Find the value of the line integral. F. dr (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x, y) = yi - xj (a) r (t) = ti + tj, osts 1
Find the value of the line integral. F. dr (Hint: If F is conservative, the integration may be easier on an alternative path.) F(x, y) = yi - xj (a) r (t) = ti + tj, osts 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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15.3 # 5
![**Finding the Value of the Line Integral**
To calculate the value of the line integral:
\[
\int_{C} F \cdot dr
\]
**Hint:** If \( F \) is conservative, the integration may be easier on an alternative path.
Given:
\[ F(x, y) = yi - xj \]
Evaluate the following paths:
(a) \( \mathbf{r_1}(t) = ti + tj, \quad 0 \leq t \leq 1 \)
[Answer Box]
(b) \( \mathbf{r_2}(t) = ti + t^2j, \quad 0 \leq t \leq 1 \)
[Answer Box]
(c) \( \mathbf{r_3}(t) = ti + t^3j, \quad 0 \leq t \leq 1 \)
[Answer Box]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2Fe9f61785-a657-41f5-b801-4ea91543a68b%2Fy4e6oog_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Value of the Line Integral**
To calculate the value of the line integral:
\[
\int_{C} F \cdot dr
\]
**Hint:** If \( F \) is conservative, the integration may be easier on an alternative path.
Given:
\[ F(x, y) = yi - xj \]
Evaluate the following paths:
(a) \( \mathbf{r_1}(t) = ti + tj, \quad 0 \leq t \leq 1 \)
[Answer Box]
(b) \( \mathbf{r_2}(t) = ti + t^2j, \quad 0 \leq t \leq 1 \)
[Answer Box]
(c) \( \mathbf{r_3}(t) = ti + t^3j, \quad 0 \leq t \leq 1 \)
[Answer Box]
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