Problem 24 - Essential Calculus

Question

Use a graphing utility to graph the polar equations and find the area of the given region. Common interior of \(r=5-3 \sin \theta\) and \(r=5-3 \cos \theta\)

Step-by-Step Solution

Answer

The short answer will be the numerical value of the area obtained after performing the above steps. The exact value will depend on the results of the integration in Step 3 and Step 4.

Step 1: Graph the Polar Equations

Use a graphing utility to plot the two polar equations \(r=5-3 \sin \theta\) and \(r=5-3 \cos \theta\). This will help to visually understand the area to be calculated.

Step 2: Find the Points of Intersection

Set the two functions equal to each other to determine the θ values where the two curves intersect: \(5-3 \sin \theta = 5-3 \cos \theta\). Solve this equation to get the values of θ.

Step 3: Evaluate the Integral

The area of the region in polar coordinates is given by integrating the square of the difference between the two polar functions over the common θ range: \[ Area = \frac{1}{2} \int_{\theta_1}^{\theta_2} ((5 - 3\sin\theta)^2 - (5 - 3\cos\theta)^2) d\theta \]

Step 4: Calculate the Area

Substitute the θ values obtained in Step 2 into the integral and calculate to get the area of the common interior.

Key Concepts

Polar Equations

Polar equations are used to express curves in the polar coordinate system, where each point is determined by a distance from a reference point and an angle from a reference direction. This can be different from the regular Cartesian coordinates most students are familiar with. A point in polar coordinates is written as \((r, \theta)\), where \(r\) is the radius or the distance from the origin, and \(\theta\) is the angle from the positive x-axis.

In our exercise, two specific polar equations are provided:

• \(r = 5 - 3 \sin \theta\)
• \(r = 5 - 3 \cos \theta\)

These expressions describe how the radii change with respect to the angle \(\theta\). The terms involving sine and cosine indicate that the curves will have symmetry properties common to these trigonometric functions.

Understanding these equations requires familiarity with how sine and cosine values fluctuate between -1 and 1, affecting the radius ranges and resulting curve shapes.

Area Calculation

Finding the area of a region described by polar equations involves integration, which is a core concept in calculus. In polar coordinates, the formula to find the area enclosed by a curve \(r(\theta)\) from an initial angle \(\theta_1\) to a final angle \(\theta_2\) is given by: \[Area = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 d\theta\]This integral takes into account how the radius changes as the angle \(\theta\) sweeps through its range.

In this specific exercise, we deal with the common interior of two polar curves. This means we are interested in the area where the curves overlap. To find this, we calculate the intersection points of the two equations, \(5 - 3 \sin \theta = 5 - 3 \cos \theta\). Solving such equations gives the values of \(\theta\) where the curves intersect, determining the limits for our integral.

Finally, we substitute these limits into the following area calculation: \[Area = \frac{1}{2} \int_{\theta_1}^{\theta_2} \left((5 - 3\sin\theta)^2 - (5 - 3\cos\theta)^2\right) d\theta\]This formulation helps us find the exact area enclosed by the curves within the overlapping region.

Graphing Utility

A graphing utility is an essential tool for working with polar equations, especially when visualizing complex curves that are not straightforward. In the context of our exercise, it assists in plotting the equations \(r=5-3\sin\theta\) and \(r=5-3\cos\theta\) to see how they interact. This visual representation can highlight symmetries and intersections, providing insight before performing any calculations.

Using a graphing utility allows:

• Checking the behavior of each polar curve as \(\theta\) varies from \(0\) to \(2\pi\)
• Identifying intersection points more easily and accurately
• Estimating areas visually to support integral calculations

Beginners or those not fully comfortable with abstract calculations might find graphing utilities invaluable for getting an intuitive sense of how polar curves behave. This understanding serves as a great foundation for further exploration into more complex polar coordinate problems.