Integrals in Cylindrical and Spherical Coordinates

These integrals are computed just like ordinary triple integrals except we multiply the integrand by $r$ (in cylindrical coordinates) or $\rho^2\sin(\phi)$ (in spherical coordinates). See the next section for a more general way of doing this. It is sometimes useful to use plots to decide how to represent the region of integration.

Let's do an integral in cylindrical coordinates.

$$\int_{-2}^2 \int_0^{\sqrt{4-x^2}}\int_0^{3}yz \mathrm{d}z \mathrm{d}y \mathrm{d}x.$$

Note that $r$ goes from 0 to 2 and $\theta$ goes from 0 to $\pi$.

Now let's do an integral in spherical coordinates.

$$\int_{-1}^1 \int_0^{\sqrt{1-x^2}}\int_0^{\sqrt{1-x^2-y^2}} xz \mathrm{d}z \mathrm{d}y \mathrm{d}x.$$

Note that $\rho$ goes from 0 to 1, $\theta$ goes from 0 to $\pi$, and $\phi$ goes from 0 to $\pi/2$.