Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.14
Obtain the average value of fr,θ,z=z2r sinθ/3 over R, the region bounded by the paraboloid z=4−x2−y2 and the plane z=0. (See Example 8.1.28.)
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∫02∫04−r2z2r2sinθ/3 dz dr dθ∫02π∫02∫04−r2r dz dr dθ = 40961058 π = 512105π
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 8.2.14(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
z2r sinθ/3
z2rsinθ3
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
0
z2r,θ
4−r2
−r2+4
r1θ
r2θ
2
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∫02∫0−r2+4z2r2sinθ3ⅆzⅆrⅆθ∫02π∫02∫0−r2+4rⅆzⅆrⅆθ
Value
StudentMultivariateCalculusFunctionAverage,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
512105π
Table 8.2.14(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 8.1.28. To integrate f over R, use the visualization task template in Table 8.2.14(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 8.2.14(b) Integration of f over R by visualization task template
Table 8.2.14(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.28
Context Panel: Evaluate and Display Inline
4096105/8 π = 512105π
Table 8.2.14(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔z2r sinθ/3:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=0..4−r2,r=0..2,θ=0..2 π,coordinates=cylindricalr,θ,z = 512105π
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=0..4−r2,r=0..2,θ=0..2 π,coordinates=cylindricalr,θ,z
Q≔4096105
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=0..4−r2,r=0..2,θ=0..2 π,coordinates=cylindricalr,θ,z
V≔8π
Divide Q by V
Q/V = 512105π
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