Final answer:
To find the area of the common interior of the polar equations, graph the equations r = 12 sin(2θ) and r = 6, then integrate with regards to θ over the interval where they overlap and multiply by 4 for symmetry.
Step-by-step explanation:
To find the area of the common interior of the polar equations r = 12 sin(2θ) and r = 6, we must first graph these equations using a graphing utility. The first equation represents a 4-leaved rose since the sine function has a coefficient of 2, which generally creates 2n petals for a sine function when n is the coefficient. The second equation is a circle with radius 6. The area of overlap is essentially the petals of the rose that lie inside the circle.
To analytically find the area of the common interior, we need to integrate with respect to θ from 0 to π/4 for a single petal of the rose that overlaps with the circle, then multiply this result by 4 to account for all four petals that intersect with the circle. The limits of integration correspond to the values of θ where the petal first and last intersects the circle. The integral is given by:
A = 4 × ∫_{0}^{π/4} ½ (r_{rose})^2 dθ where r_{rose} = 12 sin(2θ)
After integrating, we can multiply the result by 4 to get the area of all overlapping sections. Since the expression for r = 6 is a constant, it's unnecessary to involve it in the integration because it defines the boundary of overlap, not the shape within the boundary itself.