Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.4
Obtain the average value of fr,θ,z=r z2cosθ/3 over R, the wedge the planes z=y and z=0 cut from the cylinder x2+y2=4. (See Example 8.1.5.)
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
∫0π∫02∫0rsinθr2z2cosθ/3 dz dr dθ∫0π∫02∫0rsinθr dz dr dθ = 8120163 = 243320
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 8.2.4(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
r z2cosθ/3
rz2cosθ3
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
0
z2r,θ
r sinθ
rsinθ
r1θ
r2θ
2
a
b
π
π
Inert Integral: dz dr dθ
(Note automatic insertion of Jacobian.)
StudentMultivariateCalculusFunctionAverage,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=integral
∫0π∫02∫0rsinθr2z2cosθ3ⅆzⅆrⅆθ∫0π∫02∫0rsinθrⅆzⅆrⅆθ
Value
StudentMultivariateCalculusFunctionAverage,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
243320
Table 8.2.4(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 8.1.5. To integrate f over R, use the visualization task template in Table 8.2.4(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 8.2.4(b) Integration of f over R by visualization task template
Table 8.2.4(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.5
Context Panel: Evaluate and Display Inline
8120/163 = 243320
Table 8.2.4(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔r z2cosθ/3:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=0..r sinθ,r=0..2,θ=0.. π,coordinates=cylindricalr,θ,z = 243320
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..r sinθ,r=0..2,θ=0.. π,coordinates=cylindricalr,θ,z
Q≔8120
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=0..r sinθ,r=0..2,θ=0.. π,coordinates=cylindricalr,θ,z
V≔163
Divide Q by V
Q/V = 243320
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