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

Short 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-by-step solution

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.

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.

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: \[ Area = \frac{1}{2} \int_{\theta_1}^{\theta_2} ((5 - 3\sin\theta)^2 - (5 - 3\cos\theta)^2) d\theta \]

Calculate the Area

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