Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.10
Obtain the average value of fρ,φ,θ=ρ over R, the region enclosed by the surface ρ=1−cosφ, where ρ,φ,θ are the variables in spherical coordinates. (See Example 8.1.21.)
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∫01−cosφρ5/2sinφ dρ dφ dθ∫02π∫0π∫01−cosφρ2sinφ dρ dφ dθ = 128632π83 π = 1621 2
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 8.2.10(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
ρ
ρ
Region: ρ1φ,θ≤ρ≤ρ2φ,θ,φ1θ≤φ≤φ2θ,a≤θ≤b
ρ1φ,θ
0
ρ2φ,θ
1−cosφ
1−cosφ
φ1θ
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∫01−cosφρ52sinφⅆρⅆφⅆθ∫02π∫0π∫01−cosφρ2sinφⅆρⅆφⅆθ
Value
StudentMultivariateCalculusFunctionAverage,ρ=..,φ=..,θ=.., coordinates=sphericalρ,φ,θ
16221
Table 8.2.10(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 8.1.21. To integrate f over R, use the visualization task template in Table 8.2.10(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 8.2.10(b) Integration of f over R by visualization task template
Table 8.2.10(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.21
Context Panel: Evaluate and Display Inline
128632π/83 π = 16221
Table 8.2.10(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔ρ:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,ρ=0..1−cosφ,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
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=0..1−cosφ,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
Q≔1282π63
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,ρ=0..1−cosφ,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
V≔8π3
Divide Q by V
Q/V = 16221
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