91Ó°ÊÓ

Understanding polar coordinates is key to grasping how certain curves and shapes can be represented in a plane. Unlike the Cartesian coordinate system, which uses \(x\) and \(y\) to denote points on a plane, polar coordinates rely on \(r\), the radial distance from the origin, and \(\theta\), the angle from the positive \(x\)-axis.

This system is particularly useful for representing curves like circles and spirals that radiate out from a central point. To find a point on the plane:

• Determine the distance \(r\) from the origin.
• Specify the angle \(\theta\) counterclockwise from the positive \(x\) axis.

This unique method simplifies working with curves in a circular manner, allowing easier computation of various geometric properties such as area, within a given range of angles.

A limaçon is a distinctive type of curve that can have fascinating shapes, including loops, dimpled structures, and cardiod-like figures. The general equation of a limaçon in polar coordinates is given by \(r = a + b \, \cos \theta\) or \(r = a + b \, \sin \theta\). In this specific exercise, the curve is defined by \(r = \frac{1}{2} + \cos \theta\).

This equation is particularly intriguing because it describes a limaçon with an inner loop. The loop is formed when the absolute value of the coefficient of \(\cos \theta\) is greater than the constant term, \(\frac{1}{2}\) in this case. Such curves touch themselves at points marking these loops.

Visually, a limaçon with an inner loop can resemble a small oval nested inside a larger rounded shape. Identifying the points where these loops meet, as done when \(\theta\) satisfies \(\cos \theta = -\frac{1}{2}\), is essential for further calculations like finding areas or intersections.

Integration is a powerful mathematical tool used to calculate areas underneath curves, among other applications. When dealing with polar equations like the limaçon, the integration process accounts for the curve's unique radial and angular properties.

To determine the area of a region described by a polar curve, use the formula:

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

This method incorporates both the radial distance \(r\) and the change in angle \(d\theta\), integrating over the specified range of \(\theta\) values.

In this example, integrate \(r^2\) from \(\theta = \frac{2\pi}{3}\) to \(\theta = \frac{4\pi}{3}\). It involves expanding \(\left( \frac{1}{2} + \cos \theta \right)^2\), using trigonometric identities, and evaluating each component separately. The integrated result represents the total area between the loops of the limaçon, providing insight into the curvature's overall geometry.