Evaluate fxds, xds, where C is the curve r(t) = (2+t)i+(t) j+(5−3t)k, 0≤t≤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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![**Problem Statement**
Evaluate the line integral \( \int_C x \, ds \), where \( C \) is the curve defined by:
\[
\mathbf{r}(t) = (2 + t) \, \mathbf{i} + (t) \, \mathbf{j} + (5 - 3t) \, \mathbf{k}
\]
with the parameter \( t \) ranging from 0 to 1, i.e., \( 0 \leq t \leq 1 \).
**Explanation**
- **Integral to Evaluate**: The expression \( \int_C x \, ds \) represents a line integral, where the variable \( x \) is integrated over the path \( C \).
- **Curve Description**: The curve \( C \) is given by the vector function \( \mathbf{r}(t) \) which describes the position of points on the curve as a function of \( t \). The curve components are:
- \( x(t) = 2 + t \)
- \( y(t) = t \)
- \( z(t) = 5 - 3t \)
- **Parameter Range**: The parameter \( t \) varies from 0 to 1, defining the section of the curve over which the integration is to be performed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F481d8947-528e-477e-8558-b8f3bfa8246f%2F1d5aeb73-cc0b-4d6a-8379-7417349fd8f9%2Ftksvjl4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Evaluate the line integral \( \int_C x \, ds \), where \( C \) is the curve defined by:
\[
\mathbf{r}(t) = (2 + t) \, \mathbf{i} + (t) \, \mathbf{j} + (5 - 3t) \, \mathbf{k}
\]
with the parameter \( t \) ranging from 0 to 1, i.e., \( 0 \leq t \leq 1 \).
**Explanation**
- **Integral to Evaluate**: The expression \( \int_C x \, ds \) represents a line integral, where the variable \( x \) is integrated over the path \( C \).
- **Curve Description**: The curve \( C \) is given by the vector function \( \mathbf{r}(t) \) which describes the position of points on the curve as a function of \( t \). The curve components are:
- \( x(t) = 2 + t \)
- \( y(t) = t \)
- \( z(t) = 5 - 3t \)
- **Parameter Range**: The parameter \( t \) varies from 0 to 1, defining the section of the curve over which the integration is to be performed.
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