Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Example 8.1.18
Use an iterated triple integral to obtain the volume of R, the region that is bounded below by the paraboloid z=x2+y2 and above by z=4 x.
Solution
Mathematical Solution
Figure 8.1.18(a) shows the solid whose volume is obtained by iterating a triple integral in cylindrical coordinates in the order dz dr dθ.
∫02 π∫04 cosθ∫r24 r cosθr dz dr dθ = 16 π
The projection of the region R onto the xy-plane is the circle x2+y2=4 x. Converted to polar coordinates, this becomes r2=4 r cosθ, or r=4 cosθ. It is a circle with radius 2 and center 2,0.
Figure 8.1.18(a) Paraboloid cut by plane
Maple Solution - Interactive
Table 8.1.18(a) provides a solution by a task template that integrates in cylindrical coordinates and draws the region of integration.
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 8.1.18(a) Task template integrating in cylindrical coordinates
Since the iteration order can be taken as dz dr dθ, the task template in Table 8.1.18(a), using the MultiInt command from the Student MultivariateCalculus package, applies.
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
4 r cosθ
4rcosθ
r1θ
0
r2θ
4 cosθ
4cosθ
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∫04cosθ∫r24rcosθrⅆzⅆrⅆθ
Value:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z
16π
Stepwise Evaluation:
StudentMultivariateCalculusMultiInt,z=..,r=..,θ=..,coordinates=cylindricalr,θ,z,output=steps
∫02π∫04cosθ∫r24rcosθrⅆzⅆrⅆθ=∫02π∫04cosθrzz=r2..4rcosθ|rzz=r2..4rcosθⅆrⅆθ=∫02π∫04cosθr4rcosθ−r2ⅆrⅆθ=∫02π−r44+4cosθr33r=0..4cosθ|−r44+4cosθr33r=0..4cosθⅆθ=∫02π64cosθ43ⅆθ=16cosθ3+3cosθ2sinθ3+8θθ=0..2π|16cosθ3+3cosθ2sinθ3+8θθ=0..2π
Table 8.1.18(b) Task template implementing the MultiInt command iterating in the order dz dr dθ
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∫04cosθ∫r24rcosθrⅆzⅆrⅆθ=16π
Table 8.1.18(c) provides a solution from first principles.
Calculus Palette: Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫02 π∫04 cosθ∫r24 r cosθr ⅆz ⅆr ⅆθ = 16π
Table 8.1.18(c) Integration via first principles
Maple Solution - Coded
Table 8.1.18(d) obtains a solution via the MultiInt command in the Student MultivariateCalculus package. See Table 8.1.18(b) for an implementation of the integration via a task template.
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
MultiInt1,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π,coordinates=cylindricalr,θ,z,output=integral
MultiInt1,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π,coordinates=cylindricalr,θ,z = 16π
Table 8.1.18(d) MultiInt command iterating in cylindrical coordinates in the order dz dr dθ
Table 8.1.18(e) implements the iterated integration via the top-level Int and int commands.
Intr,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π=intr,z=r2..4 r cosθ,r=0..4 cosθ,θ=0..2 π
∫02π∫04cosθ∫r24rcosθrⅆzⅆrⅆθ=16π
Table 8.1.18(e) Top-level Int and int commands
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