - \sin(\theta) \). This polar plot will help visualize the region whose area is being calculated. In this case, the curve describes a limacon shape.
**Conclusion:**
By understanding these steps, you can calculate the area for a wide range of curves given in polar coordinates. Practice with different polar equations to become proficient in identifying and integrating these areas.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac048e26-edc8-46ba-8829-c16c07a7a6b4%2F9cd7ac69-d165-4173-bcb0-eddb38526eae%2Fldvr8aa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Area of a Region Defined by a Polar Equation**
**Objective:**
Learn how to compute the area enclosed by a polar coordinate equation.
**Problem Statement:**
Compute the area of the region enclosed by the polar coordinate equation \( r = 2 - \sin(\theta) \).
**Detailed Explanation:**
When dealing with polar coordinates, the area \( A \) enclosed by a curve from \( \theta = a \) to \( \theta = b \) is given by the integral:
\[
A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta
\]
For the equation \( r = 2 - \sin(\theta) \), the process to find the area involves the following steps:
1. **Determine the limits of integration**: Analyze the behavior of the equation \( r(\theta) = 2 - \sin(\theta) \) over the interval that completes one full shape (often \( 0 \) to \( 2\pi \) for common curves like limacons).
2. **Evaluate the Integral**: Substitute \( r \) into the area formula and evaluate the integral over the specified interval.
3. **Simplify and Solve**: Simplify the expression if possible and find the definite integral to compute the total area.
**Graphical Representation:**
For deeper understanding, it’s beneficial to graph the equation \( r = 2 - \sin(\theta) \). This polar plot will help visualize the region whose area is being calculated. In this case, the curve describes a limacon shape.
**Conclusion:**
By understanding these steps, you can calculate the area for a wide range of curves given in polar coordinates. Practice with different polar equations to become proficient in identifying and integrating these areas.
Expert Solution
Step 1: We have given a polar equation
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