For the polar curve r = 2 cos θ, the total arc length as θ runs from -π/2 to π/2 is?

• A. π
• B. 2π
• C. 0
• D. 4π

Answer: (B)

Arc length in polar coordinates uses the formula L = ∫ sqrt(r^2 + (dr/dθ)^2) dθ. Here r = 2 cos θ, so dr/dθ = -2 sin θ. Then r^2 + (dr/dθ)^2 = (2 cos θ)^2 + (-2 sin θ)^2 = 4(cos^2 θ + sin^2 θ) = 4, and the integrand becomes sqrt(4) = 2. Thus L = ∫{-π/2}^{π/2} 2 dθ = 2[θ]{-π/2}^{π/2} = 2(π) = 2π. This curve is a circle of radius 1 centered at (1,0), and the interval from -π/2 to π/2 traces the entire circle, whose circumference is 2π.