Describing Cylindrical Regions With Coordinate Bounds

From Planar Shapes to Coordinate Bounds

A cylindrical-coordinate description of a region is most useful when its geometry is organized around the \(z\)-axis. A disk, annulus, or sector describes the region's horizontal footprint; a range of \(z\)-values then tells how far the region extends vertically.

The essential translation is direct: the radial coordinate \(r\) measures distance from the \(z\)-axis, the angle \(\theta\) gives the direction in the \(xy\)-plane, and \(z\) remains the vertical coordinate. The main task is to choose bounds that include exactly the intended radii, directions, and heights.

Definition: A coordinate-bounded cylindrical region is a set described by simultaneous restrictions on \(r\), \(\theta\), and \(z\). A common form is \(a\le r\le b\), \(\alpha\le\theta\le\beta\), and \(c\le z\le d\), with the angle interval chosen to represent the intended directions.

Radial Bounds: Disks and Annuli

In cylindrical coordinates, \(r\ge 0\) is the distance from the \(z\)-axis. In a horizontal slice, this is the distance from the origin in the \(xy\)-plane. Consequently, a disk of radius \(R\), centered on the \(z\)-axis, has radial bounds from zero to \(R\). A planar annulus with inner radius \(a\) and outer radius \(b\), where \(0<a<b\), has radial bounds from \(a\) to \(b\).

x^2+y^2\le R^2 \quad\Longleftrightarrow\quad 0\le r\le R,

and

a^2\le x^2+y^2\le b^2 \quad\Longleftrightarrow\quad a\le r\le b.

These correspondences follow from the cylindrical-coordinate equations \(x=r\cos\theta\) and \(y=r\sin\theta\). Since \(r\ge0\), the distance \(\sqrt{x^2+y^2}\) equals \(r\), not \(|r|\). Thus \(r\le R\) includes the axis and all points at distance at most \(R\). For an annulus with \(a\le r\le b\), points at distance less than \(a\) are excluded, as are points farther than \(b\); both boundary circles are included.

When a radial boundary is not included, its inequality must be strict. For example, \(a<r\le b\) omits the inner circular boundary but includes the outer one. The same principle applies to angular and vertical bounds.

Angular Bounds: Choosing the Intended Directions

An angular interval selects directions in the horizontal plane. If \(\alpha\le\theta\le\beta\), the interval describes a sector only when its endpoints and length match the intended angular sweep. In particular, the angular width \(\beta-\alpha\) matters: it is the amount of rotation covered, not merely the pair of endpoint rays.

A full revolution can be represented by \(0\le\theta<2\pi\). The upper endpoint is omitted because the direction at \(2\pi\) is the same as the direction at \(0\). This convention gives one standard angular representative for each direction. It does not remove a ray from the geometric region: the ray at angle \(0\) is already included.

For a sector that does not cross the standard angular seam, a closed interval such as \(\pi/6\le\theta\le\pi/2\) includes both bounding rays. For a sector that crosses the seam, an interval using angles beyond \(2\pi\) can be clearer. For example, \(5\pi/3\le\theta\le2\pi\) includes both endpoint rays, with the final ray represented equivalently by \(\theta=0\). If angles are restricted to \(0\le\theta<2\pi\), the same closed sector is represented by the union of \(5\pi/3\le\theta<2\pi\) and \(\theta=0\).

Key takeaway: Choose angular bounds by tracing the sweep from the first endpoint to the second. When a sector crosses the \(0\)-to-\(2\pi\) seam, account explicitly for the direction represented at \(0\); an interval ending at \(2\pi\) with a strict upper inequality does not include that endpoint direction by itself.

Vertical Bounds and Solid Cylinders

Once the horizontal footprint has been described, vertical bounds specify the heights in the region. For example, \(c\le z\le d\), where \(c<d\), restricts the region to the horizontal slab between the planes \(z=c\) and \(z=d\), including both planes. Combining this with \(0\le r\le R\) and a full range of angles gives a solid circular cylinder of radius \(R\) between those heights.

The boundary surface of a vertical cylinder is different from the solid cylinder. The solid includes every radius from the axis to the outer edge, so its radial condition is \(0\le r\le R\). The cylindrical side alone has \(r=R\). A top or bottom face is specified by a fixed value of \(z\), together with the appropriate radial and angular restrictions.

\begin{aligned} \text{solid circular cylinder:}&\quad 0\le r\le R,\\ \text{cylindrical side:}&\quad r=R. \end{aligned}

At \(r=0\), the angle is not uniquely determined: every value of \(\theta\) represents the same point on the axis at a given height. This does not affect a full cylinder or disk, because all angular choices at \(r=0\) give the same axis points. It is still important to remember that cylindrical coordinates are not unique at the axis.

Why the Radial Translation Works

The radial rules are consequences of the coordinate equations, not separate guesses. Substitute \(x=r\cos\theta\) and \(y=r\sin\theta\) into the squared distance from the origin in the \(xy\)-plane:

\begin{aligned} x^2+y^2 &=(r\cos\theta)^2+(r\sin\theta)^2\\ &=r^2\cos^2\theta+r^2\sin^2\theta\\ &=r^2(\cos^2\theta+\sin^2\theta)\\ &=r^2. \end{aligned}

Because cylindrical coordinates use \(r\ge0\), taking the nonnegative square root gives \(\sqrt{x^2+y^2}=r\). Therefore, comparing the distance from the axis with a radius \(R\) is exactly the same as comparing \(r\) with \(R\). The angle controls which direction is selected, while \(z\) controls height; neither changes this radial identity.

Worked Examples

Worked Example: A Disk Extruded Between Two Heights

Describe the solid whose horizontal cross-section is the disk \(x^2+y^2\le9\), and whose height satisfies \(-1\le z\le4\).

First, the disk has radius \(R=3\), because \(9=3^2\). Its radial bounds are therefore \(0\le r\le3\). Since the entire disk is required, take a full revolution with \(0\le\theta<2\pi\). The vertical bounds are already given. Thus the region is

0\le r\le3,\qquad 0\le\theta<2\pi,\qquad -1\le z\le4.

To check the radial bound, take \(r=2\) and \(\theta=\pi/3\). Then \(x=2\cos(\pi/3)=1\) and \(y=2\sin(\pi/3)=\sqrt3\). Hence \(x^2+y^2=1^2+(\sqrt3)^2=1+3=4\le9\). Choosing \(z=1\) also satisfies \(-1\le1\le4\). The point lies in the described solid, as required.

Worked Example: An Annular Sector Between Horizontal Planes

Describe the portion of the annular sector defined by \(4\le x^2+y^2\le25\) and \(\pi/6\le\theta\le\pi/2\), with \(0\le z\le3\).

The squared radial inequalities give inner radius \(\sqrt4=2\) and outer radius \(\sqrt{25}=5\). Thus \(2\le r\le5\). The given angular interval runs from \(\pi/6\) to \(\pi/2\) without crossing the standard seam, and both bounding rays are included. The height bounds are \(0\le z\le3\). Together, the bounds are

2\le r\le5,\qquad \frac{\pi}{6}\le\theta\le\frac{\pi}{2},\qquad 0\le z\le3.

For a check, choose \(r=3\), \(\theta=\pi/3\), and \(z=2\). The radial coordinate is between \(2\) and \(5\), and \(\pi/6\le\pi/3\le\pi/2\), while \(0\le2\le3\). In Cartesian coordinates, \(x=3\cos(\pi/3)=3/2\) and \(y=3\sin(\pi/3)=3\sqrt3/2\). Then \(x^2+y^2=9/4+27/4=36/4=9\), and indeed \(4\le9\le25\). The selected point satisfies the original description.

Worked Example: A Sector That Crosses the Angular Seam

Describe the part of a vertical cylinder with \(1\le r\le2\) and \(-2\le z\le1\) that sweeps from angle \(5\pi/3\) through angle \(2\pi\), including both bounding rays.

The angular width is \(2\pi-5\pi/3=\pi/3\), so the sector is a \(60\)-degree sweep. To include its ending ray, use the closed interval \(5\pi/3\le\theta\le2\pi\), allowing the endpoint \(2\pi\). The full coordinate description is

1\le r\le2,\qquad \frac{5\pi}{3}\le\theta\le2\pi,\qquad -2\le z\le1.

For an interior check, choose \(r=3/2\), \(\theta=11\pi/6\), and \(z=0\). The angle satisfies \(10\pi/6\le11\pi/6\le12\pi/6\), the radius satisfies \(1\le3/2\le2\), and the height satisfies \(-2\le0\le1\). The Cartesian coordinates are \(x=(3/2)\cos(11\pi/6)=3\sqrt3/4\) and \(y=(3/2)\sin(11\pi/6)=-3/4\). Thus \(x^2+y^2=27/16+9/16=36/16=9/4=(3/2)^2\), confirming the radius.

At the ending ray, \(\theta=2\pi\) gives \(x=r\cos(2\pi)=r\) and \(y=r\sin(2\pi)=0\). This is the same direction as \(\theta=0\), so the closed interval includes that boundary ray. If using only standard angles \(0\le\theta<2\pi\), describe the same sector instead by \(5\pi/3\le\theta<2\pi\) together with \(\theta=0\); without the additional \(\theta=0\), that endpoint would be omitted.

Common Mistakes and How to Avoid Them

A frequent radial mistake is to say that \(1\le r\le3\) excludes all points within distance \(1\) of the axis. It excludes points whose distance is less than \(1\), but it includes the inner boundary: \(r=1\) satisfies both \(1\le1\) and \(1\le3\). To omit that boundary, the correct bound is \(1<r\le3\).

A common angular mistake is to claim that \(5\pi/3\le\theta<2\pi\) includes both boundary rays because \(2\pi\) and \(0\) represent the same direction. The claim fails: \(\theta=2\pi\) is excluded by the strict upper bound, while \(\theta=0\) fails the lower bound \(5\pi/3\le\theta\). Correct the description by allowing \(5\pi/3\le\theta\le2\pi\), or, if restricting to standard angles, include the separate case \(\theta=0\).

Finally, do not use \(0\le r\le R\) when the intended object is only the cylindrical side. Those bounds describe the solid footprint, including its interior. The side itself requires \(r=R\). Before finalizing a coordinate description, check the lower and upper endpoints of each coordinate against the intended boundary inclusion.

Check Your Understanding

Write coordinate bounds or explain the boundary issue in each case.

What radial bounds describe the disk \(x^2+y^2\le16\)?
What radial bounds describe an annulus with inner radius \(2\) and outer radius \(6\), including both circular boundaries?
Give cylindrical-coordinate bounds for a full solid cylinder of radius \(4\) between \(z=-3\) and \(z=2\).
Why does \(5\pi/3\le\theta<2\pi\) alone omit the ray at angle \(0\)?
What radial condition describes the cylindrical side of a cylinder of radius \(R\), rather than its solid interior?