Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Example 8.1.9
Use an iterated triple integral to obtain the volume of R, the region enclosed by the cylinders z=4−x2 and z=5 x2, and the planes y=0 and x+y=2/3.
Solution
Mathematical Solution
Figure 8.1.9(a) shows the solid whose volume is obtained by iterating a triple integral in Cartesian coordinates in the order dy dz dx.
∫−2/32/3∫5 x24−x2∫02/3−x1 dy dz dx = 329
Figure 8.1.9(a) The region R
Figure 8.1.9(b) The face y=0
The integration is from the blue face (y=0) to the red face (y=2/3−z); then from the green surface (z=5 x2) to the gold surface (z=4−x2); and finally, from x=−2/3 to x=2/3.
The bounds x=±2/3 are found by intersecting the upper and lower surfaces z=4−x2 and z=5 x2, respectively. (See Figure 8.1.9(b).) This leads to the equation 4−x2=5 x2, and to the solution x=±2/3.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
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
1→MultiInt∫−136136∫5x2−x2+4∫0136−x1ⅆyⅆzⅆx=329
Table 8.1.9(a) provides a solution by a task template that integrates in Cartesian coordinates and draws the region of integration.
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 8.1.9(a) Task template integrating in Cartesian coordinates
Table 8.1.9(b) provides a solution from first principles.
Calculus Palette: Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫−2/32/3∫5 x24−x2∫02/3−x1 ⅆy ⅆz ⅆx = 329
Table 8.1.9(b) Integration via first principles
Maple Solution - Coded
Table 8.1.9(c) obtains a solution via the MultiInt command in the Student MultivariateCalculus package.
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
MultiInt1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3,output=integral
∫−136136∫5x2−x2+4∫0136−x1ⅆyⅆzⅆx
MultiInt1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3 = 329
Table 8.1.9(c) MultiInt command iterating in Cartesian coordinates in the order dy dz dx
Table 8.1.9(d) implements the iterated integration via the top-level Int and int commands.
Int1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3=int1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3
∫−136136∫5x2−x2+4∫0136−x1ⅆyⅆzⅆx=329
Table 8.1.9(d) Top-level Int and int commands
Notice how Maple rewrites the radicals in the limits of integration.
<< Previous Example Section 8.1 Next Example >>
© 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
