Find the length of the polar curve r = √2e^θ for 0 ≤ θ ≤ π.
Correct: A
The arc length L of a polar curve r = f(θ) from θ = α to θ = β is given by the formula:
L = ∫ from α to β √(r² + (dr/dθ)²) dθ.
Given r = √2e^θ:
dr/dθ = d/dθ (√2e^θ) = √2e^θ.
Now, calculate r² + (dr/dθ)²:
r² = (√2e^θ)² = 2e^(2θ).
(dr/dθ)² = (√2e^θ)² = 2e^(2θ).
r² + (dr/dθ)² = 2e^(2θ) + 2e^(2θ) = 4e^(2θ).
Take the square root:
√(4e^(2θ)) = 2e^θ.
Now, set up the integral for the arc length from θ = 0 to θ = π:
L = ∫ from 0 to π (2e^θ) dθ.
Integrate:
L = [2e^θ] from 0 to π.
L = 2e^π - 2e^0 = 2e^π - 2(1) = 2(e^π - 1).
The final answer is A.