Chapter 7: Triple Integration
Section 7.4: Integration in Cylindrical Coordinates
Example 7.4.3
Use cylindrical coordinates to integrate the function f=1 over R, the region bounded below by z=x2+y2, above by z=1+x2+y2, and laterally by the cylinder x2+y2=1.
Solution
Mathematical Solution
Figure 7.4.3(a) shows the region R. In cylindrical coordinates, the upper surface of R is
Zr,θ=1+ r2
while the lower surface is zr,θ=r2.
The iteration in the order dz dr dθ is found to be
∫02 π∫01∫r21+r2r dz dr dθ=π
Figure 7.4.3(a) The region R
Maple Solution - Interactive
In cylindrical coordinates, the upper surface of R is Z=1+r2, while the lower surface is z=r2.
Table 7.4.3(a) provides a solution by a visualization task template. After selecting the order of iteration, the integrand and the fields for the limits of integration are given. The resulting value of the integral and a graph of the region of integration are generated by pressing the appropriate buttons.
Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cylindrical
Evaluate ∭RΨr,θ,z dv and Graph R
Volume Element dv
r dz dr dθ
r dz dθ dr
r dr dθ dz
r dr dz dθ
r dθ dr dz
r dθ dz dr
, where Ψ=
F=
G=
b=
f=
g=
a=
Table 7.4.3(a) Task template for iterating a triple integral in cylindrical coordinates
Table 7.4.3(b) provides a solution from first principles. Note that the Jacobian, r, must be inserted into the integrand.
Calculus palette: Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫02 π∫01∫r21+r2r ⅆz ⅆr ⅆθ = π
Table 7.4.3(b) Solution from first principles
Table 7.4.3(c) provides a solution by a task template that uses the MultiInt command from the Student MultivariateCalculus package, and iterates in the order dz dr dθ. The command takes a coordinate option, and hence, inserts the appropriate Jacobian automatically.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Integration≻Multiple Integration≻Cylindrical
Iterated Triple Integral in Cylindrical Coordinates
Integrand:
1
Region: z1r,θ≤z≤z2r,θ,r1θ≤r≤r2θ,a≤θ≤b
z1r,θ
r2
z2r,θ
1+r2
r2+1
r1θ
0
r2θ
a
b
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∫r2r2+1rⅆzⅆrⅆθ
Value:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
π
Stepwise Evaluation:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=steps
∫02π∫01∫r2r2+1rⅆzⅆrⅆθ=∫02π∫01rzz=r2..r2+1|rzz=r2..r2+1ⅆrⅆθ=∫02π∫01rⅆrⅆθ=∫02πr22r=0..1|r22r=0..1ⅆθ=∫02π12ⅆθ=θ2θ=0..2π|θ2θ=0..2π
Table 7.4.3(c) Solution by task template that implements the MultiInt command
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∫02π∫01∫r2r2+1rⅆzⅆrⅆθ=π
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Top-level solution using the Int and int commands
Intr,z=r2..1+ r2,r=0..1,θ=0..2 π=intr,z=r2..1+r2,r=0..1,θ=0..2 π
∫02π∫01∫r2r2+1rⅆzⅆrⅆθ=π
Solution via the MultiInt command from the Student MultivariateCalculus package
MultiInt1,z=r2..1+ r2,r=0..1,θ=0..2 π,coordinates=cylindrical,output=integral=MultiInt1,z=r2..1+ r2,r=0..1,θ=0..2 π,coordinates=cylindrical
MultiInt1,z=r2..1+ r2,r=0..1,θ=0..2 π,coordinates=cylindrical,output=steps
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