Let $f(x,y,z)=126xz-14y^2$ and let $$ be the curve given by $r(t)=($:$t,\frac{t^2}{\sqrt[\phantom{2}]{2}},\frac{t^3}{3}$:$)$ for $0t2.$ Compute ${\displaystyle \underset{}{\overset{\phantom{}}{}}}fds.$

$($Use symbolic notation and fractions where needed.$)$