Compute the line integral of the scalar function ƒ(x, y, z) = 2x² + 8z over the curve c(t) = (e¹,t²,t), 0≤ t ≤ 4 Jc f(x, y, z) ds =
Compute the line integral of the scalar function ƒ(x, y, z) = 2x² + 8z over the curve c(t) = (e¹,t²,t), 0≤ t ≤ 4 Jc f(x, y, z) ds =
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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![**Problem Statement:**
Compute the line integral of the scalar function \( f(x, y, z) = 2x^2 + 8z \) over the curve \( \mathbf{c}(t) = (e^t, t^2, t) \), where \( 0 \leq t \leq 4 \).
\[ \int_C f(x, y, z) \, ds = \boxed{} \]
**Explanation:**
This problem involves calculating a line integral along a specified curve. The scalar function given is \( f(x, y, z) = 2x^2 + 8z \).
The curve is described by the parametric functions:
- \( x(t) = e^t \)
- \( y(t) = t^2 \)
- \( z(t) = t \)
The parameter \( t \) ranges from 0 to 4. The goal is to evaluate the line integral of \( f \) along this curve \( \mathbf{c}(t) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8b731f0-d86e-4c27-b4bf-d8b1bfda93ca%2F5ac9d25e-6e8b-4e92-9b11-25f10961ad14%2F73bw4x_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Compute the line integral of the scalar function \( f(x, y, z) = 2x^2 + 8z \) over the curve \( \mathbf{c}(t) = (e^t, t^2, t) \), where \( 0 \leq t \leq 4 \).
\[ \int_C f(x, y, z) \, ds = \boxed{} \]
**Explanation:**
This problem involves calculating a line integral along a specified curve. The scalar function given is \( f(x, y, z) = 2x^2 + 8z \).
The curve is described by the parametric functions:
- \( x(t) = e^t \)
- \( y(t) = t^2 \)
- \( z(t) = t \)
The parameter \( t \) ranges from 0 to 4. The goal is to evaluate the line integral of \( f \) along this curve \( \mathbf{c}(t) \).
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