67,358 views
14 votes
14 votes
Find the area of the region enclosed by the inner loop of the curve. r = 4 + 8 sin θ

User Karl Fast
by
2.6k points

1 Answer

27 votes
27 votes

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.

User Sngreco
by
3.0k points