Calculus III: Triple Integrals in Cylindrical Coordinates

Triple integrals allow you to compute volumes, masses, and other aggregate quantities over three-dimensional regions. While Cartesian coordinates are familiar, many real-world objects—from pipes to turbines to chemical tanks—possess natural cylindrical symmetry. Mastering cylindrical coordinates transforms complex Cartesian integrals into simpler, more elegant calculations by aligning your coordinate system with the geometry of the problem. This technique is indispensable in engineering fields like fluid dynamics, electromagnetics, and mechanical design.

The Cylindrical Coordinate System and Volume Element

The cylindrical coordinate system extends polar coordinates into three dimensions by adding a $z$-axis. A point in space is represented by $(r,\theta ,z)$, where:

• $r\ge 0$ is the radial distance from the $z$-axis.
• $\theta$ is the angular coordinate measured from the positive $x$-axis, typically $0\le \theta <2\pi$.
• $z$ is the usual vertical Cartesian coordinate.

The relationship to Cartesian coordinates $(x,y,z)$ is given by: $$x=r\cos \theta ,y=r\sin \theta ,z=z.$$ The most critical concept is the cylindrical coordinate volume element, $dV$. When you integrate, you are summing over tiny volume chunks. In Cartesian coordinates, this chunk is a rectangular box: $dV=dxdydz$. In cylindrical coordinates, the "box" is a wedge-shaped fragment of a cylinder. As $r$, $\theta$, and $z$ change by small amounts $dr$, $d\theta$, and $dz$, they sweep out a volume approximated by a rectangular solid with sides of length $dr$, $rd\theta$, and $dz$. Therefore, the volume element becomes: $$dV=rdrd\theta dz.$$ The factor $r$ is the Jacobian factor for the coordinate transformation from $(x,y,z)$ to $(r,\theta ,z)$. It accounts for the fact that a change in $\theta$ produces a larger arc length for points farther from the axis ($r$ is larger). Forgetting this factor is the single most common error, as it would incorrectly compute volume.

Setting Up the Integration Limits

The general form of a triple integral in cylindrical coordinates is: $${\iiint }_{E}f(x,y,z)dV={\iiint }_{E}f(r\cos \theta ,r\sin \theta ,z)rdrd\theta dz.$$ Your primary task is to describe the 3D region $E$ with appropriate limits for $r$, $\theta$, and $z$. The process requires a clear mental picture, often aided by a sketch.

A Standard Workflow:

1. Identify the Axis of Symmetry: The $z$-axis is almost always the central axis. Your region should be "cylindrically symmetric" or easily describable relative to it.
2. Project the Region: Project $E$ onto the $xy$-plane (i.e., the $r\theta$-plane). This projection, $D$, is a polar region.
3. Determine the $\theta$ Limits: Find the smallest range of $\theta$ angles needed to sweep out the polar projection $D$. Often, for full circular regions, $\theta$ ranges from $0$ to $2\pi$.
4. Determine the $r$ Limits: For a fixed $\theta$, determine how the radial distance $r$ varies within the projection $D$. The lower limit is often 0 (starting at the $z$-axis), and the upper limit is a function of $\theta$, written as $r=g_1(\theta )$.
5. Determine the $z$ Limits: This is the crucial third dimension. For a fixed point $(r,\theta )$ in the base $D$, determine how the $z$-coordinate varies as you move vertically through the solid $E$. The lower $z$-bound is a surface, often expressed as $z=f_1(r,\theta )$, and the upper bound is another surface, $z=f_2(r,\theta )$.

The iterated integral is typically written with $z$ as the innermost integral, followed by $r$, then $\theta$, though the order can sometimes be changed if the region permits.

Converting Between Cartesian and Cylindrical Formulations

You will often need to convert between Cartesian and cylindrical formulations of surfaces and functions. This skill is essential for correctly setting up the integrand $f(x,y,z)$ and the bounding surfaces for $z$.

Key Conversions:

• The Cartesian equation for a circular cylinder of radius $a$ centered on the $z$-axis is $x^2+y^2=a^2$. In cylindrical coordinates, this simplifies dramatically to $r=a$.
• A cone centered on the $z$-axis might have equation $z=\sqrt{x^2+y^2}$. In cylindrical coordinates, this becomes $z=r$ (for the top half of the cone).
• A sphere $x^2+y^2+z^2=a^2$ becomes $r^2+z^2=a^2$.
• The integrand itself is converted: $f(x,y,z)$ becomes $f(r\cos \theta ,r\sin \theta ,z)$. A common example is $x^2+y^2$, which simplifies to $r^2$.

The power of cylindrical coordinates lies in these simplifications. A region bounded by a cylinder and two planes, which requires careful piecewise description in Cartesian coordinates, often becomes a simple rectangular region in $(r,\theta ,z)$-space.

Identifying Regions Best Suited for Cylindrical Integration

Choosing the right coordinate system saves immense effort. You should consider cylindrical coordinates when the 3D region $E$ or the function $f(x,y,z)$ exhibits axial symmetry—symmetry about a line (the $z$-axis).

Classic Region Types:

1. Regions bounded by cylinders, cones, and paraboloids centered on the $z$-axis.
2. Regions with circular cross-sections in horizontal planes (e.g., a sphere is a candidate, though spherical coordinates are often better).
3. Regions where the projection in the $xy$-plane is a circle, a portion of a circle (a washer), or a polar sector.

Worked Example: Volume of an Ice Cream Cone Find the volume of the solid bounded below by the cone $z=\sqrt{x^2+y^2}$ and above by the sphere $x^2+y^2+z^2=9$.

1. Projection: The two surfaces intersect where $z=r$ meets $r^2+z^2=9$. Substituting gives $r^2+r^2=9$, so $r=3/\sqrt{2}$. The projection $D$ onto the $xy$-plane is a disk of radius $3/\sqrt{2}$.
2. $\theta$ Limits: To cover the full disk, $\theta$ goes from $0$ to $2\pi$.
3. $r$ Limits: For a fixed $\theta$, $r$ goes from $0$ to $3/\sqrt{2}$.
4. $z$ Limits: For a fixed $(r,\theta )$, $z$ starts on the cone ($z=r$) and goes up to the sphere ($z=\sqrt{9-r^2}$).
5. Set Up and Evaluate:

$$\begin{array}{rl}V& ={\int }_0^{2\pi }{\int }_0^{3/\sqrt{2}}{\int }_r^{\sqrt{9-r^2}}rdzdrd\theta \\ & ={\int }_0^{2\pi }d\theta {\int }_0^{3/\sqrt{2}}r\left(\sqrt{9-r^2}-r\right)dr\\ & =2\pi {\left[-\frac{1}{3}(9-r^2{)}^{3/2}-\frac{r^3}{3}\right]}_0^{3/\sqrt{2}}\\ & =2\pi \left(9-\frac{9\sqrt{2}}{2}\right)=18\pi (1-\sqrt{2}/2).\end{array}$$ Notice the Jacobian factor $r$ appears immediately in the integrand.

Common Pitfalls & Engineering Considerations

1. Omitting the Jacobian Factor $r$: This mistake transforms a volume integral into an area integral in the $r\theta$-plane, yielding an incorrect result that is typically too small. Always write $dV=rdrd\theta dz$ first.
2. Incorrect $z$-Limits: The bounds for $z$ must be expressed in terms of $r$ and $\theta$. A common error is to use the Cartesian descriptions of the surfaces without converting them. For example, if the top surface is the plane $z=4$, the upper limit is simply $z=4$. But if it is the paraboloid $z=4-x^2-y^2$, you must convert it to $z=4-r^2$.
3. Misidentifying the Polar Projection $D$: Failing to correctly find the region in the $xy$-plane leads to wrong $r$ and $\theta$ limits. Always sketch the projection. For regions between two cylinders or outside a cylinder, the $r$ limits will be functions, not just constants.
4. Applying Cylindrical Coordinates Inappropriately: Not every 3D region with a circle in it is best solved with cylindrical coordinates. If the region is bounded primarily by spheres or has symmetry about a point, spherical coordinates are likely more efficient. The choice is an important engineering judgment call to minimize computational complexity.

Summary

• Cylindrical coordinates $(r,\theta ,z)$ are ideal for regions with axial symmetry, converting complex Cartesian bounds into simpler expressions like $r=a$ or $z=r$.
• The cylindrical coordinate volume element is $dV=rdrd\theta dz$. The Jacobian factor $r$ is non-negotiable and must be included in every integrand.
• Setting up the integral requires a logical process: project the 3D region onto the $xy$-plane to find the polar region $D$ (giving $r$ and $\theta$ limits), then determine how $z$ varies from the bottom to the top surface for a fixed $(r,\theta )$.
• Converting between Cartesian and cylindrical formulations for surfaces ($x^2+y^2=r^2$) and integrands is a core skill for correct setup.
• Success hinges on accurate visualization and a meticulous limit-setting process, avoiding the common traps of forgetting the Jacobian or misrepresenting the bounds in the new coordinate system.