Q 5. - Calculus

Question

Identify the quantities determined by the integral expressions

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

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

Step-by-Step Solution

Answer

The quantities determines center of mass.

Step 1: 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.

Step 2: 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.

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