where C is the curve r(t) = (sin(t), cos(t), sin(2t)), 0 ≤ t ≤ 2
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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Question
![### Evaluate the Line Integral
\[ \int_{C} \left( (y + 7 \sin(x)) \, dx + (z^2 + 8 \cos(y)) \, dy + x^3 \, dz \right) \]
where \( C \) is the curve \( \mathbf{r}(t) = (\sin(t), \cos(t), \sin(2t)) \), \( 0 \leq t \leq 2\pi \).
**Hint**: Observe that \( C \) lies on the surface \( z = 2xy \).
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \boxed{} \]
---
**Explanation**:
This problem involves evaluating a line integral over a given vector field along a specified curve. The integral involves three functions being integrated with respect to \( dx \), \( dy \), and \( dz \). The given curve \( \mathbf{r}(t) \) is a parametric function defining the path of integration from \( t = 0 \) to \( t = 2\pi \).
Take note of the hint, which indicates that the curve \( C \) lies on the surface \( z = 2xy \). This can be useful in simplifying or verifying the path of integration.
### Steps to Evaluate the Integral
1. **Parameterize the Curve**:
The curve \( \mathbf{r}(t) \) is already parameterized as \( (\sin(t), \cos(t), \sin(2t)) \).
2. **Compute \( dx, dy, dz \)**:
- \( dx = \frac{d}{dt} \sin(t) \, dt = \cos(t) \, dt \)
- \( dy = \frac{d}{dt} \cos(t) \, dt = -\sin(t) \, dt \)
- \( dz = \frac{d}{dt} \sin(2t) \, dt = 2 \cos(2t) \, dt \)
3. **Substitute \( \mathbf{r}(t) \) into the Integral**:
- \( x = \sin(t) \)
- \( y = \cos(t) \)
- \( z = \sin(2t) \)
4. **Evaluate the Integral**:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46029127-96a5-4dc2-997a-3c090d1aab2d%2F8cf9c574-d628-4320-aed1-00cf426397de%2Foo1reh3_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluate the Line Integral
\[ \int_{C} \left( (y + 7 \sin(x)) \, dx + (z^2 + 8 \cos(y)) \, dy + x^3 \, dz \right) \]
where \( C \) is the curve \( \mathbf{r}(t) = (\sin(t), \cos(t), \sin(2t)) \), \( 0 \leq t \leq 2\pi \).
**Hint**: Observe that \( C \) lies on the surface \( z = 2xy \).
\[ \int_{C} \mathbf{F} \cdot d\mathbf{r} = \boxed{} \]
---
**Explanation**:
This problem involves evaluating a line integral over a given vector field along a specified curve. The integral involves three functions being integrated with respect to \( dx \), \( dy \), and \( dz \). The given curve \( \mathbf{r}(t) \) is a parametric function defining the path of integration from \( t = 0 \) to \( t = 2\pi \).
Take note of the hint, which indicates that the curve \( C \) lies on the surface \( z = 2xy \). This can be useful in simplifying or verifying the path of integration.
### Steps to Evaluate the Integral
1. **Parameterize the Curve**:
The curve \( \mathbf{r}(t) \) is already parameterized as \( (\sin(t), \cos(t), \sin(2t)) \).
2. **Compute \( dx, dy, dz \)**:
- \( dx = \frac{d}{dt} \sin(t) \, dt = \cos(t) \, dt \)
- \( dy = \frac{d}{dt} \cos(t) \, dt = -\sin(t) \, dt \)
- \( dz = \frac{d}{dt} \sin(2t) \, dt = 2 \cos(2t) \, dt \)
3. **Substitute \( \mathbf{r}(t) \) into the Integral**:
- \( x = \sin(t) \)
- \( y = \cos(t) \)
- \( z = \sin(2t) \)
4. **Evaluate the Integral**:
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