Find the Exact Length of the Polar Curve r = 8 + 2sin(θ)

To find the length of the polar curve given by the equation r = 8 + 2sin(θ), we can use the formula for the length of a polar curve:

L = ∫_α^β √(r² + (dr/dθ)²) dθ

First, we need to differentiate r with respect to θ:

dr/dθ = 2cos(θ)

Next, we can substitute r and dr/dθ into the length formula. We find:

r² = (8 + 2sin(θ))² = 64 + 32sin(θ) + 4sin²(θ)

Now we calculate (dr/dθ)²:

(dr/dθ)² = (2cos(θ))² = 4cos²(θ)

We then substitute these into the integral:

L = ∫₀^2π √(64 + 32sin(θ) + 4sin²(θ) + 4cos²(θ)) dθ

Since sin²(θ) + cos²(θ) = 1, we simplify:

L = ∫₀^2π √(64 + 32sin(θ) + 4) dθ = ∫₀^2π √(68 + 32sin(θ)) dθ

This integral might need numerical methods or specific trigonometric identities to solve. In general, you might use techniques like substitution or numerical integration methods (like Simpson’s rule) to evaluate it depending on your needs.

Thus, the exact length of the polar curve can be approximated, or calculated using specific integration techniques to arrive at the desired answer.