Parametric, Polar and Vector-Valued Functions · Superchallenge problem

Engineers design a smooth road transition using the clothoid path $x(t)=\displaystyle\int_0^t\cos(s^2)\,ds$, $y(t)=\displaystyle\int_0^t\sin(s^2)\,ds$ (curvature changes smoothly along it, which is why it is used for highway and railway curves). Find the length of the road from $t=0$ to $t=\sqrt{\pi}$.