Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Use an iterated triple integral to obtain the volume of R, the region that is inside the cylinder x2+4 y2=4, and that is bounded above and below by the planes z=x+2, and z=0, respectively.
Figure 8.1.10(a) shows the solid whose volume is obtained by iterating a triple integral in Cartesian coordinates in the order dz dx dy.
∫−11∫−21−y222−y2∫0x+21 dz dx dy = 4 π
The decision to adopt the order chosen is cosmetic, hinging on the difference in solving for x or y in x2+4 y2=4, the equation of the ellipse giving the cross section of the cylinder. Solving for y would would result in bounds containing 1−x/22 or 4−x2/2.
Figure 8.1.10(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 8.1.10(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 8.1.10(a) Task template integrating in Cartesian coordinates
Table 8.1.10(b) provides a solution from first principles.
Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫−11∫−21−y221−y2∫0x+21 ⅆz ⅆx ⅆy = 4⁢π
Table 8.1.10(b) Integration via first principles
Maple Solution - Coded
Table 8.1.10(c) obtains a solution via the MultiInt command in the Student MultivariateCalculus package.
Install the Student MultivariateCalculus package.
MultiInt1,z=0..x+2,x=−21−y2..21−y2,y=−1..1 = 4⁢π
Table 8.1.10(c) MultiInt command iterating in Cartesian coordinates in the order dy dz dx
Table 8.1.10(d) implements the iterated integration via the top-level Int and int commands.
Table 8.1.10(d) Top-level Int and int commands
<< Previous Example Section 8.1
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. 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
What kind of issue would you like to report? (Optional)