Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Example 8.1.8
Use an iterated triple integral to obtain the volume of R, the region common to the two cylinders x2+y2=1 and x2+z2=1.
Solution
Mathematical Solution
Figure 8.1.8(a) shows the two intersecting cylinders; Figure 8.1.8(b), the actual region R; and Figure 8.1.8(c), a cut-away view in the first octant.
use plottools, plots in EX818:=module() local p1,p2; export p3; p1:=cylinder([0,0,-2],1,4); p2:=rotate(p1,0,Pi/2,0); p3:=display(p1,p2,scaling=constrained,labels=[x,y,z],axes=frame,tickmarks=[5,2,5],lightmodel=light4,orientation=[-75,65,0]); print(p3); end module: end use:
Figure 8.1.8(a) Cylinders
Figure 8.1.8(b) Region R
Figure 8.1.8(c) Cut-away
The volume of R can be found by iterating in the order dz dy dx, with the horizontal cylinder x2+z2=1 solved for z=±1−x2 and the vertical cylinder x2+y2=1 solved for y=±1−x2. As a result, the requisite volume is then
∫−11∫−1−x21−x2∫−1−x21−x21 dz dy dx = 163
Maple Solution - Interactive
Table 8.1.8(a) provides, via a visualization task template, a solution in Cartesian coordinates.
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 8.1.8(a) Solution in Cartesian coordinates via a visualization task template
Table 8.1.8(b) provides, via a visualization task template, a solution in cylindrical coordinates.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
Evaluate ∭RΨr,θ,z dv and Graph R
r dz dr dθ
r dz dθ dr
r dr dθ dz
r dr dz dθ
r dθ dr dz
r dθ dz dr
Table 8.1.8(b) Solution in cylindrical coordinates via a visualization task template
Table 8.1.8(c) provides, via a task template that implements the MultiInt command from the Student MultivariateCalculus package, a solution in Cartesian coordinates.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Multiple Integration≻Cartesian 3-D
Iterated Triple Integrals in Cartesian Coordinates
Integrand:
1
Region: z1x,y≤z≤z2x,y,y1x≤y≤y2x,a≤x≤b
z1x,y
−1−x2
−−x2+1
z2x,y
1−x2
−x2+1
y1x
y2x
a
−1
b
Inert Integral: dz dy dx
StudentMultivariateCalculusMultiInt,z=..,y=..,x=..,output=integral
∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx
Value:
StudentMultivariateCalculusMultiInt,z=..,y=..,x=..
163
Stepwise Evaluation:
StudentMultivariateCalculusMultiInt,z=..,y=..,x=..,output=steps
∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx=∫−11∫−−x2+1−x2+1zz=−−x2+1..−x2+1|zz=−−x2+1..−x2+1ⅆyⅆx=∫−11∫−−x2+1−x2+12−x2+1ⅆyⅆx=∫−112−x2+1yy=−−x2+1..−x2+1|2−x2+1yy=−−x2+1..−x2+1ⅆx=∫−11−4x2+4ⅆx=−43x3+4xx=−1..1|−43x3+4xx=−1..1
Table 8.1.8(c) Task template implementation of MultiInt solution in Cartesian coordinates
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Access the MultiInt command via the Context Panel
c=1−x2→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∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx=163
Table 8.1.8(d) provides, via a task template that implements the MultiInt command from the Student MultivariateCalculus package, a solution in cylindrical coordinates.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical
Iterated Triple Integral in Cylindrical Coordinates
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
−1−r2cos2θ
−1−r2cosθ2
z2r,θ
1−r2cos2θ
1−r2cosθ2
r1θ
0
r2θ
2 π
2π
Inert Integral: dz dr dθ
(Note automatic insertion of Jacobian.)
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=integral
∫02π∫01∫−1−r2cosθ21−r2cosθ2rⅆzⅆrⅆθ
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=steps
∫02π∫01∫−1−r2cosθ21−r2cosθ2rⅆzⅆrⅆθ=∫02π∫01rzz=−1−r2cosθ2..1−r2cosθ2|rzz=−1−r2cosθ2..1−r2cosθ2ⅆrⅆθ=∫02π∫012r1−r2cosθ2ⅆrⅆθ=∫02π−21−r2cosθ2323cosθ2r=0..1|−21−r2cosθ2323cosθ2r=0..1ⅆθ=∫02π−21−cosθ232−13cosθ2ⅆθ=2sinθ3cosθ+2cosθ2−1cosθ2+13cosθsinθ1−cosθ2θ=0..2π|2sinθ3cosθ+2cosθ2−1cosθ2+13cosθsinθ1−cosθ2θ=0..2π
Table 8.1.8(d) Task template implementation of MultiInt solution in cylindrical coordinates
C=1−r2cos2θ→assign
1→MultiInt∫02π∫01∫−1−r2cosθ21−r2cosθ2rⅆzⅆrⅆθ=163
Table 8.1.8(e) provides solutions from first principles: on the left, a solution in Cartesian coordinates; on the right, in cylindrical coordinates.
∫−11∫−1−x21−x2∫−1−x21−x21 ⅆz ⅆy ⅆx = 163
∫02 π∫01∫−1−r2cos2θ1−r2cos2θr ⅆz ⅆr ⅆθ = 163
Table 8.1.8(e) From first principles, solutions in both Cartesian and cylindrical coordinates
Maple Solution - Coded
Table 8.1.8(f) provides, from first principles using the top-level Int and int commands, solutions in Cartesian and cylindrical coordinates.
Int1,z=−1−x2..1−x2,y=−1−x2..1−x2,x=−1..1=int1,z=−1−x2..1−x2,y=−1−x2..1−x2,x=−1..1
∫−11∫−−x2+1−x2+1∫−−x2+1−x2+11ⅆzⅆyⅆx=163
Intr,z=−1−r2cos2θ..1−r2cos2θ,r=0..1,θ=0..2 π=intr,z=−1−r2cos2θ..1−r2cos2θ,r=0..1,θ=0..2 π
∫02π∫01∫−1−r2cosθ21−r2cosθ2rⅆzⅆrⅆθ=163
Table 8.1.8(f) From first principles, solutions in Cartesian and cylindrical coordinates.
Table 8.1.8(g) demonstrates the syntax applying the MultiInt command in both Cartesian and cylindrical coordinates.
Install the Student MultivariateCalculus package.
