Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.5
Obtain the average value of fx,y,z=5 x2+3 y2+z2 over R, the region common to the two cylinders x2+y2=1 and x2+z2=1. (See Example 8.1.8.)
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
∫−11∫−1−x21−x2∫−1−x21−x25x2+3y2+z2 dz dy dx∫−11∫−1−x21−x2∫−1−x21−x21 dz dy dx = 49645163 = 3115
Maple Solution - Interactive
Because the triple integral over R can be iterated in Cartesian coordinates in the order dz dy dx, the task template in Table 8.2.5(a), implementing the FunctionAverage command from the Student MultivariateCalculus package, can be used.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Average Value≻Cartesian 3-D
Average Value of a Function: fx,y,z
Function
5 x2+3 y2+z2
5x2+3y2+z2
Region: z1x,y≤z≤z2x,y,y1x≤y≤y2x,a≤x≤b
z1x,y
−1−x2
−−x2+1
z2x,y
1−x2
−x2+1
y1x
y2x
a
−1
b
1
Inert integral: dz dy dx
StudentMultivariateCalculusFunctionAverage,z=..,y=..,x=..,output=integral
∫−11∫−−x2+1−x2+1∫−−x2+1−x2+15x2+3y2+z2ⅆzⅆyⅆx∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx
Value
StudentMultivariateCalculusFunctionAverage,z=..,y=..,x=..
3115
Table 8.2.5(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 8.1.8. To integrate f over R, use the visualization task template in Table 8.2.5(b).
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 3-D
Evaluate ∭RΨx,y,z dv and Graph R
Volume Element dv
Select dvdz dy dxdz dx dydx dy dzdx dz dydy dx dzdy dz dx
, where Ψ=
F=
G=
b=
f=
g=
a=
Table 8.2.5(b) Integration of f over R by visualization task template
Table 8.2.5(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.8
Context Panel: Evaluate and Display Inline
49645/163 = 3115
Table 8.2.5(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔5 x2+3 y2+z2:
Define the bounding function g.
g≔1−x2:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=−g..g,y=−g..g,x=−1..1 = 3115
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=−g..g,y=−g..g,x=−1..1
Q≔49645
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=−g..g,y=−g..g,x=−1..1
V≔163
Divide Q by V
Q/V = 3115
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