21. Multiple Integrals in Curvilinear Coordinates

b. Integrating in Cylindrical Coordinates

2. Integral over a Cylindrical Region

We now explore integration in cylindrical coordinates with constant and non-constant limits.

Integral over a Cylindrical Box

We first want to integrate over the cylindrical box \[ r_1 \le r \le r_2 \qquad \theta_1 \le \theta \le \theta_2 \qquad z_1 \le z \le z_2 \] Using the Riemann Sum definition with \(\Delta V_{ijk}=r_i^*\,\Delta r_i\,\Delta\theta_j\,\Delta z_k\), the integral is: \[ \iiint_R f(r,\theta,z)\,dV =\lim_{p\rightarrow\infty} \lim_{q\rightarrow\infty} \lim_{s\rightarrow\infty} \sum_{i=1}^p\sum_{j=1}^q\sum_{k=1}^s f(r_i^*,\theta_j^*,z_k^*)\,r_i^*\Delta r_i\Delta\theta_j\Delta z_k \] We recognize this as the triple iterated integral: \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{z_1}^{z_2}\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2} f(r,\theta,z)\,r\,dr\,d\theta\,dz \] By Fubini's Theorem, there are five other orders of integrations obtained by reordering the differentials and the corresponding limits.

Other Integrals in Cylindrical Coordinates

In addition to integrating over cylindrical boxes, cylindrical coordinates can be used to integrate over other types of regions with appropriate definitions of the regions. The resulting integrals have the forms:

Type \(r\theta\): \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{r_1}^{r_2}\int_{g(r)}^{h(r)}\int_{p(r,\theta)}^{q(r,\theta)} f(r,\theta,z)\,r\,dz\,d\theta\,dr \] Type \(\theta r\): \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{\theta_1}^{\theta_2}\int_{g(\theta)}^{h(\theta)} \int_{p(r,\theta)}^{q(r,\theta)} f(r,\theta,z)\,r\,dz\,dr\,d\theta \] Type \(rz\): \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{r_1}^{r_2}\int_{g(r)}^{h(r)}\int_{p(r,z)}^{q(r,z)} f(r,\theta,z)\,r\,d\theta\,dz\,dr \] Type \(zr\): \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{z_1}^{z_2}\int_{g(z)}^{h(z)}\int_{p(r,z)}^{q(r,z)} f(r,\theta,z)\,r\,d\theta\,dr\,dz \] Type \(\theta z\): \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{\theta_1}^{\theta_2}\int_{g(\theta)}^{h(\theta)} \int_{p(\theta,z)}^{q(\theta,z)} f(r,\theta,z)\,r\,dr\,dz\,d\theta \] Type \(z\theta\): \[ \iiint\limits_R f(r,\theta,z)\,dV =\int_{z_1}^{z_2}\int_{g(z)}^{h(z)}\int_{p(\theta,z)}^{q(\theta,z)} f(r,\theta,z)\,r\,dr\,d\theta\,dz \]

Notice that each integral can have limits which depend only on the variables outside that integral.

The volume of the spiral ramp could be computed using the triple integral \(\displaystyle V=\int_0^2\int_0^{2\pi}\int_0^\theta 1\,r\,dz\,d\theta\,dr\) or using the double integral \(\displaystyle V=\int_0^2\int_0^{2\pi} \theta\,r\,d\theta\,dr\), but the mass must be computed using a triple integral.

The mass is given by \[\begin{aligned} M&=\iiint \delta\,dV =\int_0^2\int_0^{2\pi}\int_0^\theta 2(\theta-z)\,r\,dz\,d\theta\,dr \\ &=\int_0^2 2r\,dr\int_0^{2\pi}\int_0^\theta (\theta-z)\,dz\,d\theta \\ &=\left[r^2\dfrac{}{}\right]_0^2\int_0^{2\pi} \left[\theta z-\,\dfrac{z^2}{2}\right]_0^\theta\,d\theta \\ &=4\int_0^{2\pi} \dfrac{\theta^2}{2}\,d\theta =4\left[\dfrac{\theta^3}{6}\right]_0^{2\pi} =4\dfrac{(2\pi)^3}{6} =\dfrac{16}{3}\pi^3 \end{aligned}\]