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 area of the common region of the two equations is obtained by graphing the functions, determining their intersecting points and integrating across these points to find the area.

Step-by-step solution

Graphing Polar Equations

To visualize the intersection points of the two polar equations \( r = 5 - 3 \sin \theta \) and \( r = 5 - 3 \cos \theta \), a graphing utility can be used. From the graphs, determine the intersection points by finding the \(\theta\) values where the curves intersect.

Solve for Intersection Points

By setting the two equations equal to each other, \(5 - 3 \sin \theta = 5 - 3 \cos \theta\), it can be found that \(\sin \theta = \cos \theta\). This implies \(\theta\) = \( \frac{\pi}{4} \), \( \frac{5\pi}{4} \)

Computing the Area

The area of the common region is determined by integrating from one intersection point to the other. The area \( A \) in polar coordinates is given by \( A = \frac{1}{2} \int_{a}^{b} r^2 d\theta \) . Integrating for the two areas separately between the limits of \(\theta\) and then adding them gives the total area. The calculation is as follows: \( A = \frac{1}{2} \int_{0}^{\frac{\pi}{4}} (5 - 3 \sin \theta)^2 d\theta \) + \( \frac{1}{2} \int_{\frac{\pi}{4}}^{\frac{5\pi}{4}} (5 - 3 \cos \theta)^2 d\theta \)