Problem 32 - 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

Short Answer: The area common to the interna

Step 1: Identify Points of Intersection

To find the common interior area, first identify where the cardioids intersect. Set the equations equal, \(1 + \cos \theta = 1 - \cos \theta\) simplifying to \2 \cos \theta = 0\, which gives \theta = \frac{\pi}{2}\ and \frac{3\pi}{2}.\ Hence, these are the points of intersection.

Step 2: Set Up Area Integrals

The integral to find the area common to the cardioids involves two regions: one from \(r=1+\cos\theta\) and the other from \(r=1-\cos\theta\). Since the curves are symmetric, it suffices to find the area in one symmetric section and then double it.

Step 3: Calculate Area for First Cardioid

Calculate the integral for the first cardioid, \(1 + \cos \theta\), from \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\). The area integral is \[\frac{1}{2}\int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} (1+\cos\theta)^2 \, d\theta.\]\Simplifying and solving the integral gives \"},{

Key Concepts

Cardioid

A cardioid is a special type of polar curve that looks similar to a heart shape. The name "cardioid" comes from the Greek word "kardia," which means heart. It can be defined by equations like \(r = 1 + \cos \theta\) or \(r = 1 - \cos \theta\). Cardioids are created by tracing a path relative to a fixed point, using polar coordinates.

Cardioids have interesting properties, including:

• They are symmetric with respect to the horizontal axis or the vertical axis, depending on their equation.
• They have a pointed cusp at the pole when \(\theta = 0\) for \(1 + \cos \theta\) and at \(\theta = \pi\) for \(1 - \cos \theta\).
• The maximum distance from the pole is when the angle \(\theta\) equals the maximum value of the cosine function, typically at \(\theta = 0\) or \(\theta = \pi\).

Understanding the basic properties and shape of a cardioid is very helpful when solving problems involving areas between cardioid curves.

Area Between Curves

The area between curves is a common problem in calculus, often solved using integration. When determining the area common to the interiors of polar equations, such as two cardioids, we use the formula for the area in polar coordinates:

\[ \text{Area} = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta \]

For cardioids, the region of interest is typically the area between two curves. To find this, you:

• Find the points of intersection between the curves to define the limits of integration.
• Integrate the area formula for the first curve over the defined limits.
• Repeat the integration for the second curve, if necessary.
• The area common to both cardioids is often computed by taking advantage of their symmetry and doubling the calculated area for one symmetric segment.

Understanding these steps for calculating the area is crucial for solving similar problems in polar coordinates.

Intersection Points

Intersection points are essential in problems involving areas between curves, especially in polar coordinates. They help determine the limits for integration when finding the overlap between two polar graphs. To find intersection points of two cardioids such as \(r = 1 + \cos \theta\) and \(r = 1 - \cos \theta\), set their equations equal to each other:

\[ 1 + \cos \theta = 1 - \cos \theta \]

Simplify to solve for the angle \(\theta\), leading to the equation \(2 \cos \theta = 0\). Solving this equation provides \(\theta = \frac{\pi}{2}\) and \(\frac{3\pi}{2}\), which are the intersection points.

These points help identify the interval over which you calculate the integral. Without correctly determining these points, setting up and solving the integral can lead to incorrect results. Hence, understanding how to find and interpret intersection points is vital in problems involving polar curves.