Consider the helix represented by the vector$-$valued function $r(t)=(2cos(t),2sin(t),t).$

$($a$)$ Write the length of the arc $s$ on the helix as a function of $t$ by evaluating the integral below.

$s={\displaystyle \underset{0}{\overset{t}{}}}\sqrt[\phantom{2}]{[x^2(u){]}^2[y[u){]}^2[r^2(u){]}^2}du$

$5=$

$($b$)$ Solve for $t$ in the relationship derived in part $($a$),$ and substitute the result into the original se of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter s$.$

$$

$r(s)=$

$($c$)$ Find the coordinates of the point on the helx for the following arc lengths. $($Round your answers to three decimal places.$)$

$s=\sqrt[\phantom{2}]{5}$ and $s=4$

$5=\sqrt[\phantom{2}]{5},(x,y,z)=0)$

$)$

$($d$)$ Evaluate IIr'$($s$)$II$.$

$||r{r}^2(s)||=$