**Title: Calculating the Length of a Helix**To illustrate that the length of a smooth space curve does not depend on the parameterization used to compute it, calculate the length of one turn of the helix with the following parameterizations:a. $ \mathbf{r}(t) = \langle \cos(4t), \sin(4t), 4t \rangle $, $ 0 \leq t \leq \frac{\pi}{2} $b. $ \mathbf{r}(t) = \left\langle \cos\left(\frac{1}{2}t\right), \sin\left(\frac{1}{2}t\right), t \right\rangle $, $ 0 \leq t \leq 4\pi $c. $ \mathbf{r}(t) = \langle \cos(t), \sin(t), -t \rangle $, $ -2\pi \leq t \leq 0 $**Note:** The helix shown to the right is just one example of such a helix and does not exactly correspond to the above parameterizations.**Instruction:**Enter your answer in the answer box and then click "Check Answer."**Part to Solve:**$ a. L = $ $[ \text{Type an exact answer, using } \pi \text{ as needed}]$*2 parts remaining*

Length of the curve, rt , L=∫abrt˙dt , a≤t≤b (a) Consider the parametrisation of the helix,…

To illustrate that the length of a smooth space curve does not depend on the parameterization used to compute it, calculate the length of one turn of the helix with the following parameterizations. a. r(t) = (cos 4t)i + (sin 4t)j + 4tk, 0sts b. r(t) = | cos i+ sin -k. 0sts 4 2. c. r(t) = (cos t)i – (sin t)j – tk, - 21sts0 Note that the helix shown to the right is just one example of such a helix, and does not exactly correspond to the DATAMstrisotiono in Bata a a. L= (Type an exact answer, using t as needed.)

To illustrate that the length of a smooth space curve does not depend on the parameterization used to compute it, calculate the length of one turn of the helix with the following parameterizations. a. r(t) = (cos 4t)i + (sin 4t)j + 4tk, 0sts b. r(t) = | cos i+ sin -k. 0sts 4 2. c. r(t) = (cos t)i – (sin t)j – tk, - 21sts0 Note that the helix shown to the right is just one example of such a helix, and does not exactly correspond to the DATAMstrisotiono in Bata a a. L= (Type an exact answer, using t as needed.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Title: Calculating the Length of a Helix**

To illustrate that the length of a smooth space curve does not depend on the parameterization used to compute it, calculate the length of one turn of the helix with the following parameterizations:

a. $ \mathbf{r}(t) = \langle \cos(4t), \sin(4t), 4t \rangle $, $ 0 \leq t \leq \frac{\pi}{2} $

b. $ \mathbf{r}(t) = \left\langle \cos\left(\frac{1}{2}t\right), \sin\left(\frac{1}{2}t\right), t \right\rangle $, $ 0 \leq t \leq 4\pi $

c. $ \mathbf{r}(t) = \langle \cos(t), \sin(t), -t \rangle $, $ -2\pi \leq t \leq 0 $

**Note:** The helix shown to the right is just one example of such a helix and does not exactly correspond to the above parameterizations.

**Instruction:**
Enter your answer in the answer box and then click "Check Answer."

**Part to Solve:**
$ a. L = $ $[ \text{Type an exact answer, using } \pi \text{ as needed}]$

*2 parts remaining*

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