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Learning Objectives

By the end of this section, you will be able to:

Understand cylindrical coordinates $(r, \theta, z)$ and their relationship to Cartesian coordinates $(x, y, z)$
Explain why the volume element is $dV = r\,dr\,d\theta\,dz$ using both geometric intuition and the Jacobian determinant
Identify three-dimensional regions that are naturally suited for cylindrical coordinates (cylinders, cones, paraboloids)
Set up appropriate integration bounds for various cylindrical regions
Evaluate triple integrals in cylindrical coordinates using $\iiint_E f(x,y,z)\,dV = \iiint_E f(r\cos\theta, r\sin\theta, z)\,r\,dr\,d\theta\,dz$
Apply cylindrical coordinates to compute volumes, masses, and other physical quantities

The Big Picture: Extending Polar to 3D

"Cylindrical coordinates are the natural extension of polar coordinates to three dimensions — they turn cylinders, cones, and rotationally symmetric solids into simple rectangular boxes in (r, θ, z) space."

In the previous sections, we saw how polar coordinates simplify double integrals over circular regions. Now we extend this idea to three dimensions using cylindrical coordinates. Just as polar coordinates use $(r, \theta)$ to describe points in the plane, cylindrical coordinates add a third variable $z$ to describe points in space.

Consider the challenge of computing the volume inside a solid cylinder. In Cartesian coordinates, the boundary $x^2 + y^2 = R^2$ leads to complicated square root bounds. But in cylindrical coordinates, the same cylinder is simply $r = R$ — a constant!

When to Use Cylindrical Coordinates

Ideal for Cylindrical ✓

Solid cylinders and cylindrical shells
Cones and truncated cones
Paraboloids of revolution $z = r^2$
Helical shapes and spiral structures
Any solid with circular cross-sections parallel to the xy-plane
Integrands containing $x^2 + y^2$

Consider Other Systems ✗

Rectangular boxes (use Cartesian)
Spheres (use spherical coordinates)
Ellipsoids with three different axes
Regions bounded only by planes
Shapes without circular symmetry about z-axis

Historical Context

Cylindrical coordinates emerged naturally from the study of problems involving rotational symmetry. Leonhard Euler (1707–1783) and Joseph-Louis Lagrange (1736–1813) developed the theory of coordinate transformations in multiple dimensions, laying the groundwork for our modern understanding.

The practical power of cylindrical coordinates became evident in 19th-century physics and engineering. Carl Friedrich Gauss used them extensively in his work on electromagnetism, where the fields around wires and conducting cylinders are naturally cylindrical. James Clerk Maxwell's equations for electromagnetism are often most elegantly expressed in cylindrical form for problems with axial symmetry.

Historical Application

The famous Maxwell stress tensor, which describes electromagnetic forces on materials, takes its simplest form in cylindrical coordinates when analyzing fields around cylindrical conductors — a calculation crucial for understanding power transmission lines.

Cylindrical Coordinates Review

In cylindrical coordinates, every point in 3D space is described by three numbers:

r — the radial distance from the z-axis (always $r \ge 0$ for integration)
θ (theta) — the angle in the xy-plane, measured counterclockwise from the positive x-axis
z — the height above (or below) the xy-plane

Cylindrical ↔ Cartesian Conversion

Cylindrical → Cartesian

x = r \cos\theta

y = r \sin\theta

z = z

Cartesian → Cylindrical

r = \sqrt{x^2 + y^2}

\theta = \arctan\left(\frac{y}{x}\right)

z = z

Key identity: $x^2 + y^2 = r^2$

Notice that cylindrical coordinates are simply polar coordinates in the xy-plane with an added z-coordinate. The z-coordinate remains unchanged between systems — this is what makes cylindrical coordinates "cylindrical."

Interactive Cylindrical Coordinate Explorer

Use this interactive tool to explore how cylindrical coordinates $(r, \theta, z)$ map to Cartesian coordinates $(x, y, z)$ . Drag to rotate the 3D view, and adjust the sliders to move the point.

Interactive Cylindrical Coordinate Explorer

Drag to rotate the 3D view. Adjust sliders to change the cylindrical coordinates.

r (radial distance)2.00

θ (angle)45° (0.25π)

z (height)1.50

Cartesian Coordinates

x = r cos θ = 1.414

y = r sin θ = 1.414

z = 1.500

Volume Element

dV = r dr dθ dz

What to Explore

Keep $r$ and $z$ fixed while varying $\theta$ — the point traces a circle
Keep $\theta$ and $z$ fixed while varying $r$ — the point moves radially
Keep $r$ and $\theta$ fixed while varying $z$ — the point moves vertically

The Volume Element: Why dV = r dr dθ dz

The most critical insight for triple integrals in cylindrical coordinates is understanding the volume element. When we convert from Cartesian to cylindrical, the volume element changes from $dV = dx\,dy\,dz$ to $dV = r\,dr\,d\theta\,dz$ .

Geometric Intuition

Consider a small "box" in cylindrical coordinates defined by:

$r$ to $r + dr$ (radial direction)
$\theta$ to $\theta + d\theta$ (angular direction)
$z$ to $z + dz$ (vertical direction)

This region is a curved wedge with approximate dimensions:

\text{Volume} \approx (\text{radial width}) \times (\text{arc length}) \times (\text{height})

= (dr) \times (r\,d\theta) \times (dz) = r\,dr\,d\theta\,dz

The key insight is that the arc length at radius $r$ is $r \cdot d\theta$ , not just $d\theta$ . Elements farther from the z-axis are larger in the angular direction!

Interactive Volume Element Demonstration

This visualization shows the infinitesimal volume element in cylindrical coordinates. Adjust the parameters to see how the curved "wedge" shape depends on position.

The Cylindrical Volume Element

Explore how the infinitesimal volume element in cylindrical coordinates is a curved "wedge" with volume dV = r dr dθ dz.

r (radius)1.50

θ (angle)45°

dz (height)0.60

dθ (angle span)30°

Dimensions

Radial width (dr):0.40

Arc length (r dθ):0.785

Height (dz):0.60

Volume Element

dV = (dr)(r dθ)(dz) = r dr dθ dz

≈ 0.1885

Key Insight: The arc length at radius r is r dθ, not just dθ. This is why the Jacobian factor r appears!

The Jacobian Derivation

The factor $r$ can also be derived rigorously using the Jacobian determinant. For the transformation $x = r\cos\theta$ , $y = r\sin\theta$ , $z = z$ :

J = \frac{\partial(x, y, z)}{\partial(r, \theta, z)} = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z} \end{vmatrix}

= \begin{vmatrix} \cos\theta & -r\sin\theta & 0 \\ \sin\theta & r\cos\theta & 0 \\ 0 & 0 & 1 \end{vmatrix}

Expanding along the third row:

|J| = 1 \cdot (\cos\theta \cdot r\cos\theta - (-r\sin\theta) \cdot \sin\theta)

= r\cos^2\theta + r\sin^2\theta = r(\cos^2\theta + \sin^2\theta) = r

The Jacobian Result

dV = dx\,dy\,dz = |J|\,dr\,d\theta\,dz = r\,dr\,d\theta\,dz

The Complete Conversion Formula

Triple Integral in Cylindrical Coordinates

If $E$ is a region in 3D space that can be described in cylindrical coordinates, then:

\iiint_E f(x, y, z)\,dV = \iiint_E f(r\cos\theta, r\sin\theta, z) \cdot r\,dr\,d\theta\,dz

Step-by-Step Conversion Process

Identify cylindrical symmetry — look for regions bounded by cylinders, cones, or surfaces of revolution
Convert the boundaries to cylindrical form:
- $x^2 + y^2 = R^2$ becomes $r = R$
- $z = x^2 + y^2$ becomes $z = r^2$
- $z = \sqrt{R^2 - x^2 - y^2}$ becomes $z = \sqrt{R^2 - r^2}$
Replace the integrand: $f(x, y, z) \to f(r\cos\theta, r\sin\theta, z)$
Multiply by the Jacobian r
Determine the order of integration and set up limits
Evaluate the iterated integral

Types of Cylindrical Regions

Region	Description	Bounds (typical)	Volume
Solid Cylinder	Cylinder of radius R, height h	0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h	πR²h
Cylindrical Shell	Hollow cylinder, radii a and b	a ≤ r ≤ b, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h	π(b² - a²)h
Solid Cone	Cone with vertex at origin	0 ≤ r ≤ (R/h)z, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h	(1/3)πR²h
Paraboloid	Region under z = c - r²	0 ≤ r ≤ √c, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ c - r²	(1/2)πc²
Hemisphere	Upper half of sphere radius R	0 ≤ r ≤ √(R² - z²), 0 ≤ θ ≤ 2π, 0 ≤ z ≤ R	(2/3)πR³

Interactive Region Explorer

Explore different 3D regions and see how their boundaries are described in cylindrical coordinates. Toggle the cross-section slice to visualize the polar region at each height.

Regions Suited for Cylindrical Coordinates

Solid Cylinder

A cylinder of radius R and height h

Bounds:

0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h

Volume:πR²h

Show cross-section slice

Slice height (z)50%

Volume Element

dV = r dr dθ dz

The factor r comes from the Jacobian of the transformation

Setting Up Integration Bounds

Choosing the correct order of integration and setting up bounds requires understanding how the region is bounded at each level.

General Strategy

Sketch the region and identify its boundaries
Choose the order of integration — for many problems, integrating z first (inner integral) is convenient when z-bounds depend on r
Express each bound as a function of the outer variables
Check the limits — for full rotation, $\theta$ goes from 0 to $2\pi$

Example: Region Under a Paraboloid

Find the volume under $z = 4 - r^2$ and above the xy-plane.

Analysis:

The paraboloid opens downward and has vertex at $(0, 0, 4)$
It meets the xy-plane ( $z = 0$ ) when $r^2 = 4$ , so $r = 2$
For $0 \le r \le 2$ , $z$ ranges from 0 to $4 - r^2$
$\theta$ spans full rotation: $0 \le \theta \le 2\pi$

V = \int_0^{2\pi} \int_0^2 \int_0^{4-r^2} r\,dz\,dr\,d\theta

Worked Examples

Example 1: Volume of a Cylinder

Find the volume of a solid cylinder of radius $R$ and height $h$ .

Solution: The cylinder is described by $0 \le r \le R$ , $0 \le \theta \le 2\pi$ , $0 \le z \le h$ .

V = \int_0^{2\pi} \int_0^R \int_0^h r\,dz\,dr\,d\theta

= \int_0^{2\pi} \int_0^R r \cdot [z]_0^h \,dr\,d\theta = \int_0^{2\pi} \int_0^R rh\,dr\,d\theta

= \int_0^{2\pi} h \left[\frac{r^2}{2}\right]_0^R d\theta = \int_0^{2\pi} \frac{hR^2}{2}\,d\theta

= \frac{hR^2}{2} \cdot 2\pi = \pi R^2 h

This confirms the well-known cylinder volume formula!

Example 2: Volume of a Cone

Find the volume of a solid cone with vertex at the origin, base radius $R$ at height $h$ .

Solution: The cone surface is $z = \frac{h}{R}r$ (or equivalently, $r = \frac{R}{h}z$ ). We can set up the integral with z as the inner integral:

For fixed $r$ , $z$ ranges from $\frac{h}{R}r$ (cone surface) to $h$ (top).

V = \int_0^{2\pi} \int_0^R \int_{(h/R)r}^h r\,dz\,dr\,d\theta

= \int_0^{2\pi} \int_0^R r \left(h - \frac{h}{R}r\right) dr\,d\theta

= \int_0^{2\pi} \int_0^R \left(hr - \frac{h}{R}r^2\right) dr\,d\theta

= \int_0^{2\pi} \left[\frac{hr^2}{2} - \frac{h}{R} \cdot \frac{r^3}{3}\right]_0^R d\theta

= \int_0^{2\pi} \left(\frac{hR^2}{2} - \frac{hR^2}{3}\right) d\theta = \int_0^{2\pi} \frac{hR^2}{6}\,d\theta

= \frac{hR^2}{6} \cdot 2\pi = \frac{1}{3}\pi R^2 h

This confirms the cone volume formula: $V = \frac{1}{3}\pi R^2 h$ .

Example 3: Mass with Variable Density

Find the mass of a solid cylinder of radius 2 and height 3 if the density at any point is proportional to the distance from the axis: $\rho(r, \theta, z) = kr$ .

Solution: Mass equals the integral of density over the region.

M = \iiint_E \rho\,dV = \int_0^{2\pi} \int_0^2 \int_0^3 kr \cdot r\,dz\,dr\,d\theta

= k \int_0^{2\pi} \int_0^2 \int_0^3 r^2\,dz\,dr\,d\theta

= k \int_0^{2\pi} \int_0^2 3r^2\,dr\,d\theta = k \int_0^{2\pi} \left[r^3\right]_0^2 d\theta

= k \int_0^{2\pi} 8\,d\theta = 16\pi k

Example 4: Volume Between Two Paraboloids

Find the volume of the region between $z = r^2$ and $z = 8 - r^2$ .

Solution: First, find where the paraboloids intersect: $r^2 = 8 - r^2 \Rightarrow r = 2$ .

For $0 \le r \le 2$ , $z$ ranges from $r^2$ (lower paraboloid) to $8 - r^2$ (upper paraboloid).

V = \int_0^{2\pi} \int_0^2 \int_{r^2}^{8-r^2} r\,dz\,dr\,d\theta

= \int_0^{2\pi} \int_0^2 r(8 - r^2 - r^2)\,dr\,d\theta = \int_0^{2\pi} \int_0^2 r(8 - 2r^2)\,dr\,d\theta

= \int_0^{2\pi} \int_0^2 (8r - 2r^3)\,dr\,d\theta

= \int_0^{2\pi} \left[4r^2 - \frac{r^4}{2}\right]_0^2 d\theta = \int_0^{2\pi} (16 - 8)\,d\theta

= 8 \cdot 2\pi = 16\pi

Applications in Physics and Engineering

Center of Mass

For a solid with density $\rho(r, \theta, z)$ , the coordinates of the center of mass are:

\bar{x} = \frac{1}{M} \iiint_E x\rho\,dV = \frac{1}{M} \iiint_E r\cos\theta \cdot \rho \cdot r\,dr\,d\theta\,dz

\bar{y} = \frac{1}{M} \iiint_E y\rho\,dV = \frac{1}{M} \iiint_E r\sin\theta \cdot \rho \cdot r\,dr\,d\theta\,dz

\bar{z} = \frac{1}{M} \iiint_E z\rho\,dV = \frac{1}{M} \iiint_E z \cdot \rho \cdot r\,dr\,d\theta\,dz

Moment of Inertia

The moment of inertia about the z-axis for a solid with density $\rho$ is:

I_z = \iiint_E (x^2 + y^2)\rho\,dV = \iiint_E r^2 \cdot \rho \cdot r\,dr\,d\theta\,dz

For a uniform solid cylinder of radius $R$ , height $h$ , and total mass $M$ :

I_z = \frac{1}{2}MR^2

Electrostatics: Field of a Charged Cylinder

When calculating the electric potential or field of a uniformly charged cylinder, cylindrical coordinates allow us to exploit the symmetry. Gauss's law applications become particularly elegant in this coordinate system.

Applications in Machine Learning

3D Point Cloud Processing

LiDAR and depth sensors produce cylindrical data naturally. Cylindrical representations like PointPillars and CenterPoint use cylindrical binning for efficient 3D object detection.

Convolutional Neural Networks

Understanding how area elements scale with radius helps design rotation-equivariant convolutions and cylindrical feature extractors used in panoramic and 360° image processing.

3D Generative Models

Many 3D shape priors use cylindrical coordinate systems. NeRF-based methods for scenes with rotational symmetry leverage cylindrical parameterizations for efficiency.

Robotics and SLAM

Simultaneous localization and mapping (SLAM) often uses cylindrical coordinate representations for sensor fusion and occupancy grid mapping.

Python Implementation

Numerical Triple Integration

Numerical Triple Integrals in Cylindrical Coordinates

🐍cylindrical_integration.py

Explanation(6)

Code(104)

3Triple Integral Function

This function computes triple integrals in cylindrical coordinates using scipy's tplquad. The key is including the Jacobian factor r in the integrand.

13The Jacobian Factor

We multiply f(r, θ, z) by r to account for the volume element dV = r dr dθ dz. This is the Jacobian determinant from the coordinate transformation.

35Cylinder Volume

For a simple cylinder, r goes from 0 to R, θ from 0 to 2π, and z from 0 to h. The volume integral ∫∫∫ r dr dθ dz = πR²h.

46Cone Volume

For a cone, the z-bounds depend on r. At radius r, z ranges from (h/R)r (cone surface) to h (top). This gives volume πR²h/3.

56Paraboloid Volume

Under the paraboloid z = 4 - r², z ranges from 0 to 4 - r². Integrating gives volume 8π.

68Non-constant Density

When density ρ = r, mass M = ∫∫∫ ρ · r dr dθ dz = ∫∫∫ r² dr dθ dz, giving M = 2πhR³/3.

98 lines without explanation

1import numpy as np
2from scipy import integrate
3import matplotlib.pyplot as plt
4from mpl_toolkits.mplot3d import Axes3D
5
6def cylindrical_triple_integral(f, r_bounds, theta_bounds, z_bounds):
7    """
8    Compute a triple integral in cylindrical coordinates.
9
10    The integral is: ∫∫∫ f(r, θ, z) · r dr dθ dz
11
12    Parameters:
13    - f: function of (r, theta, z) to integrate
14    - r_bounds: tuple of (r_min_func, r_max_func) - can be constants or functions
15    - theta_bounds: tuple of (theta_min, theta_max)
16    - z_bounds: tuple of (z_min_func, z_max_func) - can depend on r
17    """
18    def integrand(z, r, theta):
19        # Include the Jacobian factor r
20        return f(r, theta, z) * r
21
22    def z_min_func(r, theta):
23        return z_bounds[0](r) if callable(z_bounds[0]) else z_bounds[0]
24
25    def z_max_func(r, theta):
26        return z_bounds[1](r) if callable(z_bounds[1]) else z_bounds[1]
27
28    def r_min_func(theta):
29        return r_bounds[0](theta) if callable(r_bounds[0]) else r_bounds[0]
30
31    def r_max_func(theta):
32        return r_bounds[1](theta) if callable(r_bounds[1]) else r_bounds[1]
33
34    result, error = integrate.tplquad(
35        integrand,
36        theta_bounds[0], theta_bounds[1],  # theta limits
37        r_min_func, r_max_func,            # r limits (can depend on theta)
38        z_min_func, z_max_func             # z limits (can depend on r, theta)
39    )
40
41    return result, error
42
43# Example 1: Volume of a cylinder (radius R=2, height h=3)
44print("Example 1: Volume of cylinder (R=2, h=3)")
45print("=" * 50)
46
47R, h = 2, 3
48volume_cylinder, _ = cylindrical_triple_integral(
49    f=lambda r, theta, z: 1,      # f = 1 gives volume
50    r_bounds=(0, R),
51    theta_bounds=(0, 2*np.pi),
52    z_bounds=(0, h)
53)
54print(f"Computed volume: {volume_cylinder:.6f}")
55print(f"Expected (πR²h): {np.pi * R**2 * h:.6f}")
56
57# Example 2: Volume of a cone (base radius R=2, height h=4)
58print("\nExample 2: Volume of cone (R=2, h=4)")
59print("=" * 50)
60
61R, h = 2, 4
62# For cone with vertex at origin: r ranges from 0 to (R/h)z
63# We integrate z first: 0 to h, then r: 0 to (R/h)z
64# Actually easier: fix z, then r goes from 0 to (R/h)z
65# But tplquad wants z as innermost. Let's reformulate:
66# z goes from (h/R)r to h for fixed r, and r from 0 to R
67
68volume_cone, _ = cylindrical_triple_integral(
69    f=lambda r, theta, z: 1,
70    r_bounds=(0, R),
71    theta_bounds=(0, 2*np.pi),
72    z_bounds=(lambda r: (h/R)*r, h)  # z from (h/R)r to h
73)
74print(f"Computed volume: {volume_cone:.6f}")
75print(f"Expected (πR²h/3): {np.pi * R**2 * h / 3:.6f}")
76
77# Example 3: Volume under paraboloid z = 4 - r²
78print("\nExample 3: Volume under paraboloid z = 4 - r²")
79print("=" * 50)
80
81volume_paraboloid, _ = cylindrical_triple_integral(
82    f=lambda r, theta, z: 1,
83    r_bounds=(0, 2),              # r from 0 to 2 (where paraboloid meets z=0)
84    theta_bounds=(0, 2*np.pi),
85    z_bounds=(0, lambda r: 4 - r**2)  # z from 0 to 4-r²
86)
87print(f"Computed volume: {volume_paraboloid:.6f}")
88print(f"Expected (8π): {8 * np.pi:.6f}")
89
90# Example 4: Mass of cylinder with density ρ = r
91print("\nExample 4: Mass of cylinder with ρ(r,θ,z) = r")
92print("=" * 50)
93
94R, h = 2, 3
95mass, _ = cylindrical_triple_integral(
96    f=lambda r, theta, z: r,      # density = r
97    r_bounds=(0, R),
98    theta_bounds=(0, 2*np.pi),
99    z_bounds=(0, h)
100)
101print(f"Computed mass: {mass:.6f}")
102# M = ∫₀^h ∫₀^2π ∫₀^R r · r dr dθ dz = h · 2π · R³/3
103expected_mass = h * 2 * np.pi * R**3 / 3
104print(f"Expected (2πhR³/3): {expected_mass:.6f}")

Visualization of Cylindrical Regions

Visualizing 3D Cylindrical Regions

🐍cylindrical_visualization.py

Explanation(5)

Code(72)

4Region Visualization

This function creates 3D visualizations of common regions that are naturally described in cylindrical coordinates.

17Cylinder

A cylinder has constant radius R. The surface is r = R for all θ and z in [0, h].

26Cone

A cone with vertex at origin has r = (R/h)z. The radius increases linearly with height.

35Paraboloid

A paraboloid z = c - r² has circular cross-sections. The radius shrinks as z increases.

46Hemisphere

A hemisphere satisfies z = √(R² - r²). Using spherical parameterization helps visualize it.

67 lines without explanation

1import numpy as np
2import matplotlib.pyplot as plt
3from mpl_toolkits.mplot3d import Axes3D
4
5def plot_cylindrical_region(region_type='cylinder', R=2, h=3, ax=None):
6    """
7    Visualize different regions suited for cylindrical coordinates.
8
9    region_type: 'cylinder', 'cone', 'paraboloid', 'hemisphere'
10    """
11    if ax is None:
12        fig = plt.figure(figsize=(10, 8))
13        ax = fig.add_subplot(111, projection='3d')
14
15    theta = np.linspace(0, 2*np.pi, 50)
16
17    if region_type == 'cylinder':
18        # Draw cylinder surface
19        z = np.linspace(0, h, 30)
20        Theta, Z = np.meshgrid(theta, z)
21        X = R * np.cos(Theta)
22        Y = R * np.sin(Theta)
23        ax.plot_surface(X, Y, Z, alpha=0.5, color='blue')
24        ax.set_title(f'Cylinder: R={R}, h={h}')
25
26    elif region_type == 'cone':
27        # Cone with vertex at origin
28        z = np.linspace(0, h, 30)
29        Theta, Z = np.meshgrid(theta, z)
30        r_vals = (R/h) * Z  # r = (R/h)z
31        X = r_vals * np.cos(Theta)
32        Y = r_vals * np.sin(Theta)
33        ax.plot_surface(X, Y, Z, alpha=0.5, color='green')
34        ax.set_title(f'Cone: R={R}, h={h}')
35
36    elif region_type == 'paraboloid':
37        # z = c - r² (paraboloid opening down)
38        c = R**2  # height at center
39        r_vals = np.linspace(0, R, 30)
40        Theta, r_grid = np.meshgrid(theta, r_vals)
41        X = r_grid * np.cos(Theta)
42        Y = r_grid * np.sin(Theta)
43        Z = c - r_grid**2
44        ax.plot_surface(X, Y, Z, alpha=0.5, color='orange')
45        ax.set_title(f'Paraboloid: z = {c} - r²')
46
47    elif region_type == 'hemisphere':
48        # Upper hemisphere: z = √(R² - r²)
49        phi = np.linspace(0, np.pi/2, 30)
50        Theta, Phi = np.meshgrid(theta, phi)
51        X = R * np.sin(Phi) * np.cos(Theta)
52        Y = R * np.sin(Phi) * np.sin(Theta)
53        Z = R * np.cos(Phi)
54        ax.plot_surface(X, Y, Z, alpha=0.5, color='purple')
55        ax.set_title(f'Hemisphere: R={R}')
56
57    ax.set_xlabel('X')
58    ax.set_ylabel('Y')
59    ax.set_zlabel('Z')
60
61    return ax
62
63# Create figure with all region types
64fig = plt.figure(figsize=(14, 10))
65regions = ['cylinder', 'cone', 'paraboloid', 'hemisphere']
66for i, region in enumerate(regions):
67    ax = fig.add_subplot(2, 2, i+1, projection='3d')
68    plot_cylindrical_region(region, R=2, h=3, ax=ax)
69
70plt.tight_layout()
71plt.savefig('cylindrical_regions.png', dpi=150)
72plt.show()

Common Mistakes to Avoid

Mistake 1: Forgetting the r Factor

The most common error is writing $dV = dr\,d\theta\,dz$ instead of $dV = r\,dr\,d\theta\,dz$ . The factor $r$ is the Jacobian and is essential!

Mistake 2: Wrong Order of Integration

When z-bounds depend on r (like $0 \le z \le 4 - r^2$ ), z must be integrated before r. Carefully consider which variables appear in which bounds.

Mistake 3: Incomplete Rotation

For a complete solid of revolution, $\theta$ goes from 0 to $2\pi$ . Using $0$ to $\pi$ gives only half the solid!

Mistake 4: Not Converting the Integrand

If the original integrand is $f(x, y, z) = x$ , you must write $f = r\cos\theta$ , not leave it as $x$ .

Mistake 5: Using Spherical When Cylindrical is Better

Spherical coordinates are better for spheres, but cylindrical coordinates are better for cylinders, cones, and paraboloids. Choose based on the region's boundaries.

Verification Strategy

After computing an integral, verify against known results:

Cylinder volume: πR²h
Cone volume: (1/3)πR²h
Hemisphere volume: (2/3)πR³

Test Your Understanding

Question 1 of 8

What is the volume element dV in cylindrical coordinates (r, θ, z)?

Summary

Triple integrals in cylindrical coordinates provide a powerful tool for computing volumes, masses, and other quantities over regions with circular cross-sections or axial symmetry.

Key Formulas

Concept	Formula	Notes
Coordinate transform	x = r cos θ, y = r sin θ, z = z	Polar in xy-plane, unchanged z
Volume element	dV = r dr dθ dz	The r is the Jacobian
Jacobian	\|∂(x,y,z)/∂(r,θ,z)\| = r	Determinant of 3×3 matrix
Key identity	x² + y² = r²	Simplifies many integrands
Full cylinder	∫∫∫ r dr dθ dz	0 ≤ r ≤ R, 0 ≤ θ ≤ 2π

Key Takeaways

Cylindrical coordinates = polar + height — they extend 2D polar coordinates to 3D by adding the z-coordinate
The Jacobian factor r is essential — it accounts for the stretching of volume elements as we move away from the z-axis
Best for rotational symmetry about z-axis — cylinders, cones, and paraboloids have simple bounds
Order of integration matters — when bounds depend on each other, the dependent variable must be integrated first (innermost)
Convert everything: the integrand, the bounds, and the volume element

The Essence of Cylindrical Coordinates:

"Polar in the xy-plane, Cartesian in z — with the Jacobian r bridging the two worlds."

Coming Next: In the next section, we'll explore Triple Integrals in Spherical Coordinates, which are ideal for spheres, cones centered at the origin, and other regions with spherical symmetry.