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Chapter 8: Applications of Triple Integration

Section 8.1: Volume

Example 8.1.19

Use an iterated triple integral to obtain the volume of $R$ , the first-octant region that lies between the cylinders $r = 1$ and $r = 3$ , and that is bounded below by the plane $z = 0$ and above, by the surface $z = 1 + x y$ .

Solution

Mathematical Solution

•	Figure 8.1.19(a) shows the solid whose volume is obtained by iterating a triple integral in cylindrical coordinates in the order $dz dr d θ$ .

$\int_{0}^{π / 2} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r dz dr d θ$ = $10 + 2 π$

Figure 8.1.19(a) Region $R$

Maple Solution - Interactive

Table 8.1.19(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 $∭_{R} Ψ (r, θ, z) dv$ and Graph $R$

Volume Element $dv$

$r dz dr d θ$

$r dz d θ dr$

$r dr d θ dz$

$r dr dz d θ$

$r d θ dr dz$

$r d θ dz dr$

, where $Ψ =$

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

Table 8.1.19(a) Task template integrating in cylindrical coordinates

Since the iteration order can be taken as $dz dr d θ$ , the task template in Table 8.1.19(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:

$1$

(1)

Region: $\{z_{1} (r, θ) \leq z \leq z_{2} (r, θ), r_{1} (θ) \leq r \leq r_{2} (θ), a \leq θ \leq b\}$

$z_{1} (r, θ)$

$0$

(2)

$z_{2} (r, θ)$

>	$1 + r^{2} \cos (θ) \sin (θ)$

$1 + r^{2} \cos (θ) \sin (θ)$

(3)

$r_{1} (θ)$

$1$

(4)

$r_{2} (θ)$

$3$

(5)

$a$

$0$

(6)

$b$

$π / 2$

$\frac{π}{2}$

(7)

Inert Integral: $dz dr d θ$

(Note automatic insertion of Jacobian.)

>	$Student [MultivariateCalculus] [MultiInt] (, z = .., r = .., θ = .., coordinates = cylindrical [r, θ, z], output = integral)$

$\int_{0}^{\frac{π}{2}} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$

(8)

Value:

>	$Student [MultivariateCalculus] [MultiInt] (, z = .., r = .., θ = .., coordinates = cylindrical [r, θ, z])$

$2 π + 10$

(9)

Stepwise Evaluation:

>	$Student [MultivariateCalculus] [MultiInt] (, z = .., r = .., θ = .., coordinates = cylindrical [r, θ, z], output = steps)$

$\begin{array}{c} \int_{0}^{\frac{π}{2}} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ \\ = & \int_{0}^{\frac{π}{2}} \int_{1}^{3} (\binom{r z}{} | \binom{}{z = 0 .. 1 + r^{2} \cos (θ) \sin (θ)}) &DifferentialD; r &DifferentialD; θ \\ = & \int_{0}^{\frac{π}{2}} \int_{1}^{3} r (1 + r^{2} \cos (θ) \sin (θ)) &DifferentialD; r &DifferentialD; θ \\ = & \int_{0}^{\frac{π}{2}} (\binom{(\frac{\cos (θ) \sin (θ) r^{4}}{4} + \frac{r^{2}}{2})}{} | \binom{}{r = 1 .. 3}) &DifferentialD; θ \\ = & \int_{0}^{\frac{π}{2}} (20 \cos (θ) \sin (θ) + 4) &DifferentialD; θ \\ = & \binom{(4 θ + 10 {\sin (θ)}^{2})}{} | \binom{}{θ = 0 .. \frac{π}{2}} \end{array}$

$2 π + 10$

(10)

Table 8.1.19(b) Task template implementing the MultiInt command iterating in the order $dz dr d θ$

Initialize

•	Tools≻Load Package: Student Multivariate Calculus

Loading Student:-MultivariateCalculus

Access the MultiInt command via the Context Panel

$C = 1 + r^{2} \cos (θ) \sin (θ)$ $\overset{assign}{\to}$

•	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

$1$ $\overset{MultiInt}{\to}$ $\int_{0}^{\frac{π}{2}} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$ $=$ $2 π + 10$

Table 8.1.19(c) provides a solution from first principles.

•	Calculus Palette: Iterated triple-integral template

•	Context Panel: Evaluate and Display Inline

$\int_{0}^{π / 2} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$ = $2 π + 10$

Table 8.1.19(c) Integration via first principles

Maple Solution - Coded

Table 8.1.19(d) obtains a solution via the MultiInt command in the Student MultivariateCalculus package. See Table 8.1.19(b) for an implementation of the integration via a task template.

•	Install the Student MultivariateCalculus package.

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

$MultiInt (1, z = 0 .. 1 + r^{2} \cos (θ) \sin (θ), r = 1 .. 3, θ = 0 .. π / 2, coordinates = cylindrical [r, θ, z], output = integral)$

$\int_{0}^{\frac{π}{2}} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$

$MultiInt (1, z = 0 .. 1 + r^{2} \cos (θ) \sin (θ), r = 1 .. 3, θ = 0 .. π / 2, coordinates = cylindrical [r, θ, z])$ =

$2 π + 10$

Table 8.1.19(d) MultiInt command iterating in cylindrical coordinates in the order $dz dr d θ$

Table 8.1.19(e) implements the iterated integration via the top-level Int and int commands.

$Int (r, [z = 0 .. 1 + r^{2} \cos (θ) \sin (θ), r = 1 .. 3, θ = 0 .. π / 2]) = int (r, [z = 0 .. 1 + r^{2} \cos (θ) \sin (θ), r = 1 .. 3, θ = 0 .. π / 2])$

$\int_{0}^{\frac{π}{2}} \int_{1}^{3} \int_{0}^{1 + r^{2} \cos (θ) \sin (θ)} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ = 2 π + 10$

Table 8.1.19(e) Top-level Int and int commands

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