Calculate length

Question

Calculate the length of the following parameterized curve over the given interval. r(theta) = <sin(theta) - theta cos(theta), cos(theta) + theta sin(theta)> , 0 <= theta <= 5

Answer 1

Answer 2

r(θ) = <sinθ - θcosθ, cosθ + θsinθ, 0 ≤ θ ≤ 5

L = ∫₀⁵√[(cosθ - cosθ + θsinθ)² + (- sinθ + sinθ + θcosθ)²]dθ = ∫₀⁵θdθ = θ²/2₀⁵ = 12.5

Answer 3

r = <sin(θ)-θcos(θ),cos(θ)+θsin(θ)>

s = int from θ= 0 to 5 of sqrt((dx/dθ)² + (dy/dθ)²)dθ

dx/dθ = θsinθ dy/dθ = θcosθ using chain rule

s = int from 0 to 5 of sqrt ( θ²cos²θ + θ²sin²θ) dθ = int 0 to 5 of θdθ = 25/2