# 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 \(\left( {x,y} \right)\) in the region \(R\) is \(\rho \left( {x,y} \right).\) The total mass of the lamina is expressed through the double integral as follows:

The static moment of the lamina about the \(x\)-axis is given by the formula

Similarly, the static moment of the lamina about the \(y\)-axis is

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

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

Similarly, the moment of inertia of a lamina about the \(y\)-axis is given by

The polar moment of inertia is

## 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

## 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

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