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
![Certainly! Below is a transcription of the provided image, suitable for an educational website.
________________________________________________________________________
**Vector Field Line Integrals**
Analyze the vector field along a specified curve and evaluate the given integrals. Consider the vector function \( \mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} - t \mathbf{k} \) for \( 0 \leq t \leq 1 \).
We need to evaluate each of the following integrals along the curve \( C \):
a. \( \int_{C} (x - y + z) \, dx \)
b. \( \int_{C} (x - y + z) \, dy \)
c. \( \int_{C} (x - y + z) \, dz \)
**Solution Steps:**
1. **Parameterize the Curve \( C \):**
Given by \( \mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} - t \mathbf{k} \):
- \( x = t^2 \)
- \( y = t \)
- \( z = -t \)
2. **Compute Derivatives for \( dx \), \( dy \), and \( dz \):**
- \( dx = 2t \, dt \)
- \( dy = dt \)
- \( dz = -dt \)
3. **Substitute \( x \), \( y \), \( z \), and their differentials in each integral:**
**a.** Integral \( \int_{C} (x - y + z) \, dx \):
\[
= \int_{0}^{1} \left( t^2 - t - t \right) \cdot 2t \, dt
= \int_{0}^{1} \left( t^2 - t \right) \cdot 2t \, dt
= \int_{0}^{1} \left( 2t^3 - 2t^2 \right) \, dt
= \left[ \frac{1}{2} t^4 - \frac{2}{3} t^3 \right]_{0}^{1}
= \frac{1}{2} - \frac{2}{3} = -\frac{1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd49838c-9fe9-4c2b-b877-1f4ac641c60c%2F69f4a3af-899b-4154-9ca5-54c23de56ffc%2Ftxo6zfq_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Below is a transcription of the provided image, suitable for an educational website.
________________________________________________________________________
**Vector Field Line Integrals**
Analyze the vector field along a specified curve and evaluate the given integrals. Consider the vector function \( \mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} - t \mathbf{k} \) for \( 0 \leq t \leq 1 \).
We need to evaluate each of the following integrals along the curve \( C \):
a. \( \int_{C} (x - y + z) \, dx \)
b. \( \int_{C} (x - y + z) \, dy \)
c. \( \int_{C} (x - y + z) \, dz \)
**Solution Steps:**
1. **Parameterize the Curve \( C \):**
Given by \( \mathbf{r}(t) = t^2 \mathbf{i} + t \mathbf{j} - t \mathbf{k} \):
- \( x = t^2 \)
- \( y = t \)
- \( z = -t \)
2. **Compute Derivatives for \( dx \), \( dy \), and \( dz \):**
- \( dx = 2t \, dt \)
- \( dy = dt \)
- \( dz = -dt \)
3. **Substitute \( x \), \( y \), \( z \), and their differentials in each integral:**
**a.** Integral \( \int_{C} (x - y + z) \, dx \):
\[
= \int_{0}^{1} \left( t^2 - t - t \right) \cdot 2t \, dt
= \int_{0}^{1} \left( t^2 - t \right) \cdot 2t \, dt
= \int_{0}^{1} \left( 2t^3 - 2t^2 \right) \, dt
= \left[ \frac{1}{2} t^4 - \frac{2}{3} t^3 \right]_{0}^{1}
= \frac{1}{2} - \frac{2}{3} = -\frac{1
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 15 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