Q 4. - Calculus

Question

Identify the quantities determined by the integral expressions. 

$\rho (x,y)$ is density function.

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

Step-by-Step Solution

Answer

The quanitities determined are moments.

Step 1: Given Information

It is given that expressions are

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

$x,y$ measured in centimeters.

Step 2: Identification of the quantities

${\iint }_{\Omega }xdA$ represents first moment about $y$ axis having constant density function $\rho (x,y)=1$

${\iint }_{\Omega }ydA$ represents first moment about $x$ axis having constant density function $\rho (x,y)=1$

Moment will me measured in grams/cm.

Hence, moment about $y$ axis is $M_y={\iint }_{\Omega }xdA$

Moment about $x$ axis is $M_x={\iint }_{\Omega }ydA$