Consider the following. C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (0, 1), starting at (0, 0) (a) Find a piecewise smooth parametrization of the path C. (t,0) r(t) = (2-t,t-1) (0,3 – t) (b) Evaluate [(x + 4√Y) ds. 0 ≤ t ≤ 1 1≤t≤2 2 ≤t≤ 3
Consider the following. C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (0, 1), starting at (0, 0) (a) Find a piecewise smooth parametrization of the path C. (t,0) r(t) = (2-t,t-1) (0,3 – t) (b) Evaluate [(x + 4√Y) ds. 0 ≤ t ≤ 1 1≤t≤2 2 ≤t≤ 3
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...
Related questions
Question
![### Mathematical Problem and Solution on Path Parametrization
**Problem Statement:**
Consider the following path \(C\), a counterclockwise path around the triangle with vertices \((0, 0)\), \((1, 0)\), and \((0, 1)\), starting at \((0, 0)\).
**Tasks:**
(a) Find a piecewise smooth parametrization of the path \(C\).
(b) Evaluate the line integral \(\int_C \left( x + 4\sqrt{y} \right) \, ds\).
**Solution to Part (a):**
The parametric function \(\mathbf{r}(t)\) for path \(C\) is defined piecewise over three intervals.
1. **Interval \(0 \leq t \leq 1\):**
\[
\mathbf{r}(t) = (t, 0)
\]
This segment corresponds to moving from \((0, 0)\) to \((1, 0)\) along the x-axis.
2. **Interval \(1 \leq t \leq 2\):**
\[
\mathbf{r}(t) = (2-t, t-1)
\]
This segment represents moving from \((1, 0)\) to \((0, 1)\).
3. **Interval \(2 \leq t \leq 3\):**
\[
\mathbf{r}(t) = (0, 3-t)
\]
This final segment moves from \((0, 1)\) back to \((0, 0)\).
**Solution to Part (b):**
Evaluate the line integral \(\int_C \left( x + 4\sqrt{y} \right) \, ds\).
(Note: Precise calculation steps and results are not provided in the text).
To determine this integral, follow the parametrizations provided and compute accordingly using standard integral methods for line integrals over parametric paths.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53d7c283-d7c7-48d5-8c8d-bb71f4fe00d9%2Fc7b4313b-2ca9-45b3-92fc-2d194c236ddd%2F8hxte2q_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Problem and Solution on Path Parametrization
**Problem Statement:**
Consider the following path \(C\), a counterclockwise path around the triangle with vertices \((0, 0)\), \((1, 0)\), and \((0, 1)\), starting at \((0, 0)\).
**Tasks:**
(a) Find a piecewise smooth parametrization of the path \(C\).
(b) Evaluate the line integral \(\int_C \left( x + 4\sqrt{y} \right) \, ds\).
**Solution to Part (a):**
The parametric function \(\mathbf{r}(t)\) for path \(C\) is defined piecewise over three intervals.
1. **Interval \(0 \leq t \leq 1\):**
\[
\mathbf{r}(t) = (t, 0)
\]
This segment corresponds to moving from \((0, 0)\) to \((1, 0)\) along the x-axis.
2. **Interval \(1 \leq t \leq 2\):**
\[
\mathbf{r}(t) = (2-t, t-1)
\]
This segment represents moving from \((1, 0)\) to \((0, 1)\).
3. **Interval \(2 \leq t \leq 3\):**
\[
\mathbf{r}(t) = (0, 3-t)
\]
This final segment moves from \((0, 1)\) back to \((0, 0)\).
**Solution to Part (b):**
Evaluate the line integral \(\int_C \left( x + 4\sqrt{y} \right) \, ds\).
(Note: Precise calculation steps and results are not provided in the text).
To determine this integral, follow the parametrizations provided and compute accordingly using standard integral methods for line integrals over parametric paths.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning