Use a graphing utility to graph the polar equations and find the area of the given region.
Common interior of r = 12 sin(2θ) and r = 6
To find the area of the common interior of the polar equations r = 12 sin(2θ) and r = 6, we can use a graphing utility to visually represent the equations and then calculate the area.
1. Open a graphing utility or software that supports polar graphing, such as Desmos or GeoGebra.
2. Enter the first polar equation, r = 12 sin(2θ), into the graphing utility.
3. Enter the second polar equation, r = 6, into the same graphing utility.
4. Adjust the viewing window of the graphing utility to ensure that both equations are clearly visible on the graph. You may need to zoom in or out to get a good view.
5. After graphing both equations, you should see two curves on the polar coordinate plane. The area we are interested in is the region where these curves overlap.
6. Use the graphing utility's tool or feature to shade the common interior region between the two curves. The specific method to shade the region may vary depending on the graphing utility you are using, but typically there will be an option to shade a region between curves or by selecting points on the graph.
7. Once the common interior region is shaded, you can estimate the area visually. Count the number of complete circles enclosed within the shaded region, as well as any partially enclosed circles. You can consider these circles as sectors of circles.
8. To calculate the area of each sector, use the formula for the area of a sector: A = 1/2 * r^2 * θ, where r is the radius and θ is the central angle in radians. In this case, the radius remains constant (6 or 12, depending on which sector you are calculating), and the central angles can be approximated visually.
9. Calculate the area of each sector within the shaded region and record the values.
10. Sum up the areas of all the sectors to find the overall area of the common interior region.
By following these steps using a graphing utility, you can graph the polar equations, visualize the common interior region, and calculate its area.