Set Pegion up in 2 2 the indegual 929 -√x²+y²+z² dv over the J. the where 2 ≤ √ 4-3²²-4² SE 1⁰k Octant 2
Set Pegion up in 2 2 the indegual 929 -√x²+y²+z² dv over the J. the where 2 ≤ √ 4-3²²-4² SE 1⁰k Octant 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
![**Setting Up the Integral**
Objective: Set up the integral \(\iiint \sqrt{x^2 + y^2 + z^2} \, dV\) over the region in the first octant where \(z \leq \sqrt{4 - x^2 - y^2}\).
**Integration limits for four possible integrals (Pick One):**
1. **Integral (i):**
\[
\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{2} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
2. **Integral (ii):**
\[
\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{4} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
3. **Integral (iii):**
\[
\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{\sqrt{2}} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
4. **Integral (iv):**
\[
\int_{0}^{\pi/2} \int_{0}^{\pi/2} \int_{0}^{2} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
Each integral is expressed in spherical coordinates, where \(\rho\) is the radial distance, \(\phi\) is the polar angle, and \(\theta\) is the azimuthal angle. The function \(\rho^3 \sin(\phi)\) appears in each integrand, reflecting the conversion from Cartesian to spherical coordinates.
**Instructions:**
- Choose one of the provided integrals that best describes the region described in the objective.
- Evaluate the integral to find the volume of the specified region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F140bd83b-645e-45c3-8644-e53276eab044%2F0d31eac2-4cb7-47c9-8e81-cafa6aae8ad6%2Fmifiywe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Setting Up the Integral**
Objective: Set up the integral \(\iiint \sqrt{x^2 + y^2 + z^2} \, dV\) over the region in the first octant where \(z \leq \sqrt{4 - x^2 - y^2}\).
**Integration limits for four possible integrals (Pick One):**
1. **Integral (i):**
\[
\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{2} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
2. **Integral (ii):**
\[
\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{4} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
3. **Integral (iii):**
\[
\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{\sqrt{2}} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
4. **Integral (iv):**
\[
\int_{0}^{\pi/2} \int_{0}^{\pi/2} \int_{0}^{2} \rho^3 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
Each integral is expressed in spherical coordinates, where \(\rho\) is the radial distance, \(\phi\) is the polar angle, and \(\theta\) is the azimuthal angle. The function \(\rho^3 \sin(\phi)\) appears in each integrand, reflecting the conversion from Cartesian to spherical coordinates.
**Instructions:**
- Choose one of the provided integrals that best describes the region described in the objective.
- Evaluate the integral to find the volume of the specified region.
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