Final answer:
To find the area inside a cardioid and outside a circle, we can use a double integral. First, determine the limits of integration for θ and r. Then, integrate with respect to r and θ over those limits to find the area.
Step-by-step explanation:
To find the area of the region inside the cardioid r=1 + cos θ and outside the circle r=3cos θ, we will use a double integral. First, we need to determine the limits of integration for θ and r. The cardioid and the circle intersect at two points, so we need to find the values of θ that correspond to those points. Let's call those values θ1 and θ2.
To find θ1 and θ2, we set the equations of the cardioid and the circle equal to each other and solve for θ. Once we have θ1 and θ2, we can integrate with respect to r from 1 + cos θ to 3cos θ, and with respect to θ from θ1 to θ2.
The area is given by the double integral: A = ∫(∫r dr dθ)
This method leverages polar coordinates to handle the region's geometry, offering a concise mathematical approach to calculate the enclosed area between the cardioid and the circle.