Density and Mass integrals

First and Second Moment Formulas

First moment about an axis. $\delta (x,y,z)$ is the density function

$$M\_y=\iiint \_Dx\delta (x,y,z)dV$$

Coordinate of the center of mass is

$$x\_c.m.=\frac{M\_y}{M}$$

Moment of Inertia about an axis. $r^2$ is the distance from the axis squared

$$I\_x=\iiint \_D{r}^2\delta (x,y,z)dV$$

Parallel Axis Theorem for Moment of Inertia

if $L\_c.m.$ is a line going through the center of mass of an object with mass $m$ and $L$ is a line parallel to it distanced $h$ units away, the moments of inertia satisfy the equation

$$I=I\_c.m.+m{h}^2$$

Spherical and Cylindrical Coordinates

Cylindrical

In cylindrical coordinates the cartesian $dzdydx$ is adapted to the cylindrical $rdzdrd\theta$ while $x=rcos\theta$ and $y=rsin\theta$

Spherical

In spherical coordinates the cartesian $dzdydx$ is adapted to ${\rho }^{2}sin(\varphi )d\rho d\varphi d\theta$ $r=\rho sin\varphi$ $x=rcos\theta$ → $x=\rho sin\varphi \cos \theta$ $y=rsin\theta$ → $x=\rho sin\varphi \sin \theta$ $\rho =\sqrt{r^2+z^2}$ The volume of an object defined in spherical coordinates is given by

$$V=\iiint \_Df(\rho ,\varphi ,\theta ){\rho }^{2}sin(\varphi )d\rho d\varphi d\theta$$

The bounds of integration should however not exceed $\alpha \le \theta \le \beta$ where $0\le \alpha -\beta \le 2\pi$