Find the mass of the portion of a sphere in the first octant with radius 1 and density function ƒ(p,0,0)= e¯º¹. (set-up triple integral then evaluate by hand)
Find the mass of the portion of a sphere in the first octant with radius 1 and density function ƒ(p,0,0)= e¯º¹. (set-up triple integral then evaluate by hand)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
Find the mass of the portion of a sphere in the first octant with radius 1 and density function \( f(\rho, \theta, \phi) = e^{-\rho^3} \). (Set up triple integral then evaluate by hand)
**Explanation:**
To solve this, we will use spherical coordinates. The first octant means all variables (ρ (rho), θ (theta), and φ (phi)) are positive within their respective limits.
1. **Limits of Integration:**
- In spherical coordinates, the limits for a sphere of radius 1 in the first octant are:
- \( \rho \) (radial distance) from 0 to 1
- \( \theta \) (azimuthal angle in xy-plane) from 0 to \(\pi/2\)
- \( \phi \) (polar angle from z-axis) from 0 to \(\pi/2\)
2. **Integral Setup:**
The mass is given by the triple integral over the volume V:
\[
\text{Mass} = \iiint_V f(\rho, \theta, \phi) \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
\]
Plugging the given density function \( f(\rho, \theta, \phi) = e^{-\rho^3} \):
\[
\text{Mass} = \int_0^{\pi/2} \int_0^{\pi/2} \int_0^1 e^{-\rho^3} \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
\]
3. **Evaluation:**
- Simplify the integrals step-by-step:
\[
\int_0^{\pi/2} \int_0^{\pi/2} \int_0^1 e^{-\rho^3} \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
\]
- First, integrate with respect to \( \rho \):
\[
\int_0^1 e^{-\rho^3} \, \rho^2 \, d\r](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14d7b245-5885-4bbf-bbb2-b3ba468485cf%2F2b406ed7-1767-4e24-b244-d51354a7bbf0%2F7gy5t8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the mass of the portion of a sphere in the first octant with radius 1 and density function \( f(\rho, \theta, \phi) = e^{-\rho^3} \). (Set up triple integral then evaluate by hand)
**Explanation:**
To solve this, we will use spherical coordinates. The first octant means all variables (ρ (rho), θ (theta), and φ (phi)) are positive within their respective limits.
1. **Limits of Integration:**
- In spherical coordinates, the limits for a sphere of radius 1 in the first octant are:
- \( \rho \) (radial distance) from 0 to 1
- \( \theta \) (azimuthal angle in xy-plane) from 0 to \(\pi/2\)
- \( \phi \) (polar angle from z-axis) from 0 to \(\pi/2\)
2. **Integral Setup:**
The mass is given by the triple integral over the volume V:
\[
\text{Mass} = \iiint_V f(\rho, \theta, \phi) \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
\]
Plugging the given density function \( f(\rho, \theta, \phi) = e^{-\rho^3} \):
\[
\text{Mass} = \int_0^{\pi/2} \int_0^{\pi/2} \int_0^1 e^{-\rho^3} \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
\]
3. **Evaluation:**
- Simplify the integrals step-by-step:
\[
\int_0^{\pi/2} \int_0^{\pi/2} \int_0^1 e^{-\rho^3} \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi
\]
- First, integrate with respect to \( \rho \):
\[
\int_0^1 e^{-\rho^3} \, \rho^2 \, d\r
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