Triple Integrals in Cylindrical Coordinates

The central idea

Cylindrical coordinates are $x=r\cos\theta$, $y=r\sin\theta$, $z=z$ with $r\ge0$. The volume element is $dV=r\,dz\,dr\,d\theta$ in one common order, with the same Jacobian $r$ as polar area. Surfaces $r=R$ are cylinders, $\theta=\theta_0$ vertical half-planes, and $z=c$ horizontal planes.

Definitions, hypotheses, and notation

A cone $z=kr$ and paraboloid $z=r^2$ become especially simple in cylindrical coordinates. Intersections of upper and lower surfaces determine radial limits; angular restrictions describe wedges. If the projection changes radial description with angle, splitting is still necessary despite the coordinate change.

Order may place $z$ inside when the solid is vertically simple, but other orders are possible. The Jacobian belongs to the coordinate transformation regardless of order, so it must not be confused with an integration bound or counted more than once.

Conceptual meaning

Cylindrical coordinates apply polar geometry in horizontal slices while leaving height unchanged. A small volume cell has radial thickness, angular arc length $r d\theta$, and vertical height $dz$, so its volume is $rdrd\theta dz$.

A dependable method and decision rule

1. Project the solid onto the $xy$-plane and describe it in polar bounds.
2. Write lower and upper $z$ surfaces over that projection.
3. Convert $x^2+y^2$ to $r^2$ and all remaining variables consistently.
4. Include the factor $r$ exactly once.
5. Choose an order compatible with the geometry and check coverage.

Fully worked example

Graphical or geometric meaning

The solid is built from vertical columns over annular sector cells. Farther from the axis, equal angular width spans a wider base, which the Jacobian $r$ records.

Common mistakes and why they fail

Verification and reasonableness checks

• Integrate one over the solid and compare with a known volume.
• Verify angular and radial bounds cover the projection once.
• Check the Jacobian and differentials have cubic-length units.

Cylindrical coordinates preserve vertical structure

Use $x=r\cos\theta$, $y=r\sin\theta$, $z=z$, and $dV=r\,dz\,dr\,d\theta$ in one common order. The Jacobian factor $r$ is essential. Cylindrical coordinates suit solids described by circles around the $z$-axis together with upper and lower surfaces. Draw the projection in the $xy$-plane to determine angular and radial bounds, then place the vertical bounds between the surfaces. Convert the entire integrand, including $x^2+y^2=r^2$. An angular interval that is too broad can duplicate the solid, while negative $r$ is normally avoided. With integrand one, the setup should reproduce a familiar cylinder or cone volume, providing a direct check.

Practice

1. Find volume of $r\le1$, $0\le z\le5$.
2. Convert $x^2+y^2=z$.
3. What is the cylindrical Jacobian?

Connections and next steps

• Double Integrals in Polar Coordinates
• Triple Integrals
• Triple Integrals in Spherical Coordinates