Q. 1 - Calculus

Question

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

Step-by-Step Solution

Answer

The expression $\frac{{\iint }_{\Omega }xdA}{{\iint }_{\Omega }dA}$ represents the center of mass located within the region $\Omega$ with constant density function $\rho (x,y)=1.$ It is measured from $y$ - axis and measured in centimeters. Its symbol is $\overline{x}.$

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

The expression $\frac{{\iint }_{\Omega }ydA}{{\iint }_{\Omega }dA}$ represents the center of mass located within the region $\Omega$ with constant density function $\rho (x,y)=1$. It is measured from $x$ - axis and measured in centimeters.

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

Step 1: Given information

The expressions are

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

Step 2: Calculation

The expression $\frac{{\iint }_{\Omega }xdA}{{\iint }_{\Omega }dA}$represents the center of mass located within the region $\Omega$ with constant density function $\rho (x,y)=1.$ It is measured from $y$ - axis and measured in centimeters. Its symbol is $\overline{x}.$

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

The expression $\frac{{\iint }_{\Omega }ydA}{{\iint }_{\Omega }dA}$ represents the center of mass located within the region $\Omega$ with constant density function $\rho (x,y)=1.$ It is measured from $x$ - axis and measured in centimeters.

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