Evaluate •10 (22² (x² + y²) ds, C is the top half of the circle with radius 5 centered at (0,0) and is traversed in the clockwise direction.
Evaluate •10 (22² (x² + y²) ds, C is the top half of the circle with radius 5 centered at (0,0) and is traversed in the clockwise direction.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
6.1.2
![**Problem:**
Evaluate
\[
\int_{C} (x^2 + y^2) \, ds
\]
where \( C \) is the top half of the circle with radius 5 centered at (0, 0) and is traversed in the clockwise direction.
---
**Explanation:**
This problem involves evaluating a line integral along the curve \( C \), which represents the top half of a circle centered at the origin with a radius of 5. The circle is traversed in a clockwise direction.
- **Equation of the Circle:**
The equation for the circle is \( x^2 + y^2 = 25 \), and we are interested in the top half, i.e., \( y \geq 0 \).
- **Line Integral:**
The integral to be evaluated is \(\int_{C} (x^2 + y^2) \, ds\), where \( ds \) is the differential arc length along the curve \( C \).
- **Parametrization:**
Parametrize the top half circle using:
\[
x = 5 \cos t, \quad y = 5 \sin t
\]
with the parameter \( t \) ranging from \( 0 \) to \( \pi \) to cover the top half from left to right typically. Since it is traversed clockwise, the parametrization would be adjusted accordingly.
- **Clockwise Direction:**
To traverse clockwise, consider reversing the parameter:
\[
x = 5 \cos t, \quad y = 5 \sin (-t)
\]
- **Integration:**
Substitute the parametrization into the integral and compute by considering the limits that correspond to the clockwise traversal.
This setup provides a detailed approach to solving the integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b463452-b960-4b00-bce0-3a69d9f467e2%2F983f904d-f442-4d1f-b9ff-52a21859d820%2Fyjys67_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Evaluate
\[
\int_{C} (x^2 + y^2) \, ds
\]
where \( C \) is the top half of the circle with radius 5 centered at (0, 0) and is traversed in the clockwise direction.
---
**Explanation:**
This problem involves evaluating a line integral along the curve \( C \), which represents the top half of a circle centered at the origin with a radius of 5. The circle is traversed in a clockwise direction.
- **Equation of the Circle:**
The equation for the circle is \( x^2 + y^2 = 25 \), and we are interested in the top half, i.e., \( y \geq 0 \).
- **Line Integral:**
The integral to be evaluated is \(\int_{C} (x^2 + y^2) \, ds\), where \( ds \) is the differential arc length along the curve \( C \).
- **Parametrization:**
Parametrize the top half circle using:
\[
x = 5 \cos t, \quad y = 5 \sin t
\]
with the parameter \( t \) ranging from \( 0 \) to \( \pi \) to cover the top half from left to right typically. Since it is traversed clockwise, the parametrization would be adjusted accordingly.
- **Clockwise Direction:**
To traverse clockwise, consider reversing the parameter:
\[
x = 5 \cos t, \quad y = 5 \sin (-t)
\]
- **Integration:**
Substitute the parametrization into the integral and compute by considering the limits that correspond to the clockwise traversal.
This setup provides a detailed approach to solving the integral.
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