   Chapter 12.5, Problem 17E

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# Investigation Consider the helix represented by the vector-valued function r ( t ) = 〈 2 cos t , 2 sin t , t 〉 .(a) Write the length of the arc s on the helix as a function of t by evaluating the integral s = ∫ 0 t [ x ' ( u ) ] 2 + [ y ' ( u ) ] 2 + [ z ' ( u ) ] 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 = 5   and   s = 4 .(d) Verify that ‖ r ' ( s ) ‖ = 1 .

(a)

To determine

To Calculate: The length of the arc s on the helix as a function of t here, the function is r(t)=2cost,2sint,t.

Explanation

Given:

The provided function is, r(t)=2cost,2sint,t.

Formula Used:

Length of the curve as:

s=0t(x(t))2+(y(t))2+(z(t))2dt

Calculation:

The helix path is,

r(t)=2cost,2sint,t

On differentiating the vector-valued function as shown below,

r(t)=2sint,2cost

(b)

To determine

To Calculate: The parametric equations of the curve r(t)=2cost,2sint,t in terms of the arc length s.

(c)

To determine

To Calculate: The coordinates of the point on the helix r(t)=2cost,2sint,t for arc lengths s=5 and s=4.

(d)

To determine

To Prove: The relation r(s)=1

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