Set up triple integrals for the volume of the sphere p = 12 in a. spherical, b. cylindrical, and c. rectangular coordinates. a. Write the triple integral in spherical coordinates. Use increasing limits of integration. 8 S S S p² sin dpd de
Set up triple integrals for the volume of the sphere p = 12 in a. spherical, b. cylindrical, and c. rectangular coordinates. a. Write the triple integral in spherical coordinates. Use increasing limits of integration. 8 S S S p² sin dpd de
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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![**Setting Up Triple Integrals for the Volume of a Sphere**
In this section, we will learn how to set up triple integrals for the volume of a sphere with a radius of ρ = 12 in three different coordinate systems: spherical, cylindrical, and rectangular.
### a. Triple Integral in Spherical Coordinates
**Problem Statement:**
Set up the triple integral for the volume of the sphere ρ = 12 using spherical coordinates.
**Solution:**
When using spherical coordinates \((ρ, φ, θ)\), the volume element \(dV\) is given by \(\rho^2 \sin(φ) dρ dφ dθ\). Given the limits for a full sphere:
- \(ρ\) ranges from 0 to 12, as the sphere has a radius of 12.
- \(φ\) ranges from 0 to \(\pi\).
- \(θ\) ranges from 0 to \(2π\).
The triple integral in spherical coordinates is set up as follows:
\[ \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{12} \rho^2 \sin(φ) dρ dφ dθ \]
In this representation:
- The outer integral \(\int_{0}^{2\pi} dθ \) corresponds to the azimuthal angle \(\theta\).
- The middle integral \( \int_{0}^{\pi} \sin(φ) dφ \) corresponds to the polar angle \(\phi\).
- The inner integral \(\int_{0}^{12} \rho^2 dρ\) corresponds to the radial distance \(\rho\).
This setup ensures that the integral captures the entire volume of the sphere using increasing limits of integration as required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1840ebe-b0df-4fe8-9210-d8e3dcfa32cc%2Fb2474480-e201-4885-864c-733ee4759b44%2Fprr18n_processed.png&w=3840&q=75)
Transcribed Image Text:**Setting Up Triple Integrals for the Volume of a Sphere**
In this section, we will learn how to set up triple integrals for the volume of a sphere with a radius of ρ = 12 in three different coordinate systems: spherical, cylindrical, and rectangular.
### a. Triple Integral in Spherical Coordinates
**Problem Statement:**
Set up the triple integral for the volume of the sphere ρ = 12 using spherical coordinates.
**Solution:**
When using spherical coordinates \((ρ, φ, θ)\), the volume element \(dV\) is given by \(\rho^2 \sin(φ) dρ dφ dθ\). Given the limits for a full sphere:
- \(ρ\) ranges from 0 to 12, as the sphere has a radius of 12.
- \(φ\) ranges from 0 to \(\pi\).
- \(θ\) ranges from 0 to \(2π\).
The triple integral in spherical coordinates is set up as follows:
\[ \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{12} \rho^2 \sin(φ) dρ dφ dθ \]
In this representation:
- The outer integral \(\int_{0}^{2\pi} dθ \) corresponds to the azimuthal angle \(\theta\).
- The middle integral \( \int_{0}^{\pi} \sin(φ) dφ \) corresponds to the polar angle \(\phi\).
- The inner integral \(\int_{0}^{12} \rho^2 dρ\) corresponds to the radial distance \(\rho\).
This setup ensures that the integral captures the entire volume of the sphere using increasing limits of integration as required.
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