As N → → ∞ ∞ this Riemann sum becomes the definite integral lim N → → ∞ ∞ Δ Δ v = v eff ∫ ∫ 0 ϕ ϕ d x 1 − − x = v eff ln ⁡ ⁡ 1 1 − − ϕ ϕ = v eff ln ⁡ ⁡ m 0 m f , since the final remaining mass of the rocket is m f = m 0 ( 1 − − ϕ ϕ ) .