4. Triple Integrals With Cylindrical Coordinates

Triple Integrals In Cylindrical Coordinates

If $E=\{(x,y,z)|(x,y)\in D,u_1(x,y)\le z\le u_2(x,y)\}$ and $D$ is given in polar coordinates by $\alpha \le \theta \le \beta$, $h_1(\theta )\le r\le h_2(\theta )$,

${\iiint }_{E}f(x,y,z)dV=$

${\int }_{\alpha }^{\beta }{\int }_{h_1(\theta )}^{h_2(\theta )}{\int }_{u_1(r\cos \theta ,r\sin \theta )}^{u_2(r\cos \theta ,r\sin \theta )}f(r\cos \theta ,r\sin \theta ,z)rdzdrd\theta$