To calculate: The length of the arc s on the helix as a function of t
The length of the curve is \(\underline{s=\ \sqrt{5t}}.\)
Used formula:
\(s=\ \int_{0}^{t}\ \sqrt{[x'(t)]^{2}\ +\ [y'(t)]^{2}\ +\ [z'(t)]^{2}\ dt}\)
Calculation:
The helix path is,
\(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\)
On differentiating this vector value function,
\(r'(t)=\ <\ -2\ \sin\ t,\ 2\ \cos\ t,\ 1\ >\)
Calculate the length of the line segment for the given interval as
\(s=\ \int_{0}^{t}\ \sqrt{[x'(t)]^{2}\ +\ [y'(t)]^{2}\ +\ [z'(t)]^{2}\ dt}\)

\(=\ \int_{0}^{t}\ \sqrt{(-2\ \sin\ t)^{2}\ +\ (2\ \cos\ t)^{2}\ +\ (1)^{2}\ dt}\)

\(=\ \int_{0}^{t}\ \sqrt{5dt}\)

\(= \sqrt{5t}\) Thus, the arc length is \(s = \sqrt{5t}.\)

\(=\ \int_{0}^{t}\ \sqrt{(-2\ \sin\ t)^{2}\ +\ (2\ \cos\ t)^{2}\ +\ (1)^{2}\ dt}\)

\(=\ \int_{0}^{t}\ \sqrt{5dt}\)

\(= \sqrt{5t}\) Thus, the arc length is \(s = \sqrt{5t}.\)