+ (sint)j +t²k for 0≤t≤3 r(t)=(cost)i F(t)=(Int)i + (21)j +1²k for 1st ≤4
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
find arc lenght for the following
![### Transcription for Educational Website
#### Vector Functions
**(c)**
\[
\vec{r}(t) = (\cos t) \, \vec{i} + (\sin t) \, \vec{j} + \frac{t^2}{3} \, \vec{k} \quad \text{for } 0 \leq t \leq 3
\]
**(d)**
\[
\vec{r}(t) = (\ln t) \, \vec{i} + (2t) \, \vec{j} + t^2 \, \vec{k} \quad \text{for } 1 \leq t \leq 4
\]
### Explanation
These expressions define vector functions \(\vec{r}(t)\) in terms of the parameter \(t\), with components along the unit vectors \(\vec{i}\), \(\vec{j}\), and \(\vec{k}\), representing directions in a three-dimensional Cartesian coordinate system.
- The function components comprise trigonometric functions \((\cos t, \sin t)\), a logarithmic function \((\ln t)\), linear terms \((2t)\), and quadratic terms \((t^2)\).
- The range of \(t\) specifies the domain for each function:
- For equation (c): \(0 \leq t \leq 3\).
- For equation (d): \(1 \leq t \leq 4\).
### Visualization
Imagine plotting these functions in a 3D space, observing the trajectory each vector traces:
- **(c)** will create a helical path affected by trigonometric circular motion in the \(xy\)-plane and a quadratic rise along the \(z\)-axis.
- **(d)** will trace a path influenced by the logarithmic variation along the \(x\)-axis, a linear movement along the \(y\)-axis, and a quadratic increase along the \(z\)-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1079eff7-cd9f-4a9b-a6b7-8587fcd056cc%2Ff142171d-8e9e-4c9e-a0ac-c201b8e2d54a%2Fy7moi0t_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Website
#### Vector Functions
**(c)**
\[
\vec{r}(t) = (\cos t) \, \vec{i} + (\sin t) \, \vec{j} + \frac{t^2}{3} \, \vec{k} \quad \text{for } 0 \leq t \leq 3
\]
**(d)**
\[
\vec{r}(t) = (\ln t) \, \vec{i} + (2t) \, \vec{j} + t^2 \, \vec{k} \quad \text{for } 1 \leq t \leq 4
\]
### Explanation
These expressions define vector functions \(\vec{r}(t)\) in terms of the parameter \(t\), with components along the unit vectors \(\vec{i}\), \(\vec{j}\), and \(\vec{k}\), representing directions in a three-dimensional Cartesian coordinate system.
- The function components comprise trigonometric functions \((\cos t, \sin t)\), a logarithmic function \((\ln t)\), linear terms \((2t)\), and quadratic terms \((t^2)\).
- The range of \(t\) specifies the domain for each function:
- For equation (c): \(0 \leq t \leq 3\).
- For equation (d): \(1 \leq t \leq 4\).
### Visualization
Imagine plotting these functions in a 3D space, observing the trajectory each vector traces:
- **(c)** will create a helical path affected by trigonometric circular motion in the \(xy\)-plane and a quadratic rise along the \(z\)-axis.
- **(d)** will trace a path influenced by the logarithmic variation along the \(x\)-axis, a linear movement along the \(y\)-axis, and a quadratic increase along the \(z\)-axis.
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