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

Common mistakes and why they fail