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Chapter 7: Triple Integration

Section 7.4: Integration in Cylindrical Coordinates

Example 7.4.7

Use cylindrical coordinates to integrate the function $f = 1$ over $R$ , the region above $z = 1 + 3 x^{2} + 3 y^{2}$ , below $2 x + 3 y + z = 11$ , and inside the cylinder $x^{2} + y^{2} = 1$ .

Solution

Mathematical Solution

•	Figure 7.4.7(a) shows the region $R$ . In cylindrical coordinates, the upper surface of $R$ is

$Z (r, θ) = 11 - r (2 \cos (θ) + 3 \sin (θ))$

while the lower surface is $z (r, θ) = 1 + 3 r^{2}$ .

•	The lateral surface is the cylinder whose cross section is a circle of radius 1.

•	The iteration in the order $dz dr d θ$ is found to be

$\int_{0}^{2 π} \int_{0}^{1} \int_{1 + 3 r^{2}}^{Z} r dz dr d θ = \frac{17}{2} π$

Figure 7.4.7(a) The region $R$

Maple Solution - Interactive

In cylindrical coordinates, the upper surface of $R$ is $Z (r, θ)$ , given by

•	Context Panel: Assign Name

$Z = 11 - r (2 \cos (θ) + 3 \sin (θ))$ $\overset{assign}{\to}$

while the lower surface is $z (r, θ) = 1 + 3 r^{2}$ . The lateral surface is the cylinder whose cross section is the circle $r = 1$ .

Table 7.4.7(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 $∭_{R} Ψ (r, θ, z) dv$ 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 7.4.7(a) Task template for iterating a triple integral in cylindrical coordinates

Table 7.4.7(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

$\int_{0}^{2 π} \int_{0}^{1} \int_{1 + 3 r^{2}}^{Z} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$ = $\frac{17}{2} π$

Table 7.4.7(b) Solution from first principles

Table 7.4.7(c) provides a solution by a task template that uses the MultiInt command from the Student MultivariateCalculus package, and iterates in the order $dz dr d θ$ . 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:

$1$

(1)

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

$z_{1} (r, θ)$

>	$1 + 3 r^{2}$

$3 r^{2} + 1$

(2)

$z_{2} (r, θ)$

$Z$

$11 - r (2 \cos (θ) + 3 \sin (θ))$

(3)

$r_{1} (θ)$

$0$

(4)

$r_{2} (θ)$

$1$

(5)

$a$

$0$

(6)

$b$

$2 π$

(7)

Inert Integral: $dz dr d θ$

(Note automatic insertion of Jacobian.)

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

$\int_{0}^{2 π} \int_{0}^{1} \int_{3 r^{2} + 1}^{11 - r (2 \cos (θ) + 3 \sin (θ))} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$

(8)

Value:

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

$\frac{17}{2} π$

(9)

Stepwise Evaluation:

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

$\begin{array}{c} \int_{0}^{2 π} \int_{0}^{1} \int_{3 r^{2} + 1}^{11 - r (2 \cos (θ) + 3 \sin (θ))} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ \\ = & \int_{0}^{2 π} \int_{0}^{1} (\binom{r z}{} | \binom{}{z = 3 r^{2} + 1 .. 11 - r (2 \cos (θ) + 3 \sin (θ))}) &DifferentialD; r &DifferentialD; θ \\ = & \int_{0}^{2 π} \int_{0}^{1} r (10 - r (2 \cos (θ) + 3 \sin (θ)) - 3 r^{2}) &DifferentialD; r &DifferentialD; θ \\ = & \int_{0}^{2 π} (\binom{(- \frac{3 r^{4}}{4} + \frac{(- 2 \cos (θ) - 3 \sin (θ)) r^{3}}{3} + 5 r^{2})}{} | \binom{}{r = 0 .. 1}) &DifferentialD; θ \\ = & \int_{0}^{2 π} (\frac{17}{4} - \frac{2 \cos (θ)}{3} - \sin (θ)) &DifferentialD; θ \\ = & \binom{(\frac{17 θ}{4} - \frac{2 \sin (θ)}{3} + \cos (θ))}{} | \binom{}{θ = 0 .. 2 π} \end{array}$

$\frac{17}{2} π$

(10)

Table 7.4.7(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

$1$ $\overset{MultiInt}{\to}$ $\int_{0}^{2 π} \int_{0}^{1} \int_{3 r^{2} + 1}^{11 - r (2 \cos (θ) + 3 \sin (θ))} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ$ $=$ $\frac{17}{2} π$

Maple Solution - Coded

Initialize

•	Install the Student MultivariateCalculus package.

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

•	Define the upper bound $Z (r, θ)$ .

$Z ≔ 11 - r (2 \cos (θ) + 3 \sin (θ)) &colon;$

Top-level solution using the Int and int commands

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

$\int_{0}^{2 π} \int_{0}^{1} \int_{3 r^{2} + 1}^{11 - r (2 \cos (θ) + 3 \sin (θ))} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ = \frac{17}{2} π$

Solution via the MultiInt command from the Student MultivariateCalculus package

$MultiInt (1, z = 1 + 3 r^{2} .. Z, r = 0 .. 1, θ = 0 .. 2 π, coordinates = cylindrical, output = integral) = MultiInt (1, z = 1 + 3 r^{2} .. Z, r = 0 .. 1, θ = 0 .. 2 π, coordinates = cylindrical)$

$\int_{0}^{2 π} \int_{0}^{1} \int_{3 r^{2} + 1}^{11 - r (2 \cos (θ) + 3 \sin (θ))} r &DifferentialD; z &DifferentialD; r &DifferentialD; θ = \frac{17}{2} π$

$MultiInt (1, z = 1 + 3 r^{2} .. Z, r = 0 .. 1, θ = 0 .. 2 π, coordinates = cylindrical, output = steps)$

$\frac{17}{2} π$

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