- Arc Length:
If $y = f(x)$, then
$\fcolorbox{white}{yellow}{$ ds = \sqrt{ dx^2 + dy^2 } = \sqrt{1+ \Big(\frac{dy}{dx}\Big)^2} dx $}$
;
- Center of mass $(\bar x, \bar y, \bar z)$:
$\fcolorbox{white}{yellow}{$ \int \bar x dM = \int x dM, \m
\int \bar y dM = \int y dM, \m \int \bar z dM = \int z dM, \m dM = \rho(x,y,z)dxdydz $}$
;
- Moment of innertia with respect to coordinate axises for an area on $x$-$y$ plane:
$\fcolorbox{white}{yellow}{$I_x = \int y^2 dM, \m I_y = \int x^2 dM, \m I_z = \int (x^2+y^2) dM $}$
In general, $\fcolorbox{white}{yellow}{$\m I = \int (\text{distance})^2 dM$}$.
- Volume by rotation: for solid volume generated by rotating $y=f(x)$ along the $x$-axis,
Volume:
$\fcolorbox{white}{yellow}{$ V = \int_a^b \pi \Big(f(x)\Big)^2 dx $}$
,
Surface area: $\fcolorbox{white}{yellow}{$ S= \int_a^b 2\pi f(x) ds = \int_a^b 2\pi f(x)\sqrt{1+(f'(x))^2}dx $}$