Let F(x, y) = (x + y, x²), and let 7(t) be the parameterization of the line segment from (0, 4) to (4, 5) that you found above. Then: F(F(t)) = = 7' (t) F(r(t)) · 7' (t) = = Finally, the line integral of ♬ along the line segment parameterized by r(t), where 0 < t < 4 [*^*F(F(t)) · F' (t)dt = is
Let F(x, y) = (x + y, x²), and let 7(t) be the parameterization of the line segment from (0, 4) to (4, 5) that you found above. Then: F(F(t)) = = 7' (t) F(r(t)) · 7' (t) = = Finally, the line integral of ♬ along the line segment parameterized by r(t), where 0 < t < 4 [*^*F(F(t)) · F' (t)dt = is
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Parameterizing a Line Segment**
**Objective**: Learn how to parameterize the line segment from the point \((0, 4)\) to the point \((4, 5)\).
### Vector Function Representation
The line segment can be described using the vector function:
\[
\vec{r}(t) = \langle t, \frac{t}{4} + 4 \rangle
\]
### Explanation
- **\(\langle t, \frac{t}{4} + 4 \rangle\)**: This notation represents a parameterized function where:
- **\(t\)** represents the x-component of the vector.
- **\(\frac{t}{4} + 4\)** represents the y-component, which is a linear function dependent on \(t\).
### Parameter Range
- \(0 \leq t \leq 4\)
This range of \(t\) ensures that the parameterized points move from \((0, 4)\) to \((4, 5)\).
### Key Points
- At \(t = 0\), the point is \((0, 4)\).
- At \(t = 4\), the point is \((4, 5)\).
This parameterization allows us to represent any point on the line segment as \(t\) varies from 0 to 4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09d105f3-6c69-4cbc-8997-6988f1733e6f%2F2fe49863-4ddd-4e03-aea6-1e5540501b0a%2F31n2t3o_processed.png&w=3840&q=75)
Transcribed Image Text:**Parameterizing a Line Segment**
**Objective**: Learn how to parameterize the line segment from the point \((0, 4)\) to the point \((4, 5)\).
### Vector Function Representation
The line segment can be described using the vector function:
\[
\vec{r}(t) = \langle t, \frac{t}{4} + 4 \rangle
\]
### Explanation
- **\(\langle t, \frac{t}{4} + 4 \rangle\)**: This notation represents a parameterized function where:
- **\(t\)** represents the x-component of the vector.
- **\(\frac{t}{4} + 4\)** represents the y-component, which is a linear function dependent on \(t\).
### Parameter Range
- \(0 \leq t \leq 4\)
This range of \(t\) ensures that the parameterized points move from \((0, 4)\) to \((4, 5)\).
### Key Points
- At \(t = 0\), the point is \((0, 4)\).
- At \(t = 4\), the point is \((4, 5)\).
This parameterization allows us to represent any point on the line segment as \(t\) varies from 0 to 4.
![Let \(\vec{F}(x, y) = \langle x + y, x^2 \rangle\), and let \(\vec{r}(t)\) be the parameterization of the line segment from \((0, 4)\) to \((4, 5)\) that you found above.
Then:
\[
\vec{F}(\vec{r}(t)) = \langle \text{____}, \text{____} \rangle
\]
\[
\vec{r}'(t) = \langle \text{____}, \text{____} \rangle
\]
\[
\vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) = \text{____}
\]
Finally, the line integral of \(\vec{F}\) along the line segment parameterized by \(\vec{r}(t)\),
where \(0 \leq t \leq \text{____}\), is
\[
\int_{0}^{4} \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt = \text{____}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09d105f3-6c69-4cbc-8997-6988f1733e6f%2F2fe49863-4ddd-4e03-aea6-1e5540501b0a%2Fa9225aq_processed.png&w=3840&q=75)
Transcribed Image Text:Let \(\vec{F}(x, y) = \langle x + y, x^2 \rangle\), and let \(\vec{r}(t)\) be the parameterization of the line segment from \((0, 4)\) to \((4, 5)\) that you found above.
Then:
\[
\vec{F}(\vec{r}(t)) = \langle \text{____}, \text{____} \rangle
\]
\[
\vec{r}'(t) = \langle \text{____}, \text{____} \rangle
\]
\[
\vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) = \text{____}
\]
Finally, the line integral of \(\vec{F}\) along the line segment parameterized by \(\vec{r}(t)\),
where \(0 \leq t \leq \text{____}\), is
\[
\int_{0}^{4} \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) \, dt = \text{____}
\]
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