Question

Identify the quantities determined by the integral expressions

$\rho (x,y)$is a density function in grams per square centimeter

$\frac{{\iint }_{\Omega }xdA}{{\iint }_{\Omega }dA}\text{and}\frac{{\iint }_{\Omega }ydA}{{\iint }_{\Omega }dA}$

Short answer

The quantities determines center of mass.

Step-by-step solution

Given Information

The given expressions are

$\frac{{\iint }_{\Omega }xdA}{{\iint }_{\Omega }dA}\text{and}\frac{{\iint }_{\Omega }ydA}{{\iint }_{\Omega }dA}$

$x,y$is in centimeter.

Identification of the quantities

$\frac{{\iint }_{\Omega }xdA}{{\iint }_{\Omega }dA}$represents center of mass within region $\Omega$having constant density function $\rho (x,y)=1$measured from $y$axis in centimeters.

$\overline{x}=\frac{{\iint }_{\Omega }xdA}{{\iint }_{\Omega }dA}$

Similarly

$\frac{{\iint }_{\Omega }ydA}{{\iint }_{\Omega }dA}$represents center of mass within region having constant density function $\rho (x,y)=1$measured from $x$axis.

$\overline{y}=\frac{{\iint }_{\Omega }ydA}{{\iint }_{\Omega }dA}$