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Polar coordinates are a two-dimensional coordinate system used to specify the location of a point by the angle and the distance from a fixed point, known as the pole. This system is different from the Cartesian coordinate system, where points are defined by

• their x-coordinate (horizontal distance from the origin)
• their y-coordinate (vertical distance from the origin)

In polar coordinates, a point is determined using:

• \(r\): the radial distance from the pole to the point, and
• \(\theta\): the angular coordinate or angle measured in radians from the positive x-axis.

This is particularly useful for complex curves and shapes, as it often simplifies the representation and calculation of areas or lengths. In this exercise, we are dealing with a polar curve described by the equation \(r = 2 - 3\sin\theta\), which depicts a ³¢¾±³¾²¹Ã§´Ç²Ô curve with an inner loop.

The ³¢¾±³¾²¹Ã§´Ç²Ô is a type of polar curve that can have an inner loop, a dimple, or no interior loops depending on its equation's coefficients. The general form of a ³¢¾±³¾²¹Ã§´Ç²Ô in polar coordinates is given by:

• \( r = a + b\sin\theta \)
• OR \( r = a + b\cos\theta \)

In the equation \( r = 2 - 3\sin\theta \), \(a = 2\) and \(b = -3\). As the magnitude of \(b\) is larger than \(a\), the ³¢¾±³¾²¹Ã§´Ç²Ô has an inner loop.A ³¢¾±³¾²¹Ã§´Ç²Ô with an inner loop is captivating because it creates a visually appealing loop within the main body of the curve when plotted. This curve intersects the pole (origin) at specific angles, known as points of intersection, which play a critical role in determining the integral limits for calculating areas.

Determining integral limits in polar coordinates involves finding the angles \(\theta\) at which the polar curve intersects the pole. For our ³¢¾±³¾²¹Ã§´Ç²Ô curve \( r = 2 - 3\sin\theta \), the integral limits are established by setting the equation equal to zero and solving for \(\theta\):

1. Set \(r = 0\)
2. Find \(\theta\) where \(2 - 3\sin\theta = 0\), which simplifies to \(\sin\theta = \frac{2}{3}\)
3. Thus, the solutions are \(\theta = \arcsin\left(\frac{2}{3}\right)\) and \(\theta = \pi - \arcsin\left(\frac{2}{3}\right)\)

These limits define the range of \(\theta\) over which the inner loop of the ³¢¾±³¾²¹Ã§´Ç²Ô is completed. Essentially, these are the bounds for integrating to calculate the area enclosed by the inner loop.

To calculate the area enclosed by a region in polar coordinates, we use the formula for the area \(A\) of a region bounded by a polar curve from angle \(\theta_1\) to \(\theta_2\):\[ A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta \]For the ³¢¾±³¾²¹Ã§´Ç²Ô \(r = 2 - 3\sin\theta \), we must square the expression and integrate over the found limits:

• Integrand: Expand \((2-3\sin\theta)^2 = 4 - 12\sin\theta + 9\sin^2\theta\)
• Integrate each term:
  • \( \int \sin\theta \, d\theta = -\cos\theta \)
  • \( \int \sin^2\theta \, d\theta = \frac{1}{2} \theta - \frac{1}{4}\sin(2\theta) \)

Substitute the limits to get the final area of the inner loop. This calculation can often require numerical or computational aids to achieve a precise estimate.