Q. 11 - Calculus

Question

Identify the quantities determined by the integral expressions in Exercises  If x and y are both measured in centimeters and $\mathrm{\rho }\left((\mathrm{x}, \mathrm{y}\right)$ is a density function in grams per square centimeter, give the units of the expression.

$\sqrt{\frac{{\iint }_{\mathrm{\Omega }} {\mathrm{x}}^2\mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}{{\iint }_{\mathrm{\Omega }} \mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}}$ and  $\sqrt{\frac{{\iint }_{\mathrm{\Omega }} {\mathrm{y}}^2\mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}{{\iint }_{\mathrm{\Omega }} \mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}}$

Step-by-Step Solution

Answer

The radius of gyration is ${\mathrm{R}}_{\mathrm{x}}=\sqrt{\frac{{\iint }_{\mathrm{\Omega }} {\mathrm{y}}^2\mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}{{\iint }_{\mathrm{\Omega }} \mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}}$ and measured in centimetres.

Step 1: Given information

The expressions are $\sqrt{\frac{{\iint }_{\mathrm{\Omega }} {\mathrm{x}}^2\mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}{{\iint }_{\mathrm{\Omega }} \mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}}$ and $\sqrt{\frac{{\iint }_{\mathrm{\Omega }} {\mathrm{y}}^2\mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}{{\iint }_{\mathrm{\Omega }} \mathrm{\rho }(\mathrm{x},\mathrm{y})\mathrm{dA}}}$

Step 2: Simplification

The objective of this problem is to identify the quantity determined by the integral expression. $x$ and $y$ are measured in centimeters and $\rho (x,y)$ is measured in grams per square centimeter. Give the units of expression.

The expressions are:

$\sqrt{\frac{{\iint }_{\Omega }{x}^2\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}}\text{ and }\sqrt{\frac{{\iint }_{\Omega }{y}^2\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}}$

The expression $\sqrt{\frac{{\iint }_{\Omega }{x}^2\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}}$ represents the radius of gyration about $y-$ axis. It is measured from y - axis and measured in centimeters. Its symbol is $R_y$.

$R_y=\sqrt{\frac{{\iint }_Q{x}^2\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}}$

The expression $\sqrt{\frac{{\iint }_{\Omega }{y}^2\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}}$ represents the radius of gyration about $x-$axis. It is measured from x - axis and measured in centimeters. Its symbol is $R_y$.

$R_x=\sqrt{\frac{{\iint }_{\Omega }{y}^2\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}}$