5. Consider the line integral Jy dx- dx + 2x dy and the three different curves (a) C₁: (1+t,1+3t) with 0≤t≤1, (b) C₂: (t, t²) with 1 ≤ t ≤ 2, (c) C3: (t, 1) for 1 ≤ t ≤ 2 and (2, t) for 1 ≤ t ≤ 4.
5. Consider the line integral Jy dx- dx + 2x dy and the three different curves (a) C₁: (1+t,1+3t) with 0≤t≤1, (b) C₂: (t, t²) with 1 ≤ t ≤ 2, (c) C3: (t, 1) for 1 ≤ t ≤ 2 and (2, t) for 1 ≤ t ≤ 4.
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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![### Line Integrals and Path Independence
**Problem 5: Consider the Line Integral**
\[
\int_C y \, dx + 2x \, dy
\]
and the three different curves:
1. **Curve \(C_1\):** \((1 + t, 1 + 3t)\) with \(0 \leq t \leq 1\),
2. **Curve \(C_2\):** \((t, t^2)\) with \(1 \leq t \leq 2\),
3. **Curve \(C_3\):**
- \((t, 1)\) for \(1 \leq t \leq 2\)
- \((2, t)\) for \(1 \leq t \leq 4\).
**Tasks:**
(a) **Sketch the Three Curves:** The curves need to be plotted on a graph to visualize their paths. The first curve \(C_1\) moves linearly, the second curve \(C_2\) forms a parabola, and the third curve \(C_3\) is composed of two line segments.
(b) **Evaluate the Integral Along Each Curve:** Compute the value of the line integral for each of the three curves defined above.
**Upshot:** Line integrals for curves connecting \(A\) to \(B\) may differ depending on the path taken. This demonstrates the dependency of line integrals on the path in non-conservative fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77548912-c51c-4c9d-8b51-f3905a3bec75%2F505d54f7-c654-406e-a970-b85ad075c7a7%2F3kj5n88_processed.png&w=3840&q=75)
Transcribed Image Text:### Line Integrals and Path Independence
**Problem 5: Consider the Line Integral**
\[
\int_C y \, dx + 2x \, dy
\]
and the three different curves:
1. **Curve \(C_1\):** \((1 + t, 1 + 3t)\) with \(0 \leq t \leq 1\),
2. **Curve \(C_2\):** \((t, t^2)\) with \(1 \leq t \leq 2\),
3. **Curve \(C_3\):**
- \((t, 1)\) for \(1 \leq t \leq 2\)
- \((2, t)\) for \(1 \leq t \leq 4\).
**Tasks:**
(a) **Sketch the Three Curves:** The curves need to be plotted on a graph to visualize their paths. The first curve \(C_1\) moves linearly, the second curve \(C_2\) forms a parabola, and the third curve \(C_3\) is composed of two line segments.
(b) **Evaluate the Integral Along Each Curve:** Compute the value of the line integral for each of the three curves defined above.
**Upshot:** Line integrals for curves connecting \(A\) to \(B\) may differ depending on the path taken. This demonstrates the dependency of line integrals on the path in non-conservative fields.
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