Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.13
Obtain the average value of fx,y,z=2 x2+3 y2+4 z2 over R, the first-octant region enclosed by the cylinder x2+z2=4 and the plane y=3. (See Example 8.1.27.)
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
∫02∫04∫04−x22x2+3y2+4z2 dz dy dx∫02∫04∫04−x21 dz dy dx = 88 π4 π = 22
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 8.2.13(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
2 x2+3 y2+4 z2
2x2+3y2+4z2
Region: z1x,y≤z≤z2x,y,y1x≤y≤y2x,a≤x≤b
z1x,y
0
z2x,y
4−x2
−x2+4
y1x
y2x
4
a
b
2
Inert integral: dz dy dx
StudentMultivariateCalculusFunctionAverage,z=..,y=..,x=..,output=integral
∫02∫04∫0−x2+42x2+3y2+4z2ⅆzⅆyⅆx∫02∫04∫0−x2+41ⅆzⅆyⅆx
Value
StudentMultivariateCalculusFunctionAverage,z=..,y=..,x=..
22
Table 8.2.13(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 8.1.27. To integrate f over R, use the visualization task template in Table 8.2.13(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 8.2.13(b) Integration of f over R by visualization task template
Table 8.2.13(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.27
Context Panel: Evaluate and Display Inline
88 π/4 π = 22
Table 8.2.13(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔2 x2+3 y2+4 z2:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=0..4−x2,y=0..4,x=0..2 = 22
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−x2,y=0..4,x=0..2
88π
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=0..4−x2,y=0..4,x=0..2
4π
Divide Q by V
Q/V = 22
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