Final answer:
To calculate the area of the region inside the circle r = 4 cos θ and outside the circle r = 2, a double integral can be used with angular bounds from -π/3 to π/3 and radial bounds from 2 to 4 cos θ.
Step-by-step explanation:
To find the area of the region inside the circle r = 4 cos θ but outside the circle r = 2, we can use a double integral. First, we establish the bounds of our double integral. The circles intersect where r = 4 cos θ = 2, which simplifies to θ = ±π/3. The bounds for θ are therefore from -π/3 to π/3. The bounds for r vary from the outer circle (r = 4 cos θ) to the inner circle (r = 2).
The double integral to find the area A is then given by:
∫-π/3π/3 ∫24 cos θ r dr dθ
To perform the integral, we integrate with respect to r first, followed by θ. The integral of r is (1/2) r2, evaluated from 2 to 4 cos θ. Then we perform the θ integration. The result will give us the area desired, where we subtract the area of the inner circle from the area enclosed by r = 4 cos θ.