Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.6
Obtain the average value of fρ,φ,θ=ρ2 over R, the region that is bounded inside by the surface ρ=1+cosφ and outside by the sphere ρ=2. (The variables ρ,φ,θ are spherical coordinates.)
(See Example 8.1.14.)
Solution
Mathematical Solution
The average value of f over R is defined as ∫∫∫Rf dv∫∫∫R1 dv. For the given values of f and R, obtain
∫02π∫0π∫1+cosφ2ρ4sinφ dρ dφ dθ∫02π∫0π∫1+cosφ2ρ2sinφ dρ dφ dθ = 643 π8 π = 83
Maple Solution - Interactive
Because the triple integral over R can be iterated in spherical coordinates in the order dρ dφ dθ, the task template in Table 8.2.6(a), implementing the FunctionAverage command from the Student MultivariateCalculus package, can be used.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Average Value≻Spherical
Average Value of a Function in Spherical Coordinates
(φ = colatitude, measured down from z-axis)
Integrand
ρ2
ρ2
Region: ρ1φ,θ≤ρ≤ρ2φ,θ,φ1θ≤φ≤φ2θ,a≤θ≤b
ρ1φ,θ
1+cosφ
1+cosφ
ρ2φ,θ
2
φ1θ
0
φ2θ
π
π
a
b
2 π
2π
Inert Integral: dρ dφ dθ
StudentMultivariateCalculusFunctionAverage,ρ=..,φ=..,θ=..,coordinates=sphericalρ,φ,θ,output=integral
∫02π∫0π∫1+cosφ2ρ4sinφⅆρⅆφⅆθ∫02π∫0π∫1+cosφ2ρ2sinφⅆρⅆφⅆθ
Value
StudentMultivariateCalculusFunctionAverage,ρ=..,φ=..,θ=.., coordinates=sphericalρ,φ,θ
83
Table 8.2.6(a) Solution by task template implementing the FunctionAverage command
To implement a solution from first principles, evaluate the integral of f over R and divide by the volume computed in Example 8.1.14. To integrate f over R, use the visualization task template in Table 8.2.6(b).
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Spherical
Evaluate ∭RΨρ,φ,θ dv and Graph R
Volume Element dv=ρ2sinφ×
dρ dφ dθ
dρ dθ dφ
dφ dρ dθ
dφ dθ dρ
dθ dφ dρ
dθ dρ dφ
, where Ψ=
F=
G=
b=
f=
g=
a=
Table 8.2.6(b) Integration of f over R by visualization task template
Table 8.2.6(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.14
Context Panel: Evaluate and Display Inline
643 π/8 π = 83
Table 8.2.6(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔ρ2:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,ρ=1+cosφ..2,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ = 83
From first principles, verify this result by integrating f over R and dividing by V, the volume of R.
Use the MultiInt command to obtain Q, the integral of f over R
Q≔MultiIntf,ρ=1+cosφ..2,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
Q≔64π3
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,ρ=1+cosφ..2,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
V≔8π
Divide Q by V
Q/V = 83
<< Previous Example Section 8.2 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
