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A limaçon is a fascinating type of curve that appears in polar coordinates. It's typically defined by an equation of the form \( r = a + b \cos n \theta \) or \( r = a + b \sin n \theta \). These curves are interesting because they can take on different shapes such as loops or dimples, depending on the values of \( a \) and \( b \).

In our exercise, the polar curve given is a limaçon with an inner loop. The equation is \( r = 2 + \cos 2\theta \), where \( a = 2 \), \( b = 1 \), and \( n = 2 \). This means that for some angles the radius \( r \) becomes small enough to form a loop inside the curve.

Understanding limaçon curves is crucial when working with polar equations, as they often require insight into the behaviors of trigonometric functions. By knowing how these curves behave, you can more easily manipulate them, recognize possible loops, and identify key points along the curve, which comes in handy for calculations like area or boundary definitions.

Calculating the area of a region defined by a polar curve is slightly different than working with rectangular coordinates. For polar coordinates, we use the formula:

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

where \( r(\theta) \) is the radius as a function of \( \theta \), and \( \alpha \), \( \beta \) are the bounds for the angle \( \theta \).

For our limaçon, \( r = 2 + \cos 2\theta \), and \( \theta \) ranges from \( 0 \) to \( 2\pi \). Plugging these into our formula, we integrate:

• First, expand \( (2 + \cos 2\theta)^2 \) to get \( 4 + 4 \cos 2\theta + \cos^2 2\theta \).
• Then use the identity \( \cos^2\theta = \frac{1 + \cos 2\theta}{2} \) to simplify the integral.
• The integration is performed over the interval from \( 0 \) to \( 2\pi \).

This process results in an area of 9 square units for the limaçon region \( R \). Understanding these mathematical steps helps solidify your grasp of how polar integrals function and how to find areas for various polar curves.

Converting polar coordinates to rectangular coordinates is a vital tool when dealing with curves, especially when you want to understand their position concerning the Cartesian plane.

The conversion is done using the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

In our example, we started with the polar equation \( r = 2 + \cos 2\theta \). To find the dimensions of the rectangle that can encapsulate the limaçon in the Cartesian plane, we look at specific \( \theta \) values:

• For \( \theta = 0 \), you get the right boundary at \( x = 3 \).
• For \( \theta = \pi \), the left boundary at \( x = 1 \).
• For \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \), the top and bottom boundaries are \( y = 2 \) and \( y = -2 \), respectively.

Thus, the smallest rectangle in Cartesian coordinates that contains the entire limaçon spans from \( x = 1 \) to \( x = 3 \) and from \( y = -2 \) to \( y = 2 \). These conversions are key when analyzing the spatial relationships of polar curves on the usual x-y axis.