Investigation Consider the helix represented by the vector-valued function (a) Write the

Chapter 12, Problem 17

(choose chapter or problem)

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

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

\(s=\int_{0}^{t} \sqrt{\left[x^{\prime}(u)\right]^{2}+\left[y^{\prime}(u)\right]^{2}+\left[z^{\prime}(u)\right]^{2}} d u\)

(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 \(s=\sqrt{5}\) and  s = 4.

(d) Verify that ||r(s)|| = 1.

 

 

Text Transcription:

r(t)=<2 cos t, 2 sin t, t>

s=Int_0^t sqrt [x’(u)]^2 + [y’(u)]^2 + [z’(u)]^2 du.

s=sqrt 5