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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![### Finding the Area of the Shaded Region
In this section, we are asked to find the area of the shaded region formed by the intersection of two polar curves.
#### Description of the Diagram:
The diagram shows two polar curves:
1. \( r = 1 + \cos \theta \)
2. \( r = 3 \cos \theta \)
These curves are plotted on a polar coordinate system. The curve \( r = 1 + \cos \theta \) is a Limacon with an inner loop, while \( r = 3 \cos \theta \) is a circle centered on the polar axis.
The shaded region is the area inside both curves, specifically the region where the two curves overlap.
#### Steps to Find the Area of the Shaded Region:
1. **Set up the integral**: To find the area of the shaded region in the polar coordinate system, we use the formula for the area enclosed by a polar curve:
\[
\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta
\]
2. **Determine the bounds of integration**: We need to find the points where the curves intersect to determine the limits of integration. These points of intersection satisfy the equation:
\[
1 + \cos \theta = 3 \cos \theta
\]
Simplifying,
\[
1 = 2 \cos \theta \implies \cos \theta = \frac{1}{2}
\]
Thus, the intersections occur at \(\theta = \pm \frac{\pi}{3}\).
3. **Calculate the area**: Use the integral formula to find the area inside each curve from \(-\frac{\pi}{3}\) to \(\frac{\pi}{3}\). The total shaded area is found by subtracting the area inside \( r = 3 \cos \theta \) from the area inside \( r = 1 + \cos \theta \) within the bounds.
4. **Evaluate the integrals**:
\[
\text{Area} = \frac{1}{2} \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left( (1 + \cos \theta)^2 - (3 \cos \theta)^2 \right) \,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faea7a391-77a2-4be9-b3a3-98e0f70bf810%2Fa998fe1a-97cf-47cc-afd9-24538e59de37%2Fhyy94db_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Area of the Shaded Region
In this section, we are asked to find the area of the shaded region formed by the intersection of two polar curves.
#### Description of the Diagram:
The diagram shows two polar curves:
1. \( r = 1 + \cos \theta \)
2. \( r = 3 \cos \theta \)
These curves are plotted on a polar coordinate system. The curve \( r = 1 + \cos \theta \) is a Limacon with an inner loop, while \( r = 3 \cos \theta \) is a circle centered on the polar axis.
The shaded region is the area inside both curves, specifically the region where the two curves overlap.
#### Steps to Find the Area of the Shaded Region:
1. **Set up the integral**: To find the area of the shaded region in the polar coordinate system, we use the formula for the area enclosed by a polar curve:
\[
\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta
\]
2. **Determine the bounds of integration**: We need to find the points where the curves intersect to determine the limits of integration. These points of intersection satisfy the equation:
\[
1 + \cos \theta = 3 \cos \theta
\]
Simplifying,
\[
1 = 2 \cos \theta \implies \cos \theta = \frac{1}{2}
\]
Thus, the intersections occur at \(\theta = \pm \frac{\pi}{3}\).
3. **Calculate the area**: Use the integral formula to find the area inside each curve from \(-\frac{\pi}{3}\) to \(\frac{\pi}{3}\). The total shaded area is found by subtracting the area inside \( r = 3 \cos \theta \) from the area inside \( r = 1 + \cos \theta \) within the bounds.
4. **Evaluate the integrals**:
\[
\text{Area} = \frac{1}{2} \int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} \left( (1 + \cos \theta)^2 - (3 \cos \theta)^2 \right) \,
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