Find the length of the loop for the given curve:
x = 12t - 4t^(3) , y = 12t^(2)

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