Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.8
Obtain the average value of fr,θ,z=z/r2 over R, the region that is bounded below by the paraboloid z=x2+y2 and above by z=4 x. (See Example 8.1.18.)
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∫04cosθ∫r24rcosθz2r dz dr dθ∫02π∫04cosθ∫r24rcosθr dz dr dθ = 12809π16 π = 809
Maple Solution - Interactive
Because the triple integral over R can be iterated in cylindrical coordinates in the order dz dr dθ, the task template in Table 8.2.8(a), implementing the FunctionAverage command from the Student MultivariateCalculus package, can be used.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Average Value≻Cylindrical
Average Value of a Function in Cylindrical Coordinates
Integrand
z/r2
z2r2
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
r2
z2r,θ
4 r cosθ
4rcosθ
r1θ
0
r2θ
4 cosθ
4cosθ
a
b
2 π
2π
Inert Integral: dz dr dθ
(Note automatic insertion of Jacobian.)
StudentMultivariateCalculusFunctionAverage,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=integral
∫02π∫04cosθ∫r24rcosθz2rⅆzⅆrⅆθ∫02π∫04cosθ∫r24rcosθrⅆzⅆrⅆθ
Value
StudentMultivariateCalculusFunctionAverage,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
809
Table 8.2.8(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 8.1.18. To integrate f over R, use the visualization task template in Table 8.2.8(b).
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
Evaluate ∭RΨr,θ,z dv and Graph R
Volume Element dv
r dz dr dθ
r dz dθ dr
r dr dθ dz
r dr dz dθ
r dθ dr dz
r dθ dz dr
, where Ψ=
F=
G=
b=
f=
g=
a=
Table 8.2.8(b) Integration of f over R by visualization task template
Table 8.2.8(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.18
Context Panel: Evaluate and Display Inline
12809π/16 π = 809
Table 8.2.8(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔z/r2:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π,coordinates=cylindricalr,θ,z
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,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π,coordinates=cylindricalr,θ,z
Q≔1280π9
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π,coordinates=cylindricalr,θ,z
V≔16π
Divide Q by V
Q/V = 809
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