15.4 Videos Guide

15.4a

• Mass of a planar lamina with shape $D$ and density $\rho(x,\ y)$

  • $m=\iint^{\ }_{D}{\rho(x,\ y)\ dA}$

• Total charge over an area with shape $D$ and charge density $\sigma(x,\ y)$

  • $Q=\iint^{\ }_{D}{\sigma(x,\ y)\ dA}$

• Moments

  • About the $x$-axis: $M_{x}=\iint^{\ }_{D}{y\ \rho(x,\ y)\ dA}$

  • About the $y$-axis: $M_{y}=\iint^{\ }_{D}{x\ \rho(x,\ y)\ dA}$

• Center of mass

  • $\left(\overline{x},\ \overline{y}\right)=\left(\frac{M_{y}}{m},\frac{M_{x}}{m}\right)$

• Moments of inertia (second moments)

  • About the $x$-axis: $I_{x}=\iint^{\ }_{D}{y^{2}\ \rho(x,\ y)\ dA}$

  • About the $y$-axis: $I_{y}=\iint^{\ }_{D}{x^{2}\ \rho(x,\ y)\ dA}$

  • About the origin: $I_{0}=\iint^{\ }_{D}{(x^{2}+y^{2})\ \rho(x,\ y)\ dA}=I_{x}+I_{y}$

• Radii of gyration

  • With respect to the $x$-axis: $\overline{\overline{y^{2}}}=\frac{I_{x}}{m}$

  • With respect to the $y$-axis: $\overline{\overline{x^{2}}}=\frac{I_{y}}{m}$

  • $\left(\overline{\overline{x}},\ \overline{\overline{y}}\right)$ is the rotational analog to the center of mass

15.4b

• Probability

  • $P\left((X,\ Y)\ \epsilon\ D\right)=\iint^{\ }_{D}{\ f(x,\ y)\ dA}$, where $f$ is the joint density function of
    $X$ and $Y$

  • Properties of a joint density function: $f(x,\ y)\geq0$ and $\iint^ {}_{\mathbb{R}^{2}}{f(x,\ y)\ dA}=1$

• Expected values for $f$ a joint density function of random variables $X$ and $Y$

  • $x$-mean: $\mu_{1}=\iint^{\ }_{\mathbb{R}^{2}}{x\ f(x,\ y)\ dA}$

  • $y$-mean: $\mu_{2}=\iint^{\ }_{\mathbb{R}^{2}}{y\ f(x,\ y)\ dA}$