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 enclosed by the cylinder x2+z2=4 and the plane y=3.
Figure 8.1.27(a) shows the solid whose volume is obtained by iterating a triple integral in Cartesian coordinates in the order dz dy dx.
∫02∫04∫04−x21 dz dy dx = 4 π
use plottools, plots in
Figure 8.1.27(a) 1st-octant portion of cylinder
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 8.1.27(a) provides a solution by a task template that integrates in Cartesian coordinates and draws the region of integration. (Unless the worksheet has been re-executed, the content of the table is a template in its end state. Clicking on any of its buttons will be futile unless it is re-initialized by re-selecting the Volume Element.)
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 8.1.27(a) Task template integrating in Cartesian coordinates
The cylinder drawn by the task template is defective because of the radical defining the upper surface (in red). Sample points are not taken sufficiently near x=2 where z=0. Hence, the gap shown in the graph.
Table 8.1.27(b) provides a solution from first principles.
Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫02∫04∫04−x21 ⅆz ⅆy ⅆx = 4⁢π
Table 8.1.27(b) Integration via first principles
Maple Solution - Coded
Table 8.1.27(c) obtains a solution via the MultiInt command in the Student MultivariateCalculus package.
Install the Student MultivariateCalculus package.
MultiInt1,z=0..4−x2,y=0..4,x=0..2 = 4⁢π
Table 8.1.27(c) MultiInt command iterating in Cartesian coordinates in the order dy dz dx
Table 8.1.27(d) implements the iterated integration via the top-level Int and int commands.
Table 8.1.27(d) Top-level Int and int commands
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