Hello I am having a difficult time solving these this problem and the three parts, can you show a full detail without missing any steps do a step by step doing this answer so I can understand how you did it can you do it step by step so I can follow along **Problem 2: Analysis of a Curve's Derivative**Given the derivative function of a curve, \( \mathbf{r'}(t) = \langle t + 1, t^3, \sin(t) \rangle \):(a) Calculate \( \int_{0}^{2} \mathbf{r'}(t) \, dt \).(b) Calculate \( \int_{0}^{2} \| \mathbf{r'}(t) \| \, dt \).(c) What is the difference between the two answers? Explain what (a) means versus (b).---**Explanation:***For part (a):*To calculate \( \int_{0}^{2} \mathbf{r'}(t) \, dt \), you need to integrate the derivative function component-wise.*For part (b):*To calculate \( \int_{0}^{2} \| \mathbf{r'}(t) \| \, dt \), you need to find the magnitude of the derivative vector function first, then integrate that magnitude over the given interval.*For part (c):*The difference between the two results lies in their interpretations:- Part (a) represents the total change in the vector function \( \mathbf{r}(t) \) over the interval [0, 2].- Part (b) represents the total arc length of the curve described by \( \mathbf{r}(t) \) over the interval [0, 2].

Suppose I have a curve with derivative function r' (t) = (t + 1, t³, sin(t)). 2 (a) Calculate f² r' (t r' (t) dt. (b) Calculate S² || r' (t)|| dt. ||r' (c) What is the difference between the two answers. Explain what (a) me

Suppose I have a curve with derivative function r' (t) = (t + 1, t³, sin(t)). 2 (a) Calculate f² r' (t r' (t) dt. (b) Calculate S² || r' (t)|| dt. ||r' (c) What is the difference between the two answers. Explain what (a) me

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

Hello I am having a difficult time solving these this problem and the three parts, can you show a full detail without missing any steps do a step by step doing this answer so I can understand how you did it

can you do it step by step so I can follow along

**Problem 2: Analysis of a Curve's Derivative**

Given the derivative function of a curve, \( \mathbf{r'}(t) = \langle t + 1, t^3, \sin(t) \rangle \):

(a) Calculate \( \int_{0}^{2} \mathbf{r'}(t) \, dt \).

(b) Calculate \( \int_{0}^{2} \| \mathbf{r'}(t) \| \, dt \).

(c) What is the difference between the two answers? Explain what (a) means versus (b).

---

**Explanation:**

*For part (a):*

To calculate \( \int_{0}^{2} \mathbf{r'}(t) \, dt \), you need to integrate the derivative function component-wise.

*For part (b):*

To calculate \( \int_{0}^{2} \| \mathbf{r'}(t) \| \, dt \), you need to find the magnitude of the derivative vector function first, then integrate that magnitude over the given interval.

*For part (c):*

The difference between the two results lies in their interpretations:
- Part (a) represents the total change in the vector function \( \mathbf{r}(t) \) over the interval [0, 2].
- Part (b) represents the total arc length of the curve described by \( \mathbf{r}(t) \) over the interval [0, 2].

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