Final answer:
The area of the region inside r=2+sin(2θ) and outside r=2+cos(2θ) is calculated using the definite integral from θ=π/4 to θ=5π/4 of the difference of the squares of the polar functions.
Step-by-step explanation:
To find the area of the region that is inside the curve r=2+sin(2θ) and outside the curve r=2+cos(2θ), we can use the provided formula A = ½∫θ1 θ2 [f(θ)² - g(θ)²]dθ, where f(θ) and g(θ) represent the polar equations of the two curves, respectively. To apply the formula, we must identify the points of intersection of the curves to determine the limits of integration (θ1 and θ2). The points of intersection occur when the two curves equal each other, in other words where 2+sin(2θ) = 2+cos(2θ). This happens at θ=π/4 and θ=5π/4. Thus, these will be the limits of integration. We then integrate to find the area:
A = ½∫5π/4 π/4 [(2+sin(2θ))² - (2+cos(2θ))²]dθ
To complete the calculation, we need to expand the squared terms, integrate term by term within the limits, and then calculate the definite integral to find the area.