Chapter 12.5, Problem 18E

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

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

$s={\displaystyle {\int }_0^t\sqrt{{[x^{\text{'}}(u)]}^2+{[y^{\text{'}}(u)]}^2+{[z^{\text{'}}(u)]}^2}}du$

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

(c) Find the coordinates of the point on the helix for arc lengths $s=\sqrt{5}\text{and}s=4$.

(d) Verify that $\Vert r^{\text{'}}(s)\Vert =1$.