Answer:
30.113
Step-by-step explanation:
r(t) = (t³, 2 sin t)
The distance traveled is the length of the path (arc length):
L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
L = ∫₁ᵖⁱ √((3t²)² + (2 cos t)²) dt
L = ∫₁ᵖⁱ √(9t⁴ + 4 cos²t) dt
Using a calculator:
L ≈ 30.113
Notice that distance is not the same as displacement. If we wanted to find the displacement:
r(π) = (π³, 0)
r(1) = (1, 2 sin 1)
Δr = √((π³ − 1)² + (0 − 2 sin 1)²)
Δr ≈ 30.053
