Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Use an iterated triple integral to obtain the volume of R, the first-octant region that is bounded by the coordinate planes, and the additional planes x=1, x+y+z=2.
Figure 8.1.29(a) shows the solid whose volume is obtained by iterating a triple integral in Cartesian coordinates in the order dy dz dx.
∫01∫02−x∫02−x−y1 dz dy dx = 76
Figure 8.1.29(a) The region R
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Access the MultiInt command via the Context Panel
Type the integrand, 1.
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in the fields of the two dialogs shown below.
Context Panel: Evaluate Integral
Table 8.1.29(a) provides a solution by a task template that integrates in Cartesian coordinates and draws the region of integration.
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=
Table 8.1.29(a) Task template integrating in Cartesian coordinates
Table 8.1.29(b) provides a solution from first principles.
Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫01∫02−x∫02−x−y1 ⅆz ⅆy ⅆx = 76
Table 8.1.29(b) Integration via first principles
Maple Solution - Coded
Table 8.1.29(c) obtains a solution via the MultiInt command in the Student MultivariateCalculus package.
Install the Student MultivariateCalculus package.
MultiInt1,z=0..2−x−y,y=0..2−x,x=0..1 = 76
Table 8.1.29(c) MultiInt command iterating in Cartesian coordinates in the order dy dz dx
Table 8.1.29(d) implements the iterated integration via the top-level Int and int commands.
Table 8.1.29(d) Top-level Int and int commands
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