Parameterize the curve [tex]C[/tex] by [tex]\mathbf r(t)=\left\langle t,\dfrac{t^2}2,\dfrac{t^3}6\right\rangle[/tex] (essentially replacing [tex]x=t[/tex] and finding equivalent expressions for [tex]y,z[/tex] in terms of [tex]t[/tex].
The length of [tex]C[/tex] is given by the line integral
[tex]\displaystyle\int_C\mathrm dS=\int_{t=0}^{t=2}\|\mathbf r'(t)\|\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^2\left\|\left\langle1,t,\dfrac{t^2}2\right\rangle\right\|\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^2\sqrt{1+t^2+\dfrac{t^4}4}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac12\int_0^2\sqrt{4+4t^2+t^4}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac12\int_0^2\sqrt{(t^2+2)^2}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac12\int_0^2(t^2+2)\,\mathrm dt[/tex]
[tex]=\dfrac12\left(\dfrac{t^3}3+2t\right)\bigg|_{t=0}^{t=2}[/tex]
[tex]=\dfrac12\left(\dfrac83+4\right)[/tex]
[tex]=\dfrac{10}3[/tex]
