Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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The figure is a polar plot showing two equations:
1. **Circle**: \( r = 2 \)
- This equation represents a circle with a radius of 2 units centered at the origin (0,0). The circle extends to points (2,0), (-2,0), (0,2), and (0,-2).
2. **Rose Curve**: \( r = 2 \cos(2\theta) \)
- This equation represents a rose curve with four petals. The petals are oriented along the axes (both x-axis and y-axis) and each petal extends to a radius of 2 units from the origin.
The plot is divided into regions with the circle overlapping the rose curve. The shaded regions are symmetrical and appear as four separate areas within the circle but outside of the petals of the rose curve.
To find the area of the shaded region, you can use a combination of integral calculus and symmetry. The area between the two polar curves in one of the four symmetrical sections can be calculated and then multiplied by four to get the total shaded area.
A detailed method to find the total area would involve setting up the relevant integral. For now, an overview of the problem setup without the integral solution is as follows:
1. **Identify the Intersection Points**: Calculate the points where the two equations intersect.
2. **Set Up the Integral**: Establish the limits of integration based on the angles of intersection.
3. **Evaluate the Integral**: Compute the definite integral to find the area between the curves in one section.
4. **Multiply by Four**: Since the graph is symmetrical with four identical shaded regions, multiply the area of one section by four to obtain the total shaded area.
This step-by-step procedure will give you the total area of the shaded regions.
For comprehensive understanding, you may refer to resources on polar coordinates, integration in polar coordinates, and properties of rose curves and circles.](https://content.bartleby.com/qna-images/question/720535be-d647-4725-8ea5-21875474b49c/73ad3c3b-e7e1-4929-a53a-29bfd0b086b9/t68wv7r_processed.png)
Transcribed Image Text:**Polar Equations and Shaded Area Calculation**
The polar equations \( r = 2 \) and \( r = 2 \cos(2\theta) \) are plotted below. Find the area of the shaded region in the figure.
The figure is a polar plot showing two equations:
1. **Circle**: \( r = 2 \)
- This equation represents a circle with a radius of 2 units centered at the origin (0,0). The circle extends to points (2,0), (-2,0), (0,2), and (0,-2).
2. **Rose Curve**: \( r = 2 \cos(2\theta) \)
- This equation represents a rose curve with four petals. The petals are oriented along the axes (both x-axis and y-axis) and each petal extends to a radius of 2 units from the origin.
The plot is divided into regions with the circle overlapping the rose curve. The shaded regions are symmetrical and appear as four separate areas within the circle but outside of the petals of the rose curve.
To find the area of the shaded region, you can use a combination of integral calculus and symmetry. The area between the two polar curves in one of the four symmetrical sections can be calculated and then multiplied by four to get the total shaded area.
A detailed method to find the total area would involve setting up the relevant integral. For now, an overview of the problem setup without the integral solution is as follows:
1. **Identify the Intersection Points**: Calculate the points where the two equations intersect.
2. **Set Up the Integral**: Establish the limits of integration based on the angles of intersection.
3. **Evaluate the Integral**: Compute the definite integral to find the area between the curves in one section.
4. **Multiply by Four**: Since the graph is symmetrical with four identical shaded regions, multiply the area of one section by four to obtain the total shaded area.
This step-by-step procedure will give you the total area of the shaded regions.
For comprehensive understanding, you may refer to resources on polar coordinates, integration in polar coordinates, and properties of rose curves and circles.
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