Cylindrical Change of Variables

System

• $x=r\cos \theta$
• $y=r\sin \theta$
• $z=z$

Usage

$${\iiint }_{E}f(x,y,z)dV={{\iint }_{D_{xy}^{\prime }}{\int }_{z_b}^{z_t}f(x,y,z)dz|}_{\text{above}}rdA$$

Natural Domain

• $r\in (0,\infty )$
• $\theta \in [0,2\pi ]$
• $z\in (-\infty ,\infty )$

Use When

• Region of integration has rotational invariance
• Ex: Paraboloids, Cones, Cylinders