91Ó°ÊÓ

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

The expressions are:

${âˆ\neg }_{\mathrm{Î\copyright }}{x}^2\mathrm{ÒÏ}(x,y)dAand{âˆ\neg }_{\mathrm{Î\copyright }}{y}^2\mathrm{ÒÏ}(x,y)dA$

The expression ${âˆ\neg }_{\mathrm{Î\copyright }}{x}^2\mathrm{ÒÏ}(x,y)dA$represents the moment of inertia of the lamina with uniform density about y-axis.

The expression ${âˆ\neg }_{\mathrm{Î\copyright }}{y}^2\mathrm{ÒÏ}(x,y)dA$represents the moment of inertia of the lamina with uniform density about x - axis.

If x and y are measured in centimeters and $\mathrm{ÒÏ}(x,y)$is measured in grams per square centimeter. Then, moment of inertia will be measured in grams square centimeter.

Moment of inertia about y - axis:

$I_y={âˆ\neg }_{\mathrm{Î\copyright }}{x}^2\mathrm{ÒÏ}(x,y)dA$

Moment of inertia about x - axis:

$I_x={âˆ\neg }_0{y}^2\mathrm{ÒÏ}(x,y)dA$