4. The upper hemisphere of the unit sphere can be parameterized, using cylindrical coordinates, as (sind) ble (r, 0) = (r cos , rsine, √1-²), tegral 0≤r≤1, 0≤0 ≤ 2T. (1-7²) 1/2 2051 Use this to find the area of the sphere. 75 A = -IN =( = 2 1/1² +r²) dr do
4. The upper hemisphere of the unit sphere can be parameterized, using cylindrical coordinates, as (sind) ble (r, 0) = (r cos , rsine, √1-²), tegral 0≤r≤1, 0≤0 ≤ 2T. (1-7²) 1/2 2051 Use this to find the area of the sphere. 75 A = -IN =( = 2 1/1² +r²) dr do
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
Please ignore all writing in pencil.
![The image contains a mathematical explanation related to parameterizing the upper hemisphere of the unit sphere using cylindrical coordinates. Here's the transcription and explanation:
---
**4. The Upper Hemisphere of the Unit Sphere:**
The upper hemisphere of the unit sphere can be parameterized using cylindrical coordinates as:
\[
\Phi(r, \theta) = (r \cos \theta, r \sin \theta, \sqrt{1 - r^2})
\]
Where:
- \(0 \leq r \leq 1\)
- \(0 \leq \theta \leq 2\pi\)
**Objective:**
Use this parameterization to find the area of the sphere.
**Mathematical Expression:**
\[
A = \int_0^{2\pi} \int_0^1 \left(\frac{1}{4(1-r^2)} + r^2\right) dr d\theta
\]
**Notes:**
- A handwritten note says "doable integral," indicating that the integral can be solved with standard methods.
- There's an expression set for \(\Phi \cdot \mathbf{r} = \langle \cos \theta, \sin \theta, \cdots \rangle\).
The given integral expression is intended to calculate the area of the upper hemisphere by integrating over the region defined by the cylindrical coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5dc3ef7d-848e-4463-9030-e4a43ebc2b5f%2F863e6af1-7e51-4c56-8bff-8d9e157e3b39%2F3nyfzzu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical explanation related to parameterizing the upper hemisphere of the unit sphere using cylindrical coordinates. Here's the transcription and explanation:
---
**4. The Upper Hemisphere of the Unit Sphere:**
The upper hemisphere of the unit sphere can be parameterized using cylindrical coordinates as:
\[
\Phi(r, \theta) = (r \cos \theta, r \sin \theta, \sqrt{1 - r^2})
\]
Where:
- \(0 \leq r \leq 1\)
- \(0 \leq \theta \leq 2\pi\)
**Objective:**
Use this parameterization to find the area of the sphere.
**Mathematical Expression:**
\[
A = \int_0^{2\pi} \int_0^1 \left(\frac{1}{4(1-r^2)} + r^2\right) dr d\theta
\]
**Notes:**
- A handwritten note says "doable integral," indicating that the integral can be solved with standard methods.
- There's an expression set for \(\Phi \cdot \mathbf{r} = \langle \cos \theta, \sin \theta, \cdots \rangle\).
The given integral expression is intended to calculate the area of the upper hemisphere by integrating over the region defined by the cylindrical coordinates.
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