- Sivi Find the arc length parameter along the curve from the point where t=0 by evaluating the integral s = |v|dt. Then 0 find the length of the indicated portion of the curve. r(t) = (e¹ cost)i + (e¹ sint)j+e¹k, - In4st≤0 The arc length parameter is s(t) = (Type an exact answer, using radicals as needed.)
- Sivi Find the arc length parameter along the curve from the point where t=0 by evaluating the integral s = |v|dt. Then 0 find the length of the indicated portion of the curve. r(t) = (e¹ cost)i + (e¹ sint)j+e¹k, - In4st≤0 The arc length parameter is s(t) = (Type an exact answer, using radicals 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
Related questions
Question
![### Arc Length Parameter Calculation
To find the arc length parameter along the curve from the point where \( t = 0 \), evaluate the integral:
\[
s = \int_0^t |\mathbf{v}| \, dr
\]
Then, find the length of the indicated portion of the curve.
### Curve Definition
The curve is defined by the vector function:
\[
\mathbf{r}(t) = \left( e^t \cos t \right) \mathbf{i} + \left( e^t \sin t \right) \mathbf{j} + e^t \mathbf{k}, \quad -\ln 4 \leq t \leq 0
\]
### Calculation of Arc Length
The task is to find the arc length parameter \( s(t) \).
**Note:** Enter an exact answer, using radicals as needed.
Place your answer in the provided box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd35cf2d-d59c-46d2-9acb-f7bd2544420e%2Fb79e1e4a-ca7a-4003-85ef-ebd8ae0a5928%2Fdxmcodn_processed.png&w=3840&q=75)
Transcribed Image Text:### Arc Length Parameter Calculation
To find the arc length parameter along the curve from the point where \( t = 0 \), evaluate the integral:
\[
s = \int_0^t |\mathbf{v}| \, dr
\]
Then, find the length of the indicated portion of the curve.
### Curve Definition
The curve is defined by the vector function:
\[
\mathbf{r}(t) = \left( e^t \cos t \right) \mathbf{i} + \left( e^t \sin t \right) \mathbf{j} + e^t \mathbf{k}, \quad -\ln 4 \leq t \leq 0
\]
### Calculation of Arc Length
The task is to find the arc length parameter \( s(t) \).
**Note:** Enter an exact answer, using radicals as needed.
Place your answer in the provided box.
![**Problem Statement:**
Find the arc length parameter along the given curve from the point where \( t = 0 \) by evaluating the integral
\[ s(t) = \int_{0}^{t} |v(\tau)| \, d\tau. \]
Then find the length of the indicated portion of the curve \( \mathbf{r}(t) = 5\cos t \, \mathbf{i} + 5\sin t \, \mathbf{j} + 9t \, \mathbf{k} \), where \( 0 \leq t \leq \frac{\pi}{3} \).
---
**Instructions:**
The arc length parameter along the curve, starting at \( t = 0 \), is \( s(t) = \) [ ].
*(Type an exact answer, using radicals as needed.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd35cf2d-d59c-46d2-9acb-f7bd2544420e%2Fb79e1e4a-ca7a-4003-85ef-ebd8ae0a5928%2Fqn7brre_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the arc length parameter along the given curve from the point where \( t = 0 \) by evaluating the integral
\[ s(t) = \int_{0}^{t} |v(\tau)| \, d\tau. \]
Then find the length of the indicated portion of the curve \( \mathbf{r}(t) = 5\cos t \, \mathbf{i} + 5\sin t \, \mathbf{j} + 9t \, \mathbf{k} \), where \( 0 \leq t \leq \frac{\pi}{3} \).
---
**Instructions:**
The arc length parameter along the curve, starting at \( t = 0 \), is \( s(t) = \) [ ].
*(Type an exact answer, using radicals as needed.)*
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