Final answer:
To find the area of the region enclosed by the inner loop of the curve, we can use the formula for the area of a polar curve. The given equation is r = 4 + 8 sin(θ). We need to find the area enclosed by the loop, so we will integrate from θ = 0 to θ = π.
Step-by-step explanation:
To find the area of the region enclosed by the inner loop of the curve, we can use the formula for the area of a polar curve. The given equation is r = 4 + 8 sin(θ).
We need to find the area enclosed by the loop, so we will integrate from θ = 0 to θ = π.
The formula for the area is A = ∫[θ1,θ2] (1/2)r²dθ. Substituting the given equation, we have A = ∫[0,π] (1/2)(4 + 8 sin(θ))²dθ.
We can simplify the integral by expanding the expression (4 + 8 sin(θ))². Then, we can integrate term by term. Finally, substitute the limits of integration and evaluate the integral to get the area.