when x = 0, the vaues of t are 0 and ± 1/√3.
The arc length,S is determined by
∫ √[(dx/dt)^2 + (dy/dt)^2] dt
= 2 ∫ √[(12 - 12t^2)^2 + (24t)^2] dt from t = 0 to t = 1/√3
= 2 ∫ 12 √[(1 - t^2)^2 + (2t)^2] dt from t = 0 to t = 1/√3
= 24 ∫ √(1 + 2t^2 + t^4) dt from t = 0 to t = 1/√3
= 24 ∫ √(1 + t^2)^2 dt from t = 0 to t = 1/√3
= 24 ∫ (1 + t^2) dt from t = 0 to t = 1/√3
= 24(t + t^3/3) from t = 0 to t = 1/√3
= 24 (1/√3) (1 + 1/9)
= 24 (√3/3) (10/9)
S = (80/9)√3
