Evaluate the line integral a) | 3x'yds, where Cis the curve r(t)=5ti + (3t-7)j. Ostsl. b) (xy + z)ds, where C is the helix r(t)=cos(t) i + sin(t)j + t k, 0sts3n C) F.dr. where the vector field F(x.y) = xfy'i -Wxj and C is the curve r(t)= 2t i + 7tj. Osts1. !!
Evaluate the line integral a) | 3x'yds, where Cis the curve r(t)=5ti + (3t-7)j. Ostsl. b) (xy + z)ds, where C is the helix r(t)=cos(t) i + sin(t)j + t k, 0sts3n C) F.dr. where the vector field F(x.y) = xfy'i -Wxj and C is the curve r(t)= 2t i + 7tj. Osts1. !!
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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Transcribed Image Text:### Evaluating Line Integrals
Here are a few problems related to the evaluation of line integrals in vector calculus:
#### Problem a
Evaluate the line integral \(\int_{C} 3x^3 y \, ds\), where \(C\) is the curve \( \mathbf{r}(t) = 5t \mathbf{i} + (3t - 7) \mathbf{j} \), \(0 \leq t \leq 1\).
#### Problem b
Evaluate the line integral \(\int_{C} (xy + z) \, ds\), where \(C\) is the helix \(\mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k}\), \(0 \leq t \leq 3\pi \).
#### Problem c
Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d\mathbf{r}\), where the vector field \(\mathbf{F}(x, y) = x^2 y^2 \mathbf{i} - y \sqrt{x} \mathbf{j}\) and \(C\) is the curve \(\mathbf{r}(t) = -2t^3 \mathbf{i} + 7t \mathbf{j}\), \(0 \leq t \leq 1\).
### Detailed Explanations:
- **Vector Fields & Curves:**
Each of these problems involves a specific type of curve \( \mathbf{r}(t) \) present within given limits for parameter \( t \). To evaluate these integrals, we typically need to parameterize the curve, find the differential element \( ds \) (or \( d\mathbf{r} \)), and substitute these into respective integrals.
- **Helix Curve:**
In Problem b, the helix is a three-dimensional curve defined by trigonometric functions which naturally wrap around a circular cylinder as \( t \) increases, making a helical shape.
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