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

To find the area of the region common to the two polar equations, graph the equations, find the intersecting points to determine the range of theta, and then evaluate the appropriate integral. The integral will involve the square of the polar equations and their difference, integrated over the interval from the lower to upper boundaries of theta found previously.

Step-by-step solution

Graph the polar equations

First, graph the two polar equations. From the graphs, identify the points where the graphs intersect and determine the interval for theta over which they intersect.

Set the two polar equations equal and solve for \(\theta\)

To find the values of theta for which the two graphs intersect, set \(5-3 \sin \theta = 5-3 \cos \theta\) and solve for \(\theta\). These solutions represent the boundaries of the region of interest.

Find the area of the common interior

The area of the interior common to two polar curves \(r_1(\theta)\) and \(r_2(\theta)\) over an interval \(\theta = [\alpha, \beta]\) is given by the formula \(\frac{1}{2} \int_{\alpha}^{\beta} |r_1^2(\theta) - r_2^2(\theta)| d\theta\). Substitute the given polar equations and integration limits into this formula and complete the integration to find the area.