Triple integrals in cylindrical coordinates

Recall that when converting from Cartesian Coordinates to Polar Coordinates with double integrals we do the following:

$\int \int_D f(x,y)dA = \int^{\theta=\beta}_{\theta=\alpha} \int^{r=g_2(\theta)}_{r=g_1(\theta)} f(r \cos \theta, r \sin \theta ) r dr d\theta$

With triple integrals, it is very similar. Instead of calling it polar coordinates, we call it cylindrical coordinates.

Suppose we want to triple integrate $f(x,y,z)$ in cylindrical coordinates in the following region of $E$.

$\alpha \leq \theta \leq \beta$
$g_1 (\theta) \leq r \leq g_2 (\theta)$
$h_1 (r \cos \theta, r \sin \theta) \leq z \leq h_2(r \cos \theta, r \sin \theta)$

Then

$\int \int \int_E f(x,y,z) dV = \int^{\beta}_{\alpha} \int^{g_2(\theta )}_{g_1(\theta )} \int^{h_2 (r \cos \theta, r \sin \theta)}_{h_1 (r \cos \theta, r \sin \theta)} f(r \cos \theta, r \sin \theta, z) rdzdrd\theta$