# Physical Applications of Double Integrals

## Mass and Static Moments of a Lamina

Suppose we have a lamina which occupies a region R in the xy-plane and is made of non-homogeneous material. Its density at a point (x, y) in the region R is ρ (x, y). The total mass of the lamina is expressed through the double integral as follows:

$m = \iint\limits_R {\rho \left( {x,y} \right)dA} .$

The static moment of the lamina about the $$x$$-axis is given by the formula

${M_x} = \iint\limits_R {y\rho \left( {x,y} \right)dA} .$

Similarly, the static moment of the lamina about the $$y$$-axis is

${M_y} = \iint\limits_R {x\rho \left( {x,y} \right)dA} .$

The coordinates of the center of mass of a lamina occupying the region $$R$$ in the $$xy$$-plane with density function $$\rho \left( {x,y} \right)$$ are described by the formulas

$\bar x = \frac{{{M_y}}}{m} = \frac{1}{m}\iint\limits_R {x\rho \left( {x,y} \right)dA} = \frac{{\iint\limits_R {x\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }},$
$\bar y = \frac{{{M_x}}}{m} = \frac{1}{m}\iint\limits_R {y\rho \left( {x,y} \right)dA} = \frac{{\iint\limits_R {y\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }}.$

When the mass density of the lamina is $$\rho \left( {x,y} \right) = 1$$ for all $$\left( {x,y} \right)$$ in the region $$R,$$ the center of mass is defined only by the shape of the region and is called the centroid of $$R.$$

## Moments of Inertia of a Lamina

The moment of inertia of a lamina about the $$x$$-axis is defined by the formula

${I_x} = \iint\limits_R {{y^2}\rho \left( {x,y} \right)dA} .$

Similarly, the moment of inertia of a lamina about the $$y$$-axis is given by

${I_y} = \iint\limits_R {{x^2}\rho \left( {x,y} \right)dA} .$

The polar moment of inertia is

${I_0} = \iint\limits_R {\left( {{x^2} + {y^2}} \right)\rho \left( {x,y} \right)dA} .$

## Charge of a Plate

Suppose electrical charge is distributed over a region which has area $$R$$ in the $$xy$$-plane and its charge density is defined by the function $${\sigma \left( {x,y} \right)}.$$ Then the total charge $$Q$$ of the plate is defined by the expression

$Q = \iint\limits_R {\sigma \left( {x,y} \right)dA} .$

## Average Value of a Function

We give here the formula for calculation of the average value of a distributed function. Let $${f \left( {x,y} \right)}$$ be a continuous function over a closed region $$R$$ in the $$xy$$-plane. The average value $$\mu$$ of the function $${f \left( {x,y} \right)}$$ in the region $$R$$ is given by the formula

$\mu = \frac{1}{S}\iint\limits_R {f\left( {x,y} \right)dA} ,$

where $$S = \iint\limits_R {dA}$$ is the area of the region of integration $$R.$$

See solved problems on Page 2.