Problem 22 - Thomas Calculus

Question

Overlapping cardioids Find the area of the region common to the interiors of the cardioids \(r=1+\cos \theta\) and \(r=1-\cos \theta\)

Step-by-Step Solution

Answer

The common area of the interiors of the cardioids is \(\frac{3\pi}{2}\) units squared.

Step 1: Identify the Intersection Points

To find the shared area of the interiors of these cardioids, we first identify where they intersect. Set the equations equal to find \(1+\cos \theta = 1-\cos \theta\), which simplifies to \(\cos \theta = 0\). So, the intersection points occur at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Step 2: Determine Area Integral Limits

The area will be split according to these intersection points. We analyze the integral from \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\) for both cardioids.

Step 3: Calculate the Area of Each Cardioid Section

The area \(A\) of a polar region is given by \(A = \frac{1}{2} \int_\alpha^\beta r^2 \, d\theta\). For \(r = 1 + \cos \theta\), evaluate the integral from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\). Similarly, compute for \(r = 1 - \cos \theta\) also from \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\).

Step 4: Symmetrical Segment Calculations

Both integrals simplify as the sections are symmetric about the x-axis. Compute separately: \[ \text{For } r=1+\cos \theta: \text{Area } = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1+\cos \theta)^2 \, d\theta \]\[ \text{For } r=1-\cos \theta: \text{Area } = \frac{1}{2} \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1-\cos \theta)^2 \, d\theta \]

Step 5: Evaluate and Subtract for Common Area

Evaluate these integrals and subtract one from the other, since the common area is the overlap between these values. Compute: \[ \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1+\cos \theta)^2 \, d\theta - \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1-\cos \theta)^2 \, d\theta \].

Step 6: Calculate Final Common Area

The final integration simplifies due to trigonometric identities and symmetry. Integrate each component and determine the overlap area.The common area is calculated by finding the enclosed area common to both functions across the intersection limits \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\).

Key Concepts

Cardioids

Cardioids are a type of curve that resembles the shape of a heart. They are often described in polar coordinates, which means each point on the curve is determined by an angle and a distance from the origin. The equation for a cardioid can appear in forms like \( r = 1 + \cos \theta \) or \( r = 1 - \cos \theta \), where \( r \) is the radial distance, and \( \theta \) is the angle in radians.
It's essential to understand that cardioids are closely related to circles and have interesting geometric properties. These curves have a loop, and their symmetry makes them particularly fascinating, showing reflective symmetry around the vertical axis for the given equations.
They are often used in mathematical exercises related to polar coordinates because they provide visual symmetry and interesting challenges related to integration and area computation.

Intersection Points

Finding intersection points is crucial when determining areas common to multiple curves. For curves like cardioids, the intersection point is where both curves meet or cross.
To find these points, set the two polar equations equal: \( 1 + \cos \theta = 1 - \cos \theta \). Simplifying this, you get \( \cos \theta = 0 \), which indicates that the points where the curves intersect are at angles \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).
The calculation of these intersection points is the first step in analyzing overlapping regions. Knowing these points helps define the limits of integration when computing areas, as these provide the range of \( \theta \) to examine.

Integral Calculus

Integral calculus is a fundamental branch of mathematics that deals with finding areas, volumes, and other concepts that arise by integrating a function. It is particularly useful when working with curves in polar coordinates.
When you're trying to find the area under a curve or between curves, you use an integral. For polar coordinates, the area \( A \) of a region can be found using the formula:

• \( A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \)

In this formula, \( r \) represents the radius described by the polar equation, and \( \alpha \) and \( \beta \) are the limits of integration, which are essentially the intersection points you calculated earlier.
This method is essential in many applications, from physics to engineering, as it allows the calculation of complex areas and volumes.

Area Computation

Computing the area of a region surrounded by curves in polar coordinates combines several mathematical concepts. Here, you focus on calculating the area of the region common to two overlapping cardioids.
This involves:

• Finding the intersection points to set the limits of integration.
• Using the area formula in polar coordinates.
• Computing the integral for each cardioid separately, often simplified by symmetry or trigonometric identities.

After evaluating each integral, subtract one result from the other to find the common region's area. This common area is important in problems requiring overlap measurement in polar forms.
Ultimately, mastering area computation with polar coordinates not only gives insight into mathematical behaviors of cardiac shapes but also builds a strong foundation in solving practical problems involving odd-shaped regions.