Chapter 7: Triple Integration
Section 7.4: Integration in Cylindrical Coordinates
Example 7.4.6
Use cylindrical coordinates to integrate the function f=1 over R, the region outside the cylinder x2+y2=1, bounded above by z=9−x2−y2, and below by z=0.
Solution
Mathematical Solution
Figure 7.4.6(a) shows the region R. In cylindrical coordinates, the upper surface of R is
Zr,θ=9−r2
while the lower surface is zr,θ= 0.
The two surfaces meet in a circle of radius 3 in the plane z=0.
The iteration in the order dz dr dθ is found to be
∫02 π∫13∫ 09−r2r dz dr dθ=32 π
Figure 7.4.6(a) The region R
Maple Solution - Interactive
In cylindrical coordinates, the upper surface of R is Zr,θ=9−r2, while the lower surface is zr,θ= 0. The two surfaces meet in a circle of radius 3 in the plane z=0. The region outside the cylinder but inside the upper surface extends from r=1 to r=3.
Table 7.4.6(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 7.4.6(a) Task template for iterating a triple integral in cylindrical coordinates
Constrained scaling is applied to the graph via its Context Panel.
Table 7.4.6(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∫13∫ 09−r2r ⅆz ⅆr ⅆθ = 32π
Table 7.4.6(b) Solution from first principles
Table 7.4.6(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
0
z2r,θ
9−r2
−r2+9
r1θ
r2θ
3
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∫13∫0−r2+9rⅆzⅆrⅆθ
Value:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
32π
Stepwise Evaluation:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=steps
∫02π∫13∫0−r2+9rⅆzⅆrⅆθ=∫02π∫13rzz=0..−r2+9|rzz=0..−r2+9ⅆrⅆθ=∫02π∫13r−r2+9ⅆrⅆθ=∫02π−14r4+92r2r=1..3|−14r4+92r2r=1..3ⅆθ=∫02π16ⅆθ=16θθ=0..2π|16θθ=0..2π
Table 7.4.6(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∫13∫0−r2+9rⅆzⅆrⅆθ=32π
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Top-level solution using the Int and int commands
Intr,z= 0..9−r2,r=1..3,θ=0..2 π=intr,z= 0..9−r2,r=1..3,θ=0..2 π
∫02π∫13∫0−r2+9rⅆzⅆrⅆθ=32π
Solution via the MultiInt command from the Student MultivariateCalculus package
MultiInt1,z= 0..9−r2,r=1..3,θ=0..2 π,coordinates=cylindrical,output=integral=MultiInt1,z= 0..9−r2,r=1..3,θ=0..2 π,coordinates=cylindrical
MultiInt1,z= 0..9−r2,r=1..3,θ=0..2 π,coordinates=cylindrical,output=steps
<< Previous Example Section 7.4 Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. 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
