Cartesian

Cylindrical

Spherical

$\int \int {\int }_R\mathrm{\δ}\left(x\,y\,z\right) \mathrm{dv}$

$\int \int {\int }_R\mathrm{\δ}\left(r\,\mathrm{\θ}\,z\right) r \mathrm{dv}\prime$

$\int \int \int \mathrm{\δ}\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right) {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right) \mathrm{dv}″$

Table 8.3.1   Total volume or mass in three-dimensional region $R$ 

First Moment

Cartesian

Cylindrical

Spherical

$M_{\mathrm{yz}}$

$\int \int {\int }_{R}x \mathrm{\δ} \mathrm{dv}$

$\int \int {\int }_{R}r \cos \left(\mathrm{\θ}\right) \mathrm{\δ} r \mathrm{dv}\prime$

$\int \int {\int }_R\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)\cos \left(\mathrm{\θ}\right) \mathrm{\δ} {\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)\mathrm{dv}″$

$M_{\mathrm{xz}}$

$\int \int {\int }_{R}y \mathrm{\δ} \mathrm{dv}$

$\int \int {\int }_{R}r \sin \left(\mathrm{\theta }\right) \mathrm{\δ} r \mathrm{dv}\prime$

$\int \int {\int }_R\mathrm{\rho } \sin \left(\mathrm{\phi }\right)\sin \left(\mathrm{\theta }\right) \mathrm{\δ} {\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)\mathrm{dv}″$

$M_{\mathrm{xy}}$

$\int \int {\int }_{R}z \mathrm{\δ} \mathrm{dv}$

$\int \int {\int }_{R}z \mathrm{\δ} r \mathrm{dv}\prime$

$\int \int {\int }_R\mathrm{\ρ} \cos \left(\mathrm{\phi }\right) \mathrm{\δ} {\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)\mathrm{dv}″$

Table 8.3.2   First moments for calculating a centroid ($\mathrm{\δ}\=\mathrm{constant}$) or a center of mass

Coordinate

Centroid

Center of Mass

$\stackrel{\&conjugate0;}{x}$ 

$\frac{M_{\mathrm{yz}}}{V}$

$\frac{M_{\mathrm{yz}}}{m}$

$\stackrel{\&conjugate0;}{y}$ 

$\frac{M_{\mathrm{xz}}}{V}$

$\frac{M_{\mathrm{xz}}}{m}$

$\stackrel{\&conjugate0;}{z}$ 

$\frac{M_{\mathrm{xy}}}{V}$

$\frac{M_{\mathrm{xy}}}{m}$

Table 8.3.3   Centroid or Center of Mass

Example 8.3.1

$R$ is the region in Example 8.1.3

$\mathrm{\delta }\left(r\,\mathrm{\theta }\,z\right)\=z r^2\sin \left(\mathrm{\theta }\/6\right)$

Example 8.3.2

$R$ is the region in Example 8.1.5

$\mathrm{\delta }\left(r\,\mathrm{\theta }\,z\right)\=r z^2\cos \left(\mathrm{\theta }\/3\right)$

Example 8.3.3

$R$ is the region in Example 8.1.8

$\mathrm{\delta }\left(x\,y\,z\right)\=3\+x\+y\+z$

Example 8.3.4

$R$ is the region in Example 8.1.14

$\mathrm{\delta }\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\={\mathrm{\rho }}^2$

Example 8.3.5

$R$ is the region in Example 8.1.15

$\mathrm{\delta }\left(r\,\mathrm{\theta }\,z\right)\=\left(r^2\+z\right) \sin \left(\mathrm{\theta }\/4\right)$

Example 8.3.6

$R$ is the region in Example 8.1.20

$\mathrm{\δ}\left(r\,\mathrm{\θ}\,z\right)\=z$

Example 8.3.7

$R$ is the region in Example 8.1.21

$\mathrm{\δ}\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\=\sqrt{\mathrm{\ρ}}$

Example 8.3.8

$R$ is the region in Example 8.1.22

$\mathrm{\delta }\left(r \,\mathrm{\theta }\,z\right)\=r z \cos \left(\mathrm{\theta }\/6\right)$

Example 8.3.9

$R$ is the region in Example 8.1.26

$\mathrm{\delta }\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\=2\+ \mathrm{\rho } \sin \left(\mathrm{\phi }\right)\cos \left(\mathrm{\theta }\right)$

Example 8.3.10

$R$ is the region in Example 8.1.27

$\mathrm{\delta }\left(x\,y\,z\right)\=2 x^2\+3 y^2\+4 z^2$

Example 8.3.11

$R$ is the region in Example 8.1.28

$\mathrm{\delta }\left(r\,\mathrm{\theta }\,z\right)\=z^{2}r \sin \left(\mathrm{\theta }\/3\right)$

Example 8.3.12

$R$ is the region in Example 8.1.29

$\mathrm{\delta }\left(x\,y\,z\right)\=x y^2{z}^3$