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Chapter 8: Applications of Triple Integration

Section 8.1: Volume

Example 8.1.14

Use an iterated triple integral to obtain the volume of $R$ , the region that is bounded inside by the surface $ρ = 1 + \cos (φ)$ and outside by the sphere $ρ = 2$ . (The variables $(ρ, φ, θ)$ are spherical coordinates.)

Solution

Mathematical Solution

•	Figure 8.1.14(a) shows the region whose volume is obtained by iterating a triple integral in spherical coordinates in the order $d ρ d φ d θ$ .

$\int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ)$ $d ρ d φ d θ$ = $8 π$

•	The sphere in Figure 8.1.14(a) has been drawn with a cut-away so that the inner surface $ρ = 1 + \cos (φ)$ is properly visible.

use plots in
module()
local p1,p2,p3;
p1:=plot3d(2,theta=0..11*Pi/8,phi=0..Pi,coords=spherical);
p2:=plot3d(1+cos(phi),theta=0 ..2*Pi,phi=0..Pi,coords=spherical,color=red);
p3:=display(p1,p2,scaling=constrained,labels=[x,y,z], orientation=[-65,85,0],axes=frame,tickmarks=[3,3,5],lightmodel=light3);
print(p3);
end module:
end use:

Figure 8.1.14(a) Surface inside sphere

Maple Solution - Interactive

Table 8.1.14(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≻Spherical

Evaluate $∭_{R} Ψ (ρ, φ, θ) dv$ and Graph $R$

Volume Element $dv = ρ^{2} \sin (φ) \times$

$d ρ d φ d θ$

$d ρ d θ d φ$

$d φ d ρ d θ$

$d φ d θ d ρ$

$d θ d φ d ρ$

$d θ d ρ d φ$

, where $Ψ =$

$F =$	$G =$	$b =$
$f =$	$g =$	$a =$

Table 8.1.14(a) Task template integrating in spherical coordinates

(The "tail" attached to the graph of $ρ = 1 + \cos (φ)$ is an artifact of how Maple composed the graphs of the surfaces. Transparency has been applied to the graph of the sphere via the Context Panel for the graph drawn by the task template. Unfortunately, this task template provided no other means of modifying the graph.)

Since the iteration order can be taken as $d ρ d φ d θ$ , the task template in Table 8.1.14(a), using the MultiInt command from the Student MultivariateCalculus package, applies.

Tools≻Tasks≻Browse:
Calculus - Multivariate≻Integration≻Multiple Integration≻Spherical

Iterated Triple Integral in Spherical Coordinates

( $φ$ = colatitude, measured down from $z$ -axis)

Integrand:

$1$

(1)

Region: $\{ρ_{1} (φ, θ) \leq ρ \leq ρ_{2} (φ, θ), φ_{1} (θ) \leq φ \leq φ_{2} (θ), a \leq θ \leq b\}$

$ρ_{1} (φ, θ)$

>	$1 + \cos (φ)$

$1 + \cos (φ)$

(2)

$ρ_{2} (φ, θ)$

$2$

(3)

$φ_{1} (θ)$

$0$

(4)

$φ_{2} (θ)$

$π$

(5)

$a$

$0$

(6)

$b$

$2 π$

(7)

Inert Integral: $d ρ d φ d θ$

>	$Student [MultivariateCalculus] [MultiInt] (, ρ = .., φ = .., θ = .., coordinates = spherical [ρ, φ, θ], output = integral)$

$\int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ$

(8)

Value:

>	$Student [MultivariateCalculus] [MultiInt] (, ρ = .., φ = .., θ = .., coordinates = spherical [ρ, φ, θ])$

$8 π$

(9)

Stepwise Evaluation:

>	$Student [MultivariateCalculus] [MultiInt] (, ρ = .., φ = .., θ = .., coordinates = spherical [ρ, φ, θ], output = steps)$

$\begin{array}{c} \int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ \\ = & \int_{0}^{2 π} \int_{0}^{π} (\binom{\frac{ρ^{3} \sin (φ)}{3}}{} | \binom{}{ρ = 1 + \cos (φ) .. 2}) &DifferentialD; φ &DifferentialD; θ \\ = & \int_{0}^{2 π} \int_{0}^{π} \frac{\sin (φ) (8 - {(1 + \cos (φ))}^{3})}{3} &DifferentialD; φ &DifferentialD; θ \\ = & \int_{0}^{2 π} (\binom{(- \frac{8 \cos (φ)}{3} + \frac{{(1 + \cos (φ))}^{4}}{12})}{} | \binom{}{φ = 0 .. π}) &DifferentialD; θ \\ = & \int_{0}^{2 π} 4 &DifferentialD; θ \\ = & \binom{4 θ}{} | \binom{}{θ = 0 .. 2 π} \end{array}$

$8 π$

(10)

Table 8.1.14(b) Task template implementing the MultiInt command iterating in the order $d ρ d φ d θ$

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

$1$ $\overset{MultiInt}{\to}$ $\int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ$ $=$ $8 π$

Table 8.1.14(c) provides a solution from first principles.

•	Calculus Palette: Iterated triple-integral template

•	Context Panel: Evaluate and Display Inline

$\int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ$ = $8 π$

Table 8.1.14(c) Integration via first principles

Maple Solution - Coded

Table 8.1.14(d) obtains a solution via the MultiInt command in the Student MultivariateCalculus package. See Table 8.1.14(b) for an implementation of this command via a task template.

•	Install the Student MultivariateCalculus package.

$with (Student :- MultivariateCalculus) &colon;$

$MultiInt (1, ρ = 1 + \cos (φ) .. 2, φ = 0 .. π, θ = 0 .. 2 π, coordinates = spherical [ρ, φ, θ], output = integral)$

$\int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ$

$MultiInt (1, ρ = 1 + \cos (φ) .. 2, φ = 0 .. π, θ = 0 .. 2 π, coordinates = spherical [ρ, φ, θ])$ = $8 π$

Table 8.1.14(d) MultiInt command iterating in spherical coordinates in the order $d ρ d φ d θ$

Table 8.1.14(e) implements the iterated integration via the top-level Int and int commands.

$Int (ρ^{2} \sin (φ), [ρ = 1 + \cos (φ) .. 2, φ = 0 .. π, θ = 0 .. 2 π]) = int (ρ^{2} \sin (φ), [ρ = 1 + \cos (φ) .. 2, φ = 0 .. π, θ = 0 .. 2 π])$

$\int_{0}^{2 π} \int_{0}^{π} \int_{1 + \cos (φ)}^{2} ρ^{2} \sin (φ) &DifferentialD; ρ &DifferentialD; φ &DifferentialD; θ = 8 π$

Table 8.1.14(e) Top-level Int and int commands

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