Consider the Parametric Equation

Question

Consider the parametric equation

x = 9(cos⁡(theta) + (theta)sin⁡(theta))

y = 9(sin⁡(theta) − (theta)cos⁡(theta))



What is the length of the curve for (theta) = 0 to (theta) = (11pi)/2?

Answer 1

Lehgth of arc L = ∫₀^11π/2[(dx/dθ)² + (dy/dθ)²]^1/2dθ = ∫₀^11π/29[(θcosθ) + (θsinθ)²]^1/2dθ = 9θ²/2₀^11π/2= 1089π²/8

Answer 2

ds = sqrt((dx/dT)²+(dy/dT)²)dT

dx/dT = 9TcosT and dy/dT = 9TsinT

ds = sqrt(9²T²(cos²T +sin²T))dT = 9TdT

Just integrate 9TdT from 0 to 11pi/7

Take care