Question

In Exercises 49 and 50 , use the integration capabilities of a graphing utility to approximate to two decimal places the area of the region bounded by the graph of the polar equation. \(r=\frac{2}{3-2 \sin \theta}\)

Short answer

Because the limits of integration (i.e., the values of 'a' and 'b') weren't given in the problem, the exact numerical result isn't provided in this solution. It will depend on the region one is interested in, which could be derived from the graph created in step 1. However, the main procedure to solve this exercise involves plotting the graph using the given polar equation, setting up the integral using the formula for area in polar coordinates, and then obtaining the numerical result using the integration functionality of the graphing utility.

Step-by-step solution

Plot the polar equation

Enter the polar equation \(r = \frac{2}{3-2 \sin \theta}\) into the graphing utility. Examine the graph to identify the region for which the area needs to be found. Also, determine the appropriate limits of integration.

Set up the integration

To find the area of the region as explained in step 1, set up the integral as follows:\( A = \frac{1}{2} \int_{a}^{b} (\frac{2}{3-2 \sin \theta})^2 d\theta\) Where 'a' and 'b' are the limits of integration, which corresponds to the range of theta values which cover the region of interest.

Calculation Using a Graphing Utility

Use the graphing utility's integration function to calculate the integral value. Enter the integral with the appropriate limits derived from step 1. The graphing utility will perform the numerical integration and return an approximate value. Keep in mind to round the final answer to two decimal places.

Key concepts

Integration of Polar Equations

Integration in polar coordinates is a tool used to compute areas bound by polar curves. Instead of rectangles, integration in polar coordinates uses sectors of circles. For any polar equation, like the given equation \( r = \frac{2}{3-2 \sin \theta} \), the area \( A \) of a region can be calculated using the formula:
\[ A = \frac{1}{2} \int_{a}^{b} r^2 d\theta \]
This represents the sum of infinitely many infinitesimal sectors of circles, from \( \theta = a \) to \( \theta = b \). Every small piece of area is a fraction of a circle's complete area where \( r \) varies as a function of \( \theta \). Simplifying and integrating such expressions require a good grasp of calculus and the ability to work with algebraic manipulations.

Numerical Integration

Numerical integration becomes essential when an integral cannot be solved analytically or when an approximation is sufficient. This technique is particularly useful when dealing with complex equations such as polar equations. A graphing utility or calculator can perform numerical integration, often using methods like Simpson’s Rule or the Trapezoidal Rule.
These methods approximate the area under the curve by summing up areas of simple geometric shapes that fit inside the curve. The accuracy of the numerical integration improves with the number of subdivisions or trapezoids used. High-end calculators and computer software can handle these calculations effortlessly, offering a practical way to estimate areas bounded by sophisticated curves.

Graphing Utility Usage

A graphing utility can be an invaluable tool for visualizing polar equations and performing calculations such as integration. When approaching a problem like finding the area of a region bounded by a polar curve, the graphing utility serves two primary functions:

• Visualization: It provides a graphical representation of the polar curve, which helps to identify the bounded region and any symmetries or repeating patterns.
• Calculation: It can compute integrals numerically, a function crucial for applying the integration of polar equations to find areas.

Understanding how to effectively use these features is key. For instance, zooming in on specific areas of the curve can help determine accurate limits of integration, and familiarizing oneself with how the utility inputs integrals can avoid possible syntax errors during calculation.

Limits of Integration

Identifying the correct limits of integration is crucial when calculating areas in polar coordinates. These limits, \( a \) and \( b \), correspond to the angular range over which the curve extends around the pole. To determine the limits of integration for a given polar equation:

• Graph the equation, either by hand or through a graphing utility.
• Analyze the resulting curve to find the angles where the curve starts and ends, or completes a region.
• Use these angles as your limits of integration, ensuring that the entire area of interest is covered.

The correct identification of limits will help to avoid underestimating or overestimating the area being calculated. In the given exercise, careful examination of the polar graph helps determine the precise limits for the region defined by \( r = \frac{2}{3-2 \sin \theta} \).