Answer:
18π
Explanation:
Arc length for a parametric equation is:
s = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
dx/dt = -9 sin(9t)
dy/dt = 9 cos(9t)
s = ∫₀²ᵖⁱ √((-9 sin(9t))² + (9 cos(9t))²) dt
s = ∫₀²ᵖⁱ √(81 sin²(9t) + 81 cos²(9t)) dt
s = ∫₀²ᵖⁱ 9 dt
s = 9t |₀²ᵖⁱ
s = 18π
You can check your answer with geometry. The curve is a circle centered at the origin with radius 1. From 0 to 2π, the particle makes 9 revolutions, or travels 18π radians. So the arc length is 1 × 18π = 18π.