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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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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6f07dba-75ce-4887-9617-f6bb8f9dcf1f%2Ff747df45-70cc-4e3d-9cae-d56dd593f54b%2Fs48bqz8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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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