5. Triple Integrals With Spherical Coordinates

Triple Integral In Spherical Coordinates

If $E$ is given in spherical coordinates by

$c\le Ø\le d,\alpha \le \theta \le \beta ,g_1(\theta ,Ø)\le \theta \le g_2(\theta ,Ø),$

then

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

${\int }_c^d{\int }_{\alpha }^{\beta }{\int }_{g_1(\theta ,Ø)}^{g_2(\theta ,Ø)}f(\rho \sin Ø\cos \theta ,\rho \sin Ø\sin \theta ,\rho \cos Ø){\rho }^2\sin Ød\rho d\theta dØ$