Chapter 8: Applications of Triple Integration
Section 8.2: Average Value
Example 8.2.15
Obtain the average value of fx,y,z=x y2z3 over R, the first-octant region that is bounded by the coordinate planes, and the additional planes x=1, x+y+z=2. (See Example 8.1.29.)
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
∫01∫02−x∫02−x−yxy2z3 dz dy dx∫01∫02−x∫02−x−y1 dz dy dx = 2511512076 = 25117640
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 8.2.15(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
x y2z3
xy2z3
Region: z1x,y≤z≤z2x,y,y1x≤y≤y2x,a≤x≤b
z1x,y
0
z2x,y
2−x−y
y1x
y2x
2−x
a
b
1
Inert integral: dz dy dx
StudentMultivariateCalculusFunctionAverage,z=..,y=..,x=..,output=integral
∫01∫02−x∫02−x−yxy2z3ⅆzⅆyⅆx∫01∫02−x∫02−x−y1ⅆzⅆyⅆx
Value
StudentMultivariateCalculusFunctionAverage,z=..,y=..,x=..
25117640
Table 8.2.15(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 8.1.29. To integrate f over R, use the visualization task template in Table 8.2.15(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 8.2.15(b) Integration of f over R by visualization task template
Table 8.2.15(c) completes the solution from first principles.
Copy and paste the value of ∫∫∫Rf dv
Divide by the volume of R from Example 8.1.29
Context Panel: Evaluate and Display Inline
25115120/76 = 25117640
Table 8.2.15(c) Completion of the solution from first principles
Maple Solution - Coded
Initialize
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define the function f.
f≔x y2 z3:
Apply the FunctionAverage command from the Student MultivariateCalculus package
FunctionAveragef,z=0..2−x−y,y=0..2−x,x=0..1 = 25117640
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..2−x−y,y=0..2−x,x=0..1
Q≔25115120
Use the MultiInt command to obtain V, the volume of R
V≔MultiInt1,z=0..2−x−y,y=0..2−x,x=0..1
V≔76
Divide Q by V
Q/V = 25117640
<< Previous Example Section 8.2 Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
