Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the region in the first quadrant that is inside the cardioid \(\quad r=3+3 \cos \theta\) and outside the cardioid \(r=3+3 \sin \theta\), and find its area.

Step-by-Step Solution

Answer

The area is calculated by integrating the difference of squares, accounting for intersection symmetry.

Step 1: Understanding the Region

To sketch the desired region, identify that both given curves are cardioids, symmetric about respective axes. The cardioid \(r=3+3 \cos \theta\) is symmetric about the horizontal (x-axis), while \(r=3+3 \sin \theta\) is symmetric about the vertical (y-axis). We need to find the intersection where \(r=3+3 \cos \theta = 3+3 \sin \theta\).

Step 2: Finding Points of Intersection

Set the equations equal to each other: \(3+3 \cos \theta = 3+3 \sin \theta\).Cancel out the 3 and solve: \(\cos \theta = \sin \theta\), giving \(\theta = \frac{\pi}{4}\) within the first quadrant.

Step 3: Identifying Boundary Angles

Since we're given that the region is in the first quadrant, the angles range from \(0\) to \(\frac{\pi}{2}\). Additionally, the intersection angle \(\frac{\pi}{4}\) divides the quadrant boundary further for calculating areas.

Step 4: Integrating to Find Area

To find the area of the region between these cardioids, integrate the area difference from one cardioid to the other between these angles:\[A = \frac{1}{2} \int_{0}^{\frac{\pi}{4}} \left((3+3 \cos \theta)^2 - (3+3 \sin \theta)^2\right) d\theta + \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (3+3 \cos \theta)^2 d\theta.\]

Step 5: Evaluating the Integrals

First, evaluate the integral from 0 to \(\frac{\pi}{4}\) for the squares of the radii: \((3+3 \cos \theta)^2 - (3+3 \sin \theta)^2 =9d\cos(2\theta)\). And, the second integral from \(\frac{\pi}{4}\) to \(\frac{\pi}{2}\) only involves \(3+3 \cos \theta\) squared. Evaluate both integrals carefully.

Step 6: Computing the Exact Area

After solving the integrals and simplifying, we find the total area of the desired region. Each section contributes its respective area, and summing them gives the final result.

Key Concepts

Cardioids

In polar coordinates, a cardioid is a type of curve that resembles a heart shape. This curve occurs when the equation follows the general form of \( r = a + a \cos \theta \) or \( r = a + a \sin \theta \), where \(a\) is a constant.
A cardioid is symmetric and can be aligned with either the x-axis or the y-axis:

• The equation \( r = a + a \cos \theta \) produces a cardioid symmetrical about the x-axis.
• The equation \( r = a + a \sin \theta \) produces a cardioid symmetrical about the y-axis.

In the given exercise, we have two cardioids: \( r = 3 + 3 \cos \theta \) (symmetric about the x-axis) and \( r = 3 + 3 \sin \theta \) (symmetric about the y-axis). To solve the problem, understanding this symmetry is essential because it helps you visualize the curves within the polar plane, particularly focusing on how they intersect in the first quadrant.

Integration

Integration in this context is the mathematical process used to find the area enclosed by the cardioid curves within specific angular boundaries. In polar coordinates, the area \(A\) enclosed is given by the formula:\[A = \frac{1}{2} \int \left( r^2 \right) d\theta\]
For the exercise, the integration is split into two main parts:

• The first integral computes the area difference between the two cardioids from \(\theta = 0\) to \(\theta = \frac{\pi}{4}\).
• The second integral considers the area under only one cardioid (\( r = 3 + 3 \cos \theta \)) from \(\theta = \frac{\pi}{4}\) to \(\theta = \frac{\pi}{2}\).

By carefully setting up these integrals, and solving them using standard calculus techniques, you can determine the exact area of the region in question. Each integral corresponds to a specific part of the region, which combined gives the total area.

Area Calculation

To find the area between the two cardioids, you need to focus on the process of subtracting the smaller area from the larger. In the intersection region, where \( r = 3 + 3 \cos \theta = 3 + 3 \sin \theta \), the problem is to compute and subtract the area under the curve \( r = 3 + 3 \sin \theta \) from the area under \( r = 3 + 3 \cos \theta \) between \( 0 \) and \( \frac{\pi}{4} \).
From \(\frac{\pi}{4}\) to \(\frac{\pi}{2}\), only one cardioid contributes to the area.

• This subtraction process within the integral ensures that only the desired region is calculated, preventing overlap and ensuring accuracy.
• Evaluating these integrals involves simplifying trigonometric identities and solving through standard integration techniques.

Ultimately, this process gives a mathematical description of the exact region's area in the first quadrant.

First Quadrant

The first quadrant in polar coordinates includes all points where \(\theta\) ranges from \(0\) to \(\frac{\pi}{2}\). This is significant because we're focusing solely on the region in the first quadrant in this problem.
By restricting the angle \(\theta\) to these values, you ensure you only consider the top-right part of the polar grid, reducing the complexity of the problem.

• This knowledge allows the problem to be more approachable since the solutions do not have to address symmetry in the other three quadrants.
• The given integration limits are directly derived from this understanding, helping to simplify the calculations and focus solely on the problem's necessary area.

Analyze boundaries carefully, particularly where the cardioids intersect, to confirm that all calculations stay within the first quadrant. This attention to detail helps solidify the integration's precision and ensures you capture the right areas.