Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Example 8.1.7
Use an iterated triple integral to obtain the volume of R, the region enclosed by the cylinder y2+4 z2=16 and the planes x=1 and x+y=5.
Solution
Mathematical Solution
Figure 8.1.7(a) shows the solid whose volume is obtained by iterating a triple integral in Cartesian coordinates in the order dx dy dz.
∫−22∫−24−z224−z2∫05−y1 dx dy dz = 40 π
The bounds on y are determined by solving the equation of the cylinder for y=±16−4 z2, and factoring this to y=±24−z2.
Iterating in the order dx dz dy also works, with z=±16−y2/2.
Figure 8.1.7(a) Sliced elliptic cylinder
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Access the MultiInt command via the Context Panel
Z=24−z2→assign
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∫−22∫−2−z2+42−z2+4∫05−y1ⅆxⅆyⅆz=40π
Table 8.1.7(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 8.1.7(a) Task template integrating in Cartesian coordinates
Table 8.1.7(b) provides solutions from first principles.
Calculus Palette: Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫−22∫−24−z224−z2∫05−y1 ⅆx ⅆy ⅆz = 40π
∫−44∫−16−y2/216−y2/2∫05−y1 ⅆx ⅆz ⅆy = 40π
Table 8.1.7(b) Integration via first principles
Maple Solution - Coded
Table 8.1.7(c) obtains a solution via the MultiInt command in the Student MultivariateCalculus package.
Student:-MultivariateCalculus:-MultiInt1,x=0..5−y,y=−24−z2..24−z2,z=−2..2
40π
Table 8.1.7(c) MultiInt command iterating in Cartesian coordinates in the order dy dz dx
Table 8.1.7(d) implements the iterated integration via the top-level Int and int commands.
Int1,x=0..5−y,y=−24−z2..24−z2,z=−2..2=int1,x=0..5−y,y=−24−z2..24−z2,z=−2..2
∫−22∫−2−z2+42−z2+4∫05−y1ⅆxⅆyⅆz=40π
Table 8.1.7(d) Top-level Int and int commands
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