Triple Integrals in Cylindrical Coordinates

Definition A point $P = (x, y, z)$ in a right-handed coordinate system has cylindrical coordinates $(r, \theta, z)$, where $(r, \theta)$ are polar coordinates of the projection $(x, y, 0)$ and $z$ is unchanged.

We take $r \ge 0$, so $r = \sqrt{x^2 + y^2}$ is the distance from $P$ to the $z$-axis, and $\theta$ is measured counterclockwise from the positive $x$-axis when viewed from above. The angle is determined only up to multiples of $2\pi$, and not at all on the $z$-axis. The coordinate surfaces are the cylinder $r = c$ for $c > 0$, the half-plane $\theta = c$ bounded by the $z$-axis, and the horizontal plane $z = c$.

Definition The small solid with $r_0 \le r \le r_0 + \Delta r$, $\theta_0 \le \theta \le \theta_0 + \Delta \theta$ and $z_0 \le z \le z_0 + \Delta z$, with $r_0 \ge 0$ and $0 < \Delta \theta \le 2\pi$, is a cylindrical wedge. Its base is a piece of an annulus with area $\frac{1}{2}\big((r_0 + \Delta r)^2 - r_0^2\big) \Delta \theta$, so its volume is exactly

Riemann sums over such wedges lead to the volume element $dV = r \, dz \, dr \, d\theta$. The same factor is the Jacobian, since the change of variables has determinant $\frac{\partial(x, y, z)}{\partial(r, \theta, z)} = r \ge 0$.

Definition Let $E$ be a solid of type 1 whose projection $D$ onto the $xy$-plane is described in polar coordinates by

with $0 \le \beta - \alpha \le 2\pi$ and $h_1$, $h_2$ continuous with $0 \le h_1 \le h_2$. Let $E$ lie between the graphs $z = u_1(x, y)$ and $z = u_2(x, y)$, with $u_1 \le u_2$ continuous on $D$. If $f$ is continuous on $E$, then

where $U_i = u_i(r \cos \theta, r \sin \theta)$ and $f$ is evaluated at $(r \cos \theta, r \sin \theta, z)$.

Definition To find the bounds of a solid in cylindrical coordinates, write each bounding surface in terms of $r$, $\theta$ and $z$, using $x^2 + y^2 = r^2$. Decide which surface is the bottom and which is the top for each point of the shadow. The shadow’s outer edge is usually where the bottom and top surfaces meet, found by setting their $z$ values equal.

Definition For $f$ continuous on the solid, Fubini’s theorem, applied to the integrand $f r$ over the region the solid occupies in $(r, \theta, z)$ space, lets the integration in $r$, $\theta$ and $z$ be done in any order, provided the bounds of each inner variable are rebuilt from the inequalities that describe the solid. When the outer variable is $z$, the slice at height $z$ is a horizontal region, and if the surface bounding that slice changes at some height, the integral must be split there.

Definition Suppose $f$ is continuous on a solid $E$. If $E$ is unchanged by the reflection $(x, y, z) \mapsto (-x, y, z)$ and $f(-x, y, z) = -f(x, y, z)$ on $E$, then $\iiint_E f \, dV = 0$. The same holds with the roles of $x$ and $y$ exchanged, for the reflection $(x, y, z) \mapsto (x, -y, z)$ and $f$ odd in $y$. If $E$ is unchanged by every rotation about the $z$-axis, so that its bounds do not involve $\theta$ and $\theta$ runs over a full turn, and $f$ depends only on $r$ and $z$, then the $\theta$ integral gives a factor $2\pi$. For such a solid with a density that depends only on $r$ and $z$, the center of mass lies on the $z$-axis.

Definition For a solid $E$ with continuous density $\delta \ge 0$, the mass and the moment of inertia about the $z$-axis are

because $x^2 + y^2 = r^2$ is the squared distance to the $z$-axis.