Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.12
Obtain the average value of fρ,φ,θ=ρ3/2 over R, the region interior to the surface ρ=2 sinφ, where ρ,φ,θ are the variables of spherical coordinates. (See Example 8.1.26.)
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∫02sinφρ7/2sinφ dρ dφ dθ∫02π∫0π∫02sinφρ2sinφ dρ dφ dθ
=1280231EllipticK12π2 π2
=640231 πEllipticK12
≐1.6352
The EllipticK function is one of the special functions of applied mathematics. Known to Maple, it is typically studied in a course in complex variables.
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 8.2.12(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
ρ3/2
ρ32
Region: ρ1φ,θ≤ρ≤ρ2φ,θ,φ1θ≤φ≤φ2θ,a≤θ≤b
ρ1φ,θ
0
ρ2φ,θ
2 sinφ
2sinφ
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∫02sinφρ72sinφⅆρⅆφⅆθ∫02π∫0π∫02sinφρ2sinφⅆρⅆφⅆθ
Value
StudentMultivariateCalculusFunctionAverage,ρ=..,φ=..,θ=.., coordinates=sphericalρ,φ,θ
640EllipticK22231π
Table 8.2.12(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 8.1.26. To integrate f over R, use the visualization task template in Table 8.2.12(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 8.2.12(b) Integration of f over R by visualization task template
Table 8.2.12(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.26
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
1280231EllipticK122π/2 π2 = 640EllipticK22231π→at 5 digits1.6352
Table 8.2.12(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔ρ3/2:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,ρ=0..2 sinφ,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..2 sinφ,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
Q≔1280EllipticK22π231
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,ρ=0..2 sinφ,φ=0..π,θ=0..2 π,coordinates=sphericalρ,φ,θ
V≔2π2
Divide Q by V and obtain a floating=point approximation via the evalf command
Q/V = 640EllipticK22231π
evalfQ/V = 1.635103860
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