arc length?

y' = (2/3)x^(-1/3)
(y')² = 4/(9x^2/3)

∫ [-1 to 8] (1 + (y')²)^(1/2) dx
= ∫ [-1 to 8] (1 + 4/(9x^2/3))^(1/2) dx
= ∫ [-1 to 8] ( (9x^(2/3)+4)/(9x^2/3) )^(1/2) dx
= ∫ [-1 to 8] (1/(3x^(1/3))) * (9x^(2/3)+4)^(1/2) dx
= ∫ [-1 to 8] (3/(3x^(1/3))) * (x^(2/3)+4/9)^(1/2) dx
= ∫ [-1 to 8] (1/x^(1/3)) * (x^(2/3)+4/9)^(1/2) dx
= (x^(2/3)+4/9)^(3/2) from x= -1 to 8

we can split it to (-1,0) and (0,8)
since y=x^(2/3) an even function
(-1,0) = (0,1)

That is,
= (x^(2/3)+4/9)^(3/2) from x=0 to 8 + (x^(2/3)+4/9)^(3/2) from x=0 to 1
= [(40/9)^(3/2) - 8/27] + [(13/9)^(3/2) - 8/27]
= 9.07342 + 1.43971
= 10.51313