Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.3
Obtain the average value of fr,θ,z=z r2sinθ/6 over R, the region interior to the cylinder x2+y2=9 that is bounded below by the xy-plane, and above by the paraboloid z=x2+y2. (See Example 8.1.3.)
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∫03∫0r2zr3sinθ/6 dz dr dθ∫02π∫03∫0r2rdz dr dθ = 1968316812π = 2438π
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 8.2.3(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 r2sinθ/6
zr2sinθ6
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
0
z2r,θ
r2
r1θ
r2θ
3
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∫03∫0r2zr3sinθ6ⅆzⅆrⅆθ∫02π∫03∫0r2rⅆzⅆrⅆθ
Value
StudentMultivariateCalculusFunctionAverage,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
2438π
Table 8.2.3(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 8.1.3. To integrate f over R, use the visualization task template in Table 8.2.3(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 8.2.3(b) Integration of f over R by visualization task template
Table 8.2.3(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.3
Context Panel: Evaluate and Display Inline
1968316/812π = 2438π
Table 8.2.3(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 r2sinθ/6:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=0..r2,r=0..3,θ=0..2 π,coordinates=cylindricalr,θ,z = 2438π
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..r2,r=0..3,θ=0..2 π,coordinates=cylindricalr,θ,z
Q≔1968316
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=0..r2,r=0..3,θ=0..2 π,coordinates=cylindricalr,θ,z
V≔81π2
Divide Q by V
Q/V = 2438π
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