Find the area of the region that lies inside the first curve and outside the second curve. r = 19 sin(0), r = 10 sin(0)
Find the area of the region that lies inside the first curve and outside the second curve. r = 19 sin(0), r = 10 sin(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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![**Educational Resource: Calculating Area Between Polar Curves**
---
**Problem Statement:**
Find the area of the region that lies inside the first curve and outside the second curve.
Given polar equations:
- \( r = 19 \sin(\theta) \)
- \( r = 10 - \sin(\theta) \)
**Explanation:**
To solve this problem, you need to compute the area that is enclosed by the first curve but not by the second curve.
**Steps:**
1. **Understanding the Curves:**
- \( r = 19 \sin(\theta) \): This represents a limaçon with an inner loop.
- \( r = 10 - \sin(\theta) \): This curve represents another limaçon which may form different shapes depending on the range of \(\theta\).
2. **Intersection Points:**
- Determine the points of intersection by setting the equations equal:
\[
19 \sin(\theta) = 10 - \sin(\theta)
\]
- Solve for \(\theta\) to find the limits of integration.
3. **Calculate the Area:**
- Use the formula for the area enclosed by polar curves:
\[
A = \frac{1}{2} \int_{\theta_1}^{\theta_2} (r_1^2 - r_2^2) \, d\theta
\]
- Here, \( r_1 = 19 \sin(\theta) \) and \( r_2 = 10 - \sin(\theta) \).
4. **Integration Boundaries:**
- The limits of integration \(\theta_1\) and \(\theta_2\) are the solutions where the two curves intersect.
5. **Graphical Representation:**
- A graph would illustrate the curves in the polar coordinate system, showing the area to be calculated. This involves plotting both curves to identify the region of interest.
By following these steps, the area of the region inside the first curve and outside the second can be accurately calculated. Understanding these principles is crucial for solving such polar coordinate problems effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff42c59a5-908c-4bcd-9e36-90d243a82d72%2F666f01d8-23cc-4a51-bb02-2ce1c418ceae%2Fhbnsox8_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Resource: Calculating Area Between Polar Curves**
---
**Problem Statement:**
Find the area of the region that lies inside the first curve and outside the second curve.
Given polar equations:
- \( r = 19 \sin(\theta) \)
- \( r = 10 - \sin(\theta) \)
**Explanation:**
To solve this problem, you need to compute the area that is enclosed by the first curve but not by the second curve.
**Steps:**
1. **Understanding the Curves:**
- \( r = 19 \sin(\theta) \): This represents a limaçon with an inner loop.
- \( r = 10 - \sin(\theta) \): This curve represents another limaçon which may form different shapes depending on the range of \(\theta\).
2. **Intersection Points:**
- Determine the points of intersection by setting the equations equal:
\[
19 \sin(\theta) = 10 - \sin(\theta)
\]
- Solve for \(\theta\) to find the limits of integration.
3. **Calculate the Area:**
- Use the formula for the area enclosed by polar curves:
\[
A = \frac{1}{2} \int_{\theta_1}^{\theta_2} (r_1^2 - r_2^2) \, d\theta
\]
- Here, \( r_1 = 19 \sin(\theta) \) and \( r_2 = 10 - \sin(\theta) \).
4. **Integration Boundaries:**
- The limits of integration \(\theta_1\) and \(\theta_2\) are the solutions where the two curves intersect.
5. **Graphical Representation:**
- A graph would illustrate the curves in the polar coordinate system, showing the area to be calculated. This involves plotting both curves to identify the region of interest.
By following these steps, the area of the region inside the first curve and outside the second can be accurately calculated. Understanding these principles is crucial for solving such polar coordinate problems effectively.
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