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
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